
Citation 
 Permanent Link:
 http://digital.auraria.edu/AA00003159/00001
Material Information
 Title:
 Truncated quadrics and elliptic curves
 Creator:
 Flink, Stephen C
 Publication Date:
 2009
 Language:
 English
 Physical Description:
 xii, 106 leaves : ; 28 cm
Subjects
 Subjects / Keywords:
 Quadrics ( lcsh )
Curves, Elliptic ( lcsh ) Curves, Elliptic ( fast ) Quadrics ( fast )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Bibliography:
 Includes bibliographical references (leaves 105106).
 General Note:
 Department of Mathematical and Statistical Sciences
 Statement of Responsibility:
 by Stephen C. Flink.
Record Information
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 University of Colorado Denver
 Holding Location:
 Auraria Library
 Rights Management:
 All applicable rights reserved by the source institution and holding location.
 Resource Identifier:
 435778680 ( OCLC )
ocn435778680
 Classification:
 LD1193.L622 2009d F54 ( lcc )

Full Text 
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TRUNCATED QUADRICS AND ELLIPTIC CURVES
by
Stephen C. Flink
B.S., University of Colorado at Denver, 2000
M.S., University of Colorado at Denver, 2004
A thesis submitted to the
University of Colorado Denver
in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Applied Mathematics
2009
This thesis for the Doctor of Philosophy
degree by
Stephen C. Flink
has been approved
by
Ellen Gethner
3o Mf.iL Zof
Date'
Mjcfkrel S. Jacobson
Flink, Stephen C. (Ph.D., Applied Mathematics)
Truncated Quadrics and Elliptic Curves
Thesis directed by Professor Stanley E. Payne
ABSTRACT
Let p be an odd prime and let q = pe. Let Â£ be an elliptic quadric in
PG(3,q). The quadric Â£ carries the structure of the projective line PG(l,q2),
and the points of Â£ may be put in a onetoone correspondence with the points
of PG( 1, q2) in a manner that preserves the structure of the quadric, in terms of
the respective automorphism groups. In this thesis, we consider the geometric
properties of the subset Â£n of Â£ whose points correspond in this way to the
nonzero squares in the Galois Field Fg2. In the course of determining the number
of points of Â£n on certain hyperplanes of PG(3,q), there arise two families
of elliptic curves. The HasseWeil theorem is invoked to give bounds on the
cardinalities of plane intersections with Â£n. Empirical results for small values of
q show that these bounds are the best possible. The theory of elliptic curves over
finite fields is used to establish proofs of other properties of the plane sections
of Â£.
A substructure TCn of the hyperbolic quadric K in PG(3,q) is defined and
studied. Ka is analogous to and has geometric properties very similar to those of
Â£d. As with our examination of Â£a, there arise two families of elliptic curves, and
m
the HasseWeil theorem implies bounds on the cardinalities of plane intersections
with 1Kd. We find that the two families of curves which arise in the study of !Kn
are identical with the two families of elliptic curves from the study of Â£n. We
call these families of curves EÂ£, parameterized by F*. and Eg, parameterized by
F,\{0,1,1}.
Assume now that q is not a power of 3. We show that the set of curves Eg
is symmetric in the sense that the curves in this set with q + 1 + t points are
in onetoone correspondence with the curves in this set with q + 1 t points.
When q = 3 mod 4, the set of curves Eg exhibits the same symmetry.
This abstract accurately represents the content of the candidates thesis. I
recommend its publication.
Signed
IV
DEDICATION
For Greg
ACKNOWLEDGMENT
This thesis would not have been possible without the assistance of the
George Galloway trust. I would also like to thank the mathematics faculty
at CU Denver for awarding me a Lynn Bateman Memorial Teaching Award for
the Fall semester of 2006.
Many people have provided encouragement and useful advice. Their influ
ence may have been inadvertent and seemed small at the time and may have
come in the course of working together or from offhand conversations. Thank
you to John Wilson, Leanne Holder, Mark Miller, Janice Dugger, Art Busch,
Oscar Jenkins, Dave Brown, Frank Thraxton, Sandy Barrett, Victoria Naman,
Georgia Meginity, Shannon Raptis and Francis Conry. Special thanks to Rob
Rostermundt for our many conversations and for being the designated driver af
ter I completed my comprehensive exam. From the distant past, I need to thank
Leona Jackson, Bill Blair, Rick Schmidt, James Plastino and Mark Chamberlin.
In no particular order, math courses from Dave Wilson, Hugh Bradley, Tom
Kammerling, Markus Emsermann, Rich Lundgen, Dave Fisher, Tom Russell,
Bill Cherowitzo, Bruce MacMillan, and Brooks Reid have been particularly en
lightening. Thanks again to Bruce MacMillan for allowing me to sit in on his
Calculus courses during the 200405 academic year.
I would like to thank the members of my thesis committee for taking the time
to examine this work and for their helpful feedback. Thank you in particular to
Stan Payne for his guidance in my research.
Most of all, I would like to thank Yongxia for her love, support and patience.
CONTENTS
Figures .................................................................. ix
Tables..................................................................... x
Chapter
1. Introduction.......................................................... 1
2. Two Quadrics in PG(3, q)............................................ 8
2.1 Definitions and Preliminaries ........................................ 8
2.2 An Elliptic Quadric.................................................. 11
2.3 A Ruled Quadric in PG(3, q) ...................................... 17
3. The Actions of TwoPoint Stabilizers.............................. 19
3.1 The Stabilizer of Two Points of Â£................................. 19
3.2 Orbits of Planes Under Gi......................................... 21
3.3 The Group Stabilizing Two Points of IK ........................... 23
3.4 Point Sets Associated with Squares................................... 28
3.4.1 The Square Points of Â£........................................... 28
3.4.3 The Square Points of K.......................................... 32
4. Truncated Quadrics and Elliptic Curves............................... 37
4.1 Point Counts on Planes Meeting Â£a.................................. 37
4.2 Some Background on Elliptic Curves................................. 41
4.3 Birational Transformations Between Quartics and Cubics............ 43
4.3.1 Elliptic Curves From Orbits 0gD.................................... 43
vii
4.3.3 Elliptic Curves From Orbits 0^a..................................... 47
4.3.4 Elliptic Curves from Orbits 0^ ..................................... 48
4.3.5 Elliptic Curves from Orbits 0^, ................................... 50
4.3.6 Families of Curves.................................................. 51
4.4 Summary of Plane Intersections with Truncated Quadrics............. 52
4.4.1 Other Plane Orbit Types ............................................ 52
4.4.2 Summary of Plane Intersections for Truncated Quadrics............... 53
4.5 Sums of Point Counts ............................................... 56
4.6 The Invariant j and Symmetry of Incidences.......................... 58
4.6.1 Admissible Changes of Variables .................................... 59
4.6.4 The Invariant j..................................................... 61
Appendix
A. Additional Results .................................................... 66
A.l q 1 Elliptic Quadrics Intersecting in 2 Points.................... 66
A. 2 Addition on the Curves............................................... 67
B. Tables of Elliptic Curves.............................................. 69
B. l Tables for Elliptic Curves .......................................... 69
C. Programs .............................................................. 98
C.l C++ Programs........................................................... 98
C.2 Sage Programs ........................................................ 104
References................................................................ 105
viii
FIGURES
Figure
2.1 Tangent planes to Â£ and the lines l and l1........................... 15
3.1 A diagram of the elliptic quadric from the point of view of the sta
bilizer of two points................................................... 31
3.2 A diagram of the hyperbolic quadric from the point of view of the
stabilizer of two points not on any line of the quadric.............. 35
IX
TABLES
Table
4.1 Intersection numbers of Â£D with planes in IPg for q 263 ............ 38
4.2 The number of Fgrational points on elliptic curves E = E^n for
u) G F263, their jinvariants and multiplicities in Eg................ 65
B.l Point counts and jinvariants for curves over F3...................... 70
B.2 Point counts and jinvariants for curves over F5...................... 70
B.3 Point counts and jinvariants for curves over F7...................... 70
B.4 Point counts and jinvariants for curves over Fn ..................... 70
B.5 Point counts and jinvariants for curves over F13 71
B.6 Point counts and jinvariants for curves over F47 71
B.7 Point counts and jinvariants for curves over F19 71
B.8 Point counts and jinvariants for curves over F23 72
B.9 Point counts and jinvariants for curves over F29 72
B.10 Point counts and jinvariants for curves over F3i .................... 72
B.ll Point counts and jinvariants for curves over F37 73
B.12 Point counts and jinvariants for curves over F41 73
B.13 Point counts and jinvariants for curves over F43 73
B.14 Point counts and jinvariants for curves over F47 74
B.15 Point counts and jinvariants for curves over F53 74
B.16 Point counts and ^invariants for curves over F59 74
x
B.17 Point counts and jinvariants for curves over Fgi ..................... 75
B.18 Point counts and jinvariants for curves over Fg7 .................... 75
B.19 Point counts and jinvariants for curves over F7J .................... 76
B.20 Point counts and jinvariants for curves over F73 76
B.21 Point counts and jinvariants for curves over F79 77
B.22 Point counts and jinvariants for curves over Fg3 .................... 77
B.23 Point counts and jinvariants for curves over Fg9 .................... 78
B.24 Point counts and jinvariants for curves over F97 78
B.25 Point counts and jinvariants for curves over F101.................... 79
B.26 Point counts and jinvariants for curves over F103.................... 79
B.27 Point counts and jinvariants for curves over F107.................... 80
B.28 Point counts and jinvariants for curves over F109.................... 80
B.29 Point counts and jinvariants for curves over F113.................... 81
B.30 Point counts and jinvariants for curves over F127.................... 82
B.31 Point counts and jinvariants for curves over F131.................... 83
B.32 Point counts and jinvariants for curves over F137.................... 84
B.33 Point counts and jinvariants for curves over Fi39.................... 85
B.34 Point counts and jinvariants for curves over F149.................... 86
B.35 Point counts and jinvariants for curves over F151.................... 87
B.36 Point counts and jinvariants for curves over F157.................... 88
B.37 Point counts and jinvariants for curves over F163.................... 89
B.38 Point counts and jinvariants for curves over Fi67.................... 90
B.39 Point counts and jinvariants for curves over F173.................... 91
B.40 Point counts and jinvariants for curves over Fi79.................... 92
xi
B.41 Point counts and jinvariants for curves over Fisi..................... 93
B.42 Point counts and jinvariants for curves over Figi..................... 94
B.43 Point counts and jinvariants for curves over F193..................... 95
B.44 Point counts and jinvariants for curves over F197..................... 96
B.45 Point counts and jinvariants for curves over F199..................... 97
xii
1. Introduction
The focus of this dissertation is on a question in threedimensional projec
tive geometry, PG(3,q), when Fg is an arbitrary finite field of odd order. Our
principal problem is to determine geometric properties of an object with a natu
ral algebraic definition, but whose geometric structure is not obvious. We begin
with a nondegenerate quadric Q in PG(3,q). We partition the points of Q so
that the partite classes correspond in a natural way with the elements of the
extended field Â¥q = F9 U {oo}. We study the geometry of Qn, the set of points
associated with the nonzero squares in F9 under this partition. We call Qa a
truncated quadric. Our problem is to describe Qn in terms of plane intersections.
The first step is to classify all orbits of planes under the action of the group stabi
lizing our partition. We find that the most interesting orbits fall into two classes
for each of the two types of quadric under consideration. Counting points of
Qd on planes in these orbits is shown to be equivalent to counting points on
certain nonsingular elliptic curves. These are algebraic curves of genus 1, for
which there exists a large and rich theory. We apply the HasseWeil Theorem,
a deep result which gives bounds on the number of points on algebraic curves
over finite fields. This theorem, in turn, gives us bounds on the cardinalities of
plane intersections with the truncated quadrics. We present numerical results
establishing the tightness of these bounds. We give a simple characterization of
the families of elliptic curves which arise when considering each type of orbit.
Finally, we explain a symmetry of incidence numbers which occurs in certain
1
families of orbits of planes, which have the property that for every plane meet
ing the truncated quadric in ^ + t points, there is a plane in the same family
meeting it in t points.
In Chapter 2, we recall basic notions from projective geometry over finite
fields. We review results which establish that the elliptic quadric is the unique
example of an ovoid in PG(3, q) when q is odd. The points of an elliptic quadric
in PG{3, q) may be put in onetoone correspondence with the points of the
projective line PG(l,q2) so that each reflects the structural properties of the
other. We point out that this is analogous to the identification of the points of
the extended complex plane with the unit sphere via stereographic projection.
Choosing a specific elliptic quadric Â£, we show how to make the correspon
dence between the points of Â£ and PG(l,q2) explicit. We find that the norm
map from F*2 to F* partitions the points of Â£ \ {0, oo} into q 1 disjoint ovals.
This partition is the same as the partition induced by a certain flock of Â£. Fix
the points 0 and oo of Â£, and define the line to be the line of intersection of the
tangent planes to Â£ at 0 and oo. For each of the q 1 nontangent planes on
P. we choose representative vectors parameterized by the nonzero elements of
Fq. This choice of representatives is such that for each value a G F*, the points
of Â£ on the plane with representative a are the elements of F*2 with norm a.
In Section 2.3, we choose a specific hyperbolic quadric At in PG(3, q) whose
algebraic form bears a strong resemblance with the form chosen for the elliptic
quadric Â£. In Â£, the points corresponding to Fq U oo form an oval o, and
Â£ fl AC = o. If we remove a certain pair of points from At, the remaining points
may be partitioned into q + 1 subsets. In fact, the set of q + 1 planes we used
2
to partition Â£ are used to partition the points of J~C. in a manner different from
but analogous to the partition for Â£. We find that a subset of the points of fit
has a natural partition into q 1 arcs of size q 1.
In Chapter 3, we examine the actions of the stabilizers of the elliptic and
hyperbolic quadrics on lines and planes in PG(3,q). Recall our identification
of the points of the elliptic quadric Â£ with the points of the projective line
PG( 1, q2), and so with the set Fq2 = Fq2 U oo. We are interested in the action of
the group Gi stabilizing the points 0 and oo of the quadric, as this is the group
stabilizing our partition.
In Section 3.2, Theorem 3.2.1 describes the orbits of all planes of PG(3,q)
under the action of G;. In section 3.3, we describe the geometry of the hyperbolic
quadric 'K and the action of the stabilizer Hi of 0 and oo in a manner analogous
to the work in Sections 3.1 and 3.2 for Â£. Theorem 3.3.1 describes all orbits of
planes under the action of Ht.
In section 3.4, we define our truncated quadrics Â£n and TCn. These are
the subsets of Â£ and TC which correspond to squares in Fg under the respective
partitions of the quadrics described in Chapter 2. The truncated elliptic quadric
Â£n is stabilized by G;D, a group of index 2 in Gi. The truncated hyperbolic
quadric TÂ£D is stabilized by a group Hia of index 2 in Hi. Having described the
actions of Gi and Hi, it is straightforward to describe the actions of Gia and
Hia on the planes of PG(3,q). We do so in Theorems 3.4.2 and 3.4.4.
For each of the stabilizing groups Gia and Hia, there are orbits of planes
whose intersections with the truncated quadric are easily described. For exam
ple, when q 1 of the q + 1 planes on a line are all in the same orbit, the planes
3
are known to partition a point set of a certain size (say \(q2 1)), and so must
each contain (g + l) points of the truncated quadric. Define an interesting orbit
of planes (under the action of Gia or Hia) to be one whose intersection numbers
with their respective truncated quadric are not immediately obtained by appli
cation of the orbitstabilizer theorem. We find that it is sufficient to consider two
families of interesting orbits for each of the truncated quadrics: {O^ : u e
and {OgQ : u
{0$& : u F*} for !Hn. These interesting orbits of planes have the property
that no more than two planes in such an orbit meet in the same line. For each
family of orbits, we choose a line such that each orbit of planes in the family has
2 representatives in the set of planes on that line. In the remainder of the the
sis, we describe the intersection of planes from these classes with our truncated
quadrics and examine some of the consequences of that description.
In Section 4.1, we offer a summary of results from algebraic geometry in
general and from the theory of elliptic curves in particular, which we will use in
subsequent proofs. Most important for us is the HasseWeil theorem, stated in
Theorem 4.2.2, which gives bounds on the number of points on an elliptic curve.
Also important is Theorem 4.2.1, which states that every curve is birationally
equivalent to a unique nonsingular curve. In Section 4.2, wre begin with a plane
aw in an interesting orbit under Gia. Some algebra on the equations de
scribing the intersection of au with Â£n yields a plane curve CF whose F9rational
points are in 2:1 proportion with the points of a^, D Â£n. The equation for the
curve CF is of the form s2 =
roots. In Section 4.3, we show that this curve is birationally equivalent to a
4
curve whose equation is of the form y2 = /(x), where / is a cubic in x without
repeated roots. This is a standard form for an elliptic curve, and we may bring
to bear upon our problem substantial theoretical machinery. In Theorem 4.3.1,
an application of the HasseWeil theorem, we find that
q 1 _ . o l
2 VQ \au hi Â£q <  h y/q
In section 4.4, we consider intersections of truncated quadrics with representative
planes from our other interesting orbits under Gia and Hia. We again obtain
elliptic curves from the equations for plane intersections and again apply the
HasseWeil theorem.
Also in Section 4.3, we show that the two families of curves which arise from
Â£n are identical to the two families of curves which arise from 7Ca. In Section
4.4, we use our use results from Sections 4.2 and 4.3 to describe bounds of point
counts on planes in other orbits. The idea here is that Gia stabilizes both Â£n
and Â£^ = Â£ \ (Â£n U {0,oo}), and we are able to find an element p of Gi which
interchange Â£a and Â£^. Then p also interchanges with a plane with the
property that aw fl Â£n + p{aUJ) fl Â£a = q + 1. The situation is again similar
for Ka. We summarize point counts of plane intersections with the truncated
quadrics in Theorems 4.4.3 and 4.4.4.
In section 4.5, we explain a property of some of the families of orbits of
planes under Gia. The property occurs for families of interesting orbits whenever
q = 3 mod 4 and in certain cases when q = 1 mod 4. Let 7 be such a family of
orbits. Then whenever there is an orbit 0 G? whose planes meet Â£a in ^ +1
points, there is a different orbit O' 6 7 whose planes meet Â£n in ^ t points.
The proof that these planes come in these complementary pairs is accomplished
5
using properties of a numeric invariant of the associate elliptic curves.
Appendix A.l contains a minor result which is not used in the main part of
the thesis, but which might be of independent interest. Here we describe a set
of q 1 elliptic quadrics whose pairwise intersection is {0, oo}, all of which are
stabilized by G;. A description of an analogous family of hyperbolic quadrics
whose pairwise intersection is a set of four points of PG(3, q) not all in one plane
is given in Section 6.1 of [20].
Appendix B contains tables of elliptic curves for each of the two families
described in Theorem 4.4.3, for odd primes q < 200. For a given q and for
each of the two families of curves, the table gives the jinvariant of each curve,
the number of F9rational points on the curve, and the number of times the
curve occurs in that family. Appendix C.l contains a program in C++ written
to produce the tables in Appendix B. Appendix C.2 contains a short program
in SAGE which uses some builtin elliptic curve functions. This program was
written to verify the accuracy of the C++ program in section C.l.
The work presented in this thesis originated with an attempt to construct
sets of points in PG(n, q) with the property that the cardinalities of hyperplane
intersections with the set take on relatively few values. We see from Theorems
4.5.1 and 4.5.2 and from the tables in Appendix B that the truncated quadrics
are highly irregular with respect to plane intersections. Let S be a subset of
PG(n, q) such that the cardinalities of hyperplane intersections with S take on
few values. If these cardinalities take on k values, we call S a kintersection set.
Nondegenerate quadrics provide examples of 2intersection sets in PG(n, q) when
n is odd. Some of our early motivation came from the construction of Brouwer
6
in [2], which yields 2intersection sets by removing points from nondegenerate
quadrics in odddimensional projective space. Another important work on 2
intersection sets is [4], which lists all such sets known at that time and explains
how 2intersection sets may be used to construct strongly regular graphs and
2weight codes.
The literature on quadrics in finite projective space is large and varied. We
acknowledge the influence of several works whose results and terminology did not
directly come into play in the final version of this thesis. Elliptic and hyperbolic
quadrics give examples of finite circle planes. A standard reference on the circle
planes associated with elliptic quadrics is Chapter 6 of [11], The Ph.D. thesis
of Orr [19], as reworked in [20], and the paper [3] by Bruck helped us to find a
proper frame of reference and to steer clear of some false conjectures.
7
2. Two Quadrics in PG(3,q)
2.1 Definitions and Preliminaries
In this section, we give a brief review of some relevant results from finite
projective geometry. We follow Stan Paynes book [20] for theorems and no
tation. Much of what we describe here may be found in [7], which is a good
introductory text on projective geometry.
Let V be a vector space of dimension n + 1, for some 0 < n < oo over
a field F. The projective geometry 7(V) is the geometry whose rdimensional
subspaces are defined to be the r+1dimensional subspaces of V, for r = 0,..., n.
Incidence in IP(V) is defined by subspace containment in V. We call the 0
, 1 and 2dimensional subspaces of (V) the points, lines and planes of the
geometry, and an (n l)dimensional subspace is a hyperplane. When F is a
field with q elements, we write PG(n, q) to denote 7(V). For our purposes, the
term projective plane will refer to PG(2, q), although there exist other examples
of projective planes. We refer the reader to the early chapters of [20] and to
Chapter 1 of [7] for an axiomatic approach to projective geometry.
In PG(n,q), we use a row vector x = (x0,x\,... ,xn) to represent a point
and a column vector w = [yo,yi,. .. ,yn]T a hyperplane in PG(n,q), with the
understanding that for the vector representing a point or hyperplane, we may
substitute any nonzero scalar multiple of that vector. The vector representing
7T generates the null space of all the points incident with n. That is, the point
x is incident with the hyperplane 7r whenever X7r = 0.
When q is the power of an odd prime, of the nonzero elements of the Ga
lois field Fg are squares and are nonsquares. We will let and 0 respectively
denote the set of nonzero squares and the set of nonsquares in Â¥q. Throughout
the main text of this thesis, we use 77 to denote a fixed but arbitrary nonsquare
in Â¥q.
An oval in a projective plane ir of order q is a set of q + 1 points, no three
collinear. Let P be a point on an oval o. Of the q + 1 lines through P, q are
secant lines, meeting 0 in exactly one other point. The remaining line through
P is the tangent line to o at P. It can be shown that when q is odd, each point
of 7r \ 0 is on 0 or 2 tangent lines. When q is even, one may show that there is a
point N such that all tangent lines to 0 meet at N. That is, when q is even, an
oval may be extended to a hyperoval, a set of q + 2 points, no three on a line.
A quadric Q in PG(n, q) is a set of points x = (x0,xi,..., xn) satisfying a
homogeneous quadratic equation
n n
/(To, xn) = ^2 aijXiXJ = 0
i=0 j=i
for some a^ E Â¥q not all zero. Let A = (a^) be the upper triangular matrix such
that f(x0,..., xn) = xAxt, and let B = A + AT. A point of Q is a singular point
provided Bx.T = 0 and xAxT = 0. We say that the quadric Q is degenerate (or
singular) if it has a singular point.
In PG(2,q), a nondegenerate quadric is a conic. Every conic is an example
of an oval. When q is odd, the converse is true as well.
Theorem 2.1.1 (Segre) Every oval in PG(2,q), q odd, is a conic.
9
Our primary interest is in quadrics in PG(3. q). For a point x on a nonsin
gular quadric Q, define the tangent hyperplane to Q at x to be the hyperplane
x1 := Bxt. It can be shown that in PG(3, q), there are exactly two nonisomor
phic nonsingular quadrics. When for all x G Q, xx intersects Q in more than
just x, it happens that xx D Q is the union of two lines which meet at x. In this
case, Q is a hyperbolic quadric. The quadric Q is an elliptic quadric if for each
xeQ,Qnx1 = x. In this case, for each point x of Q, every line meeting x and
not contained in x1 meets Q \ {x} in exactly one point.
A set Â§ of points of PG(3, q) is called an ovoid if it satisfies the following:
1. Each line meets S in at most 2 points.
2. Each point of S lies on exactly q + 1 tangent lines, all of which lie on a
plane.
Let P be a point on an ovoid S. Since P is on exactly q + 1 tangent lines, the
remaining q2 lines through P must meet S in exactly one other point. Thus S
has exactly q2 + 1 points.
A kcap in PG(3, q) is a set K of k points, no three on a line. We can
think of fccaps as the convex objects in PG(3, q). We can show that ovoids are
examples of maximal fccaps.
Lemma 2.1.2 If q is odd and K is a kcap in PG(3,q), then k < q2 + 1.
Proof: Let P and Q be points of K and consider a plane tt on the line PQ.
We claim that \tt n K\ < q + 1. The q + 1 lines of 7r meeting P each meet K
in at most one other point, so \tv fl K\ < q + 2. Suppose \tt D K\ = q + 2. If R
10
is a point of it not on K, then each line in 7r through R must meet K in 0 or 2
points. But this is not possible, since q + 2 is odd. Thus \n D K\ < q + 1. Thus
the planes on PQ contain at most (q + l)(q 1) + 2 points of K.
One may show that every q2 + 1cap K has the property that for every point P
of K there is a unique plane Tip such that TTPnK = P. That is, every q2 + 1cap
is an ovoid. We see from the definition that every elliptic quadric is an ovoid.
When q is odd, the converse is true as well.
Theorem 2.1.3 (Barlotti, Panella) When q is odd, every ovoid in PG(3,q)
is an elliptic quadric.
Hence in PG(3,q), q odd, the study of ovoids and of fccaps of maximum size
is reduced to the study of elliptic quadrics. Further, all elliptic quadrics in
PG(3,q) are projectively equivalent. See Theorem 5.4.4 in [20] for proof of this
fact. Thus the study of ovoids is reduced to the study of an essentially unique
object for each odd prime power q.
2.2 An Elliptic Quadric
Let 7] be a nonsquare in Fg, q odd, and let
Then
0 1 0 0
1 0 0 0
0 0 2 0
0 0 0 2 T)
Â£ = {x = (x0, Xi,X2, x3) : xExt = 0}
11
is an elliptic quadric. An explicit representation of the points is
Â£ = {(1, s2 7]t2, s,t) : s,t Â£ Fg} U {(0,1,0,0)}.
For a point x, the unique tangent plane to Â£ at x is the plane Â£xr.
Using the explicit form, we may define addition and multiplication on the
points of Â£ \ {(0,1, 0, 0)} by
(1, s2rjt2, s, t) + (1, u2rjv2, u, v) = (1, (s + u)2~rj{t + v)2, s + u,t + v) (2.1)
and
(1, s2r]t2, s, t)( 1, u2~j]V2, u, v) = (1, (s2rjt2){u2~r]v2), su + rjtv, sv + tu).{2.2)
Note that {1, y7??} are a basis for Fg2 over Fg, that is, we may represent Fg2 =
{s + y/rjt : s, t G Fg} and it is easy to show that
Q : Fg2 > Â£ \ {(0,1, 0, 0)}
9 : s + y/rjt i> (1, s2 rjt2, s, t)
is a field isomorphism, using operations defined above.
We claim that = y/rj. The map r i> rq on Fg2 is a field automorphism
fixing only Fg C Fg2. Then (s2 r]t2)q = (s + y/rjqt)(s ^qt). Thus s2 rjt2 =
2/Fq{s + ,/rjt)i where NF^/Fq is the relative norm from Fg2 to Fg. Given a fixed
basis for the vector space F^> underlying PG( 1, q2), we choose representatives for
the points of the projective line so that PG(1, q2) = {(1, a) : a G Fg2} U {(0,1)}.
We identify the points of Â£ with the points of PG( 1, q2) by
(l,s2 rjt2,sp) ^ (l,s + yPjt) (2.3)
12
and
(0,1,0,0) (0,1) = (oo). (2.4)
Note that s2 rjt2 is a square in Â¥q if and only if s + yfrjt is a square in F2.
To see this, let a be a primitive element of F92. Then aq+1 generates F* C F*2
Suppose s + y/rjt = an. Then s2 rjt2 = (s + y/rjt)(s y/rjt) = (ag+1)n.
We will say that a point P = (1, s2 rjt2, s, t) of Â£ is square if s2 rjt2 Â£
and that P is a nonsquare if s2 rjt2 6 0. Wre label the points (1,0, 0,0) and
(0,1,0, 0) with 0 and oo, respectively. Define
Â£a = {(1, s2 rjt2, s, t) : s2 rjt2 G } (2.5)
and
= {(1, s2 ?/f2, s, t) : s2 rjt2 G 0} (2.6)
To better motivate our definition of Â£D, consider the sphere Z of radius 1
whose center is at the origin in R3. Let P = (0, 0,1) and let II be the plane
z = 0. Define the map taking a point Q = (x, y. z) on Z to the
p:Z\{P}^n
P:(x,y,z)^yL,JL, o).
This is stereographic projection of Z\ {P} onto the xy plane. If we identify each
point (x, y, 0) with the complex number x + yi, we have identified the points of
% \ {P} with the complex numbers. Let
PC = {(1,C) :C e}u{(o,i)}
13
be the complex projective line. Then the map
p:Z\{P}^ PC
p : (0, 0,1) (0,1) = oo
identifies the points of Z with the points of the complex projective line. The unit
sphere, so identified with the extended complex numbers C U oo is the Riemann
sphere. The map p is analogous to our identification of the points of Â£ with the
projective line PG{ 1, q2). Then the set of points
is analogous to Â£n.
Although we will not require it later, we may make explicit the correspon
dence between the points of Â£ \ {oo} and the points of an affine plane. We
project from oo = (0,1,0,0) through the remaining points of Â£ to the plane
7T = [0,1,0, 0]r. The line ((0,1,0,0), (1, s2 pt2. s, t)) meets 7r in the point
(1,0, s, t). No points of Â£ are mapped to the line at infinity ((0,0,1, 0), (0, 0,0,1))
of 7r.
Let l = ((1,0, 0, 0), (0,1, 0, 0)). The tangent planes to Â£ at the points
(1, 0, 0, 0) and (0,1,0, 0) are [0,1, 0, 0}T and [1,0, 0, 0]T respectively, which meet
in the line lx = ((0,0,0,1), (0,0,1,0)). The planes on l1 different from the
tangents [1,0, 0, 0]T and [0,1, 0, 0]r are
= {{x, y, z) G Z \ {P} : z > 0}
(2.7)
lA = {7rc = [l,c1,0,0]T:ceF;}
(2.8)
14
Figure 2.1: Tangent planes to Â£ and to at the points (1, 0,0, 0) and (0,1, 0,0)
are the planes [0,1, 0, 0]T and [1, 0,0. 0]T, respectively. The oval o = {(1, s2, s, 0) :
s E Fg} U {(0,1, 0,0)} in the plane [0, 0, 0,1]T is the intersection of our chosen
elliptic quadric Â£ with our chosen hyperbolic quadric TC. The lines l and l1 are
both fixed by G;. The planes in Vg all meet in the line and the planes in Fg
all meet in l.
15
each of which meets Â£ in the oval {(1, s2 r/t2, s,t) : s2 r]t2 = c}. This implies
that for c e F*, there are q + 1 pairs (u,v) such that u2 T)v2 = c. For future
reference, define
Vkn = {[l,c\0,0]T:cen}
and
Ve0 = {[1. c1, 0, 0]T : c Â£ 0}.
The planes on / are
Fe = {aa = [0,0,1, a]T : a F?} U {a^ = [0, 0, 0,1]T}.
The plane meets Â£ in the oval
o={(l,s2,s,0) :sGF9}U{(0,1,0,0)} (2.9)
and for the remaining planes in Fe,
aa fl Â£ = {(1, s2 r]t2, s,t) : s = ta} U {(0,1, 0,0)}. (2.10)
Note that if {ta)2 rjt2 = r2 then (vta)2 rj{vt)2 = (vr)2 and that both
(1. {ta)2 rjt2, ta, t) and (1, {vta)2 r/{vt)2, vta, vt) are on aa. It follows
that, for b Â£ Fq, the points of Â£ \ {(1, 0,0, 0), (0,1, 0,0)} on are either all
square or all nonsquare. As with Vg, define a partition of Fz by
Fza = {a : aa n Â£n ^ 0} and FÂ£0 = {aa :aan8,0^ 0}.
We refer to Figure 2.1 for a schematic of the relationship between the points
of Â£, Vg, and FÂ£.
2.3 A Ruled Quadric in PG{3,q)
16
We describe a particular hyperbolic quadric IK in PG(3, q), with an emphasis
on the geometric similarities IK has with Â£. For a more complete description of
hyperbolic quadrics in PG(3,q), we refer to sections 6.1 and 6.2 of [20].
Put
H
0 1 0 0
1 0 0 0
0 0 2 0
0 0 0 2
and let
IK = {x : x.ffxT = 0} = {x : xQX\ x\ + 3:3 = 0}
The points may be described explicitly:
IK = {(1, s2 t2, s,t) : s,t G F9} U {(0, l.s,s):sG F*}
U{(0,1 s, s) : s G F*}
U{(0,0,1,1), (0,0,1,1), (0,1, 0,0)}.
IK contains the oval o = {(l.s2, s,0) : s Fg} U {(0,1,0)0)} in the plane
[0,0,0,1]T, in common with Â£ and indeed TCflÂ£ = o. Let / = ((1, 0,0,0), (0,1,0, 0))
as in our description of Â£. The line lL = ((0, 0,1,0), (0,0, 0,1)) meets IK in the
points (0, 0,1,1) and (0,0,1,1), and the planes [1, 0, 0, 0]T and [0,1,0,0]T on
P are tangent at the points (1,0, 0, 0) and (0,1, 0. 0) respectively. The remain
ing planes on P are V^ = {7rc = [1, c_1,0, 0]T : c G F9}, each of which meet IK
in the q + 1 points of an oval. In particular,
IK fl 7rc = {(1, s2 t2, s, t) : s2 t1
c}U{(0,0,1,1),(0,0,1,1)}. (2.11)
17
Let P be a point of J~C that is not on [1,0, 0, 0]r or [0,1,0, 0]T. Then P has
the form (1, a2 b2, a. b) for some a, b E Fq, a ^ b. We say that P is square if
a2 b2 E , and nonsquare if a2 b2 E 0. Now define partitions of and F^c
by
b'Kn = {[1, c~\0,0]T : c E } and = {[1, c_1,0,0]T :cE0}.
and
F:kd = {aa : aa n IKq 7^ 0} and = {aa ' OLa fl 7^ 0}.
Then the square points of TC are all on the planes of Wen and are all on the
planes of F^n and the nonsquare points of IK are all on the planes of V^0 and
are all on the planes of F^. The reasoning is the same as with that for Â£ in
the previous section.
18
3. The Actions of TwoPoint Stabilizers
In this chapter, we will determine the actions of the stabilizers of the trun
cated quadrics Â£n and !KD on the planes of PG(3,q). This will reduce our
problem of finding cardinalities of plane intersections with the quadrics to one
involving relatively few orbit types.
3.1 The Stabilizer of Two Points of Â£
In section 6.4 of [20], the stabilizer in GL(A.q) of the elliptic quadric Â£ is
found to be the group G generated by
/ 1 a2 rjb2 a b \
< 0 1 0 0 : a,b eÂ¥g
0 2 a 1 0
< 0 2b 0 1 >
{Wb\}
1 0 0 0 \
o to Tjb2 0 0 a,b E Â¥q, (a,b) ^ (0,0)
0 0 a b
0 0 rjb a >
0 1 0 0 1 0 0 0
N = 1 0 0 0 M = 0 1 0 0
0 0 1 0 0 0 1 0
0 0 0 1 0 0 0 1
19
Recall the identification of the points of Â£ with the points of PG( 1, g2) given in
equations 2.3 and 2.4. We note simply that the set {[r^]} is a group isomorphic
to the additive group of F?2 fixing (0,1,0,0) = (oo), and {[^a.f>]} is a group
isomorphic to the multiplicative group of fixing (1,0. 0, 0) = 0 and (0,1, 0,0).
The element N interchanges 0 and (oo), and it is easy to show that G is 3
transitive on the points of Â£. It is possible to show that G', the commutator
subgroup of G is isomorphic to PSL{2, g2). We refer to Chapter 12 of [24] for
details.
Henceforth, we drop the brackets and write ipa_b for the matrix [(pa,b\
The stabilizer of the pair of points {(1, 0, 0, 0), (0,1,0, 0)} is the group Gi
generated by M, N and {(pa,b} Note that the points of Â£\{(1,0, 0,0), (0,1,0,0)}
are the nonzero elements of Fg2 under the isomorphism 8. The group G stabi
lizing the elliptic quadric contains PSL(2, g2) as a subgroup. See p. 190 of [24]
for a proof.
Observe that {q?a,b} and form distinct subgroups of Gi, each of
order g2 1 and which intersect in {tpa,b ' Q2 V^2 = 1}, a subgroup of order
g + 1. Thus {{yy.b}, M{ipafi}M) is a group of order = (Q + J)(? l)2
The element N normalizes and thus Gi is a group of or
der 2(g + l)(g l)2. Note that A {a2I : a Â£ F*} is a subgroup of Gt
fixing all points. Consider the line m = ((1,1,1, 0), (1,1, 1, 0)) secant to
Â£. The element ipSjt takes m to ((l,s2 rjt2, s, t), (1, s2 r/t2. s, t)), so
the orbit of m under Gi has order at least On the other hand, m
is stabilized by Gm = (M, N, A, v^no) < Gi, a group of order 4(g 1), so
{{(1, s2 rjt2, s, t), (1, s2 rjt2, s, t)} : (s, t) ^ (0, 0)} constitutes the entire
20
orbit of {(1,1,1,0). (1,1, 1, 0)}, under the action of Gi.
Similarly, we see that the line m! = ((1,77,1,0), (1,77, 1,0)) is exterior to Â£
and is in an orbit of size under Gi.
3.2 Orbits of Planes Under Gi
Let Â£<2 be the set of lines in the orbit of m under Gi and let Â£g be the orbit
of m! under G[. Choose u 7^ 0 and consider the plane [1, 1, 0, u]T containing m.
It is easy to check that under the stabilizer in Gi of m, the orbit of [1, 1, 0, u]T
is {[1, 1, 0, cu]T}, and that the orbit under the action of the stabilizer of m! of
[1, 7]_1, 0, lu]t is {[1, ??_1, 0, o;]}. Define Tg to be the set of lines generated
by one of (1,0,0,0), (0,1,0,0) and a point of lL. It is straightforward to show
that Gi acts transitively on the points of lL, and that lL is fixed by G;. Because
N E Gi interchanges (1, 0, 0,0) and (0,1,0, 0), it follows that Tg is a single orbit
of lines under Gi.
Theorem 3.2.1 The orbits of planes of PG(3,q) under the action of Gi, the
stabilizer of {(1, 0, 0,0), (0,1, 0,0)} are as follows:
1. The two tangent planes [0,1,0,0]r and [l,0,0,0]r to Â£ at (1,0,0,0) and
(0,1,0,0), respectively.
2. The set of q2 1 planes tangent to Â£ \ {(1, 0, 0,0), (0,1, 0, 0)}.
3. The q 1 planes Vg = {7rc = [1, c1, 0,0]T : c F*}.
4. The q + 1 planes Fg = {7Q.b = [0, 0, a, b]T : a.b E Fq}1.
1 We shall see that this is a useful way to represent these planes
21
5. The planes not tangent to Â£ and meeting Â£ in exactly one of (1,0, 0,0),
(0,1,0,0) form an orbit of size 2(q2 1). These are the planes which are
not tangent to Â£ and not in Fg and which contain a line o/Tg. Call this
orbit Og.
6. The orbits Ogn, to E F* of size q2 1 such that [1, 1,0,lu]t E Ofn.
These are the planes not in Fg or Vg containing a single line of Fg. Call
this orbit 0 g;.
7. The ^ orbits Og^, u E F* of size q2 1 such that [1, r/_1, 0,u]T E
uj 7^ 2. These are the planes not in Fg or Vg and not tangent to Â£ which
contain a single line of.Fg.
Proof: The stabilizer Gi of {(1, 0, 0, 0), (0,1,0, 0)} is transitive on
Â£\{(1,0,0,0),(0,1,0,0)}
and therefore on the tangent planes to the points different from {(1,0, 0,0), (0,1, 0, 0)}
as well. The lines l = ((1,0, 0,0), (0,1,0, 0)) and lx = ((0, 0,1, 0), (0,0, 0,1)) are
fixed by Gi and it is easy to check that the group is transitive on the sets Vg
and Fg.
The subgroup of Gi fixing (0,1, 0, 0) is generated by M and {(pa,b} and is seen
to be transitive on the points of Zx. The stabilizer in (M, {zuatb}) of the plane
[0, 0,0, l]r contains all elements [^s,o]> s E F*, and [(^s,o][l, 0,0, c]T = [1, 0,0, sc]T,
so Gi is transitive on planes of the set Og which contain (0,1, 0, 0). Finally, N
interchanges (1, 0, 0, 0) and (0,1, 0, 0), so the planes in Og form a single orbit.
Each line of Fg is in exactly one plane of Fg and in exactly one plane of Vg.
The matrix M stabilizes m and interchanges [1, 1, 0,u;]r and [1, 1, 0, u]T.
22
Since Gi is transitive on Â£g, Ogn has order at least q2 1. The stabilizer of m
contains A, the matrix AIN and [^lo] which generate a group of order 2(q 1)
which fixes [1, 1, 0,o;]T. Thus 0Â£n is an orbit of size q2 1 and there are ^
such orbits.
Similarly, the orbit Â£g under G; of m' gives ^ orbits of size q2 1. We
check whether a plane in an orbit Og^ can be a tangent plane, that is, whether
(1. ?]_1, 0,uj)E E Â£. We find that when lo = 2, the planes are tangent to the
respective points (1, rj, 0, 1). The remaining ^ orbits are each of size q2 1.
The total number of planes accounted for by the union of these orbits is
q3 + q2 + q + 1 which is all of the planes in PG(3, q).
3.3 The Group Stabilizing Two Points of fit
We turn now to the hyperbolic quadric "K defined in Section 2.3. We seek
similarities between fit and Â£ in terms of the stabilizers in each case of the points
(1,0,0,0) and (0,1, 0,0).
The stabilizer in PGG+(4, q) of the pair {(1, 0,0, 0), (0,1, 0, 0)} contains the
group generated by the matrices
0 1 0 0 1 0 0 0
1 0 0 0 M = 0 1 0 0
0 0 1 0 0 0 1 0
0 0 0 1 0 0 0 1
23
and the set
/ 1 0 0 0 \
0 a2 b2 0 0 : a2 b2 0
0 0 a b
0 0 b a J
a.b} ^
Let K = ({wafi}) Note first that {c^a,6}I = (q l)2 and that N does not
normalize K, and the intersection of K and NKN consists of the ma$ such that
a2 b2 = 1. We see that the plane i\\ meets fit in the oval {(1, a2 b2, a, b) :
a2 b2 = 1} U {(0, 0,1,1), (0, 0,1. 1)}, which implies that a2 b2 = 1 has q 1
solutions (a, b). Thus the group Hx = (K, NKN) has order = (q
l)3, and (fV, K, NKN)\ = 2(q l)3. The group HK is normalized by the matrix
Af, so that (A4,N:K) is a matrix group of order 4(q l)3, and HK contains
the scalar matrices a21 and Na2I, a e F*, and these are the only elements
of this group which act as the identity on all points of tt1. Thus the group
of homographies in PGL(3,q) fixing fit and stabilizing {(1, 0, 0, 0), (0,1, 0, 0)}
has at least 4(q l)2 elements. In chapter 6, section 1 of [20], it is shown
that the complete group of homographies of the hyperbolic quadric in PG(3, q),
denoted PGO+(4, q), has order 2(q3 q)2, so we wish to show that the orbit of
{(1, 0, 0, 0), (0,1, 0, 0)} has order \{q + 1 )2q2. The stabilizer in PGO+(4, q) of
the point (0,1, 0, 0) contains a subgroup generated by matrices
{Ba} =
1 0 a 2 a 2 >
< 0 1 0 0 : a e Fq
0 a l 0
< 0 a 0 l >
24
and
/ 1 0 a 2 a 2
0 1 0 0 : a G Fq
0 a 1 0
k. 0 a 0 1
which contains {wa^} and is seen to be transitive on points of fit not on the
tangent plane (0,1, 0, 0)x. The matrix N interchanges (0,1, 0,0) and (1,0, 0, 0)
and we find that PGO+(4, q) acts transitively on points of TC and on pairs {P, Q}
of points such that P Q1. There are ~(q + 1 )2q2 such pairs, and we conclude
that the complete stabilizer of two points of J~C not on a line contained in TC has
order 4(ql)2. Thus, when working with the stabilizer of {(1, 0, 0, 0), (0,1,0,0)},
it is sufficient to use the matrix group of order 4(q l)3 generated by M, N and
{zuatb}. Call this group Ht.
Now consider the line m = ((1,1,1,0), (1,1, 1,0)) secant to TC. The ele
ment wa}b takes m to ((1, a2 62, a, b), (1, a2 b2, a, b)). Let be the lines in
the orbit of m under Hi. Let be the orbit of m' = {(1,77,1,0), (1,77, 1,0)).
The fines in and are those on the intersections of planes in F^ with
some plane of V^. Let be the lines on one of (1,0,0,0) or (0,1,0,0) and
on a point of lx \ {(0,0,1,1), (0,0,1, 1)}. Let be the set of lines gener
ated by one of (0,0,1,1) or (0,0,1, 1) and a point of ((1,0, 0, 0), (0,1, 0,0)) \
{(1,0, 0, 0), (0,1, 0, 0)}. Our next theorem describes the action of Hi on the
planes of PG(3,q).
Theorem 3.3.1 The orbits of planes under the action of Hi are as follows:
25
1. The tangent planes [0,1,0, 0]r and [1, 0, 0, 0]T to TC at the points (1,0,0,0)
and (0,1,0,0), respectively, form an orbit of size 2.
2. The tangent planes [0,0,1, 1]T and [0, 0,1, l]r to TC at the points (0, 0,1,1)
and (0, 0,1, 1), respectively, form an orbit of size 2.
3. The planes Vx = {nc} = {[1, c_1, 0, 0]r : c E F*}, form an orbit of size
q1.
f. The planes Fx on the line ((1, 0, 0, 0), (0,1, 0, 0)) distinct from [0,0,1,1]T
and [0, 0,1, 1]T form an orbit of size q 1.
5. The planes containing exactly one line of 7x form a single orbit of size
2(q l)2. Call this orbit 0xi
6. The planes containing exactly one line of form a single orbit of size
2(q l)2. Call this orbit 0^.
7. The planes on exactly one of the lines ((1, 0, 0, 0), (0, 0,1,1)),
((0,1, 0, 0), (0, 0,1,1)), ((0,1, 0, 0), (0, 0,1, 1)), or {(1, 0,0, 0), (0, 0,1, 1))
form an orbit Qif of size 4(q 1).
8. The single orbit of tangent planes to points ofTi not on [1, 0,0,0]T or
[0,l,0,0]r
9. The ^ orbits Off, uj G F*, u ^ 2, of size (q l)2 such that
[1,1, 0, w] G 0^n.
10. The orbits Off, u E F* of size (q l)2 such that [1, ry1,0, uj] G Off.
26
Proof: Hi stabilizes both l and H, so the first four sets are seen to be orbits,
as H is transitive on each of these sets. For items 5 and 6, we need to show
that Hi acts transitively on the indicated sets of planes. For item 5, note that
{tuatb} fixes (1,0, 0, 0) and is transitive on points of lJ~\ {(0,0,1,1), (0, 0,1, 1)}.
Then we see that ws\o takes the plane [0,1,1,0]T on ((1,0, 0,0). (0, 0, 0,1)) to
[0,1,5,0]. Thus the planes containing exactly one line of T;h form a single orbit.
Item 6 is handled similarly.
For the orbit 0^, observe that Hi is transitive on the 4 lines described
and since the pairs {(1, 0, 0, 0), (0,1,0, 0)} and {(0, 0,1,1), (0.0,1, 1)} each
are orbits, these 4 lines form a single orbit. Note that msgo takes the plane
[0,1,1, 1]T on ((1.0,0, 0), (0,0,1,1)) to [0,1, S, 8}T. Thus Hi is transitive on
all planes in the set 0^, and the remaining planes on the 4 lines in the statement
are tangents to "H whose orbits are described in 1 and 2, so 0^ is a single orbit.
For the orbits in 9, recall that Hi is transitive on lines ((1, a2b2, a. b), (1, a2
b2, a, 6)), (a, b) ^ (0,0) and that the stabilizer of ((1,1,1, 0), (1,1, 1, 0)) is
generated by M, N, ro_10 and A and so the orbit under this group of a repre
sentative plane, say [1, 1, 0, u)]T, uj 7^ 0 contains only itself and [1, 1,0, u]T,
giving the stated orbit size. Now check that (1,1, 0, u)H = (1, 1, 0, 2uj) is
a point on D~C iff uj = 1.
As Hi acts transitively on the points of IK \ {(1, 0,0,0), (0,1, 0, 0)}, it also
acts transitively on the corresponding tangent planes.
The lines form a single orbit under Hi, as {(0, 0,1,1), (0, 0,1, 1)} and
the points of ((1, 0, 0, 0), (0,1, 0,0))\{(1,0,0, 0), (0,1,0, 0)} form orbits of points,
and the stabilizer of (0, 0,1,1) contains the set {wafi}, which is transitive on the
27
points of {(1,0, 0, 0), (0,1, 0, 0)) \ {(1, 0,0,0), (0,1,0, 0)}. It is straightforward to
check that for each c F*, the set {wa^b : a? b2 = 1} contains an element wa^
such that a b = c, so that this set is transitive on planes [0,1, tu. eoY on the
line ((1,0, 0,0), (0,0,1,1)).
3.4 Point Sets Associated with Squares
In this section, we define the subsets of the points of Â£ and TC corresponding
under the partitions induced by the planes in Vg and tbr which correspond to
the nonzero squares in F9. These subsets are the truncated quadrics Â£D and
3.4.1 The Square Points of Â£
The group Gi contains a subgroup Gia of index 2 generated by M, N and
{[
Gi = Gia U
and acted regularly upon by {ifa,b 2 vb2 E }, namely
Â£a = {(1, s2 r]t2, s, t) : s2 r/t2 E }. (3.1)
We can view a point (1, s2 rjt2, s. t) E Â£ \ {(1, 0, 0,0), (0,1,0, 0)} as being either
square or nonsquare depending upon whether s2 rjt2 is square or nonsquare.
Similarly, a line ((1, s2 r]t2, s, t), (1, s2 rjt2, s, t)} e in the orbit of m
meets Â£ in two points, both of which are either square or nonsquare. The lines
in the orbit of m are all the secant lines to Â£ which contain a single point of lL.
A plane [0, 0, a, b]T E Tg meets Â£ in the q + 1 points
{(1, u, c, d) : u = c2 T)d2 and c/d = b/a} U {(1, 0,0, 0), (0,1, 0, 0)}
28
and we see that for a particular choice of plane in Fg, u is either always a
square or always a nonsquare, so there are two orbits of planes in Fg under Gia.
Each plane of Fg meets lL in a single point, so there are two orbits of points
of P under Gia as well. It is also easy to show that there are two orbits of
planes in Vg under Gia and since each plane of Vg meets l in a unique point,
there are 3 orbits of points of /, including {(1, 0,0, 0), (0,1, 0, 0)}. The orbit FÂ£
of m under Gi splits into two orbits Fgn and F^ under the action of G^a.
Similarly, the orbit F^ splits into two orbits Fgn and Fgn. Let Fg be as defined
in Theorem 3.2.1. Each plane [0, 0. a, 6]T G Fg contains (1,0, 0,0), (0,1,0,0) and
the q 1 points {(1, c2 rjd2, c, d) : ca + bd 0} of Â£. Because c2 r]d2 G iff
(re)2 rj(rd)2 G , we see that a plane in Fg meets Â£n in q 1 points, or it meets
Â£ \ {(1, 0,0,0), (0,1,0, 0)} in q 1 points. Call a plane of Fg square if it meets
Â£n and nonsquare otherwise. That is, put Fga = {a G Fga : a fl Â£n ^ 0} and
Fg0 = {a G F : a fl Â£D = 0}. The orbits of planes of PG(3, q) are bipartitions
of the orbits under Gi, with the exception of {[1, 0.0,0]T, [0,1,0, 0]r}, which
remains an orbit under the action of Gia
Theorem 3.4.2 The orbits of planes of PG(3,q) under the action of Gia are
as follows:
1. The tangent planes [0,1, 0,0]T and [1, 0,0, 0]T to (1, 0, 0,0) and (0,1,0, 0)
form an orbit of size 2.
2. The set of tangent planes to Â£n form a single orbit.
3. The set of tangent planes to Â£^ form a single orbit.
29
4 The planes V^n = {[1, c_1,0, 0]T : c G G F*} form an orbit of size
5. The planes Vg0 = {[1, c1, 0, 0]r : c G jTl G F*} /orm an orbit of size
5. FÂ£d, i/ie 21 square planes of form a single orbit.
7. FÂ£(Zj, the nonsquare planes of Fe form a single orbit.
8. The planes not tangent to E and meeting E in exactly one of (1,0, 0,0),
(0,1,0,0) and containing a line in a plane of FÂ£d form an orbit of size
q2 1. Call this orbit 0Â£(n.
9. The planes not tangent to E and meeting E in exactly one of (1,0, 0,0),
(0,1,0, 0) and containing a line in a plane of FÂ£(Zj form an orbit of size
q2 1. Call this orbit 0Â£;^.
10. The orbits 0Â£g, for some wGFJ of size q2p such that
= [l,l,0,u;]T G 0^
and is on a line of Â£Â£ in a plane o/FÂ£a. From the q 1 planes on a line in
Â£<Â£ and not in VÂ£ or FÂ£, one may choose 2 representatives of each orbit.
11. The 2=1 orbits 0Â£g, for some u GF of size which consist of planes
containing one line in From the q 1 planes on a line in Â£Â£n and
not in VÂ£ or FÂ£, one may choose 2 representatives of each orbit.
12. The 2=i orbits 0Â£^, for some of size 2=i which consist of planes
containing one line in FÂ£^. From the q 1 planes on a line in and
not in Ve or FÂ£, one may choose 2 representatives of each orbit.
30
13. The orbits Ogj, for some oj 6 F* of size which consist of planes
containing one line in From the q 1 planes on a line in and
not in Vg or Fg, one may choose 2 representatives of each orbit.
Proof: Based on Theorem 3.2.1 and the discussion preceding the statement of
the theorem, it is straightforward to verify the assertions.
Vg : q1 planes meeting in lL
Fg : <7+1 planes
meeting
in l
Figure 3.1: A diagram of the elliptic quadric from the point of view of the
stabilizer of two points.
Figure 3.1 is a diagram of Â£ from the point of view of the stabilizer
G[ of {(1, 0, 0, 0), (0,1, 0, 0)}. The horizontal lines represent planes in Fg,
and the vertical lines represent planes in Vg. The planes Vs meet in =
((0,0,1,1), (0, 0,1, 1)) and the planes Fg meet in l = ((1, 0, 0, 0), (0,1, 0, 0)).
The points of Â£\{(1, 0, 0, 0), (0,1, 0,0)} are on the lines in orbits Â£Â£ = VgnnFgD
31
and Lg = Ve0 n F&0 under the action of G;n. Each dot represents a
line of Â£2 containing a pair of points (l,a2 ?/&2,a,6), (1, a2 r/fe2, a, b),
(a,b) / (0,0). The intersections without dots represent the lines in the orbit of
m' = {(1,?7,1,0), (l,r/, 1,0)), Â£g under G(. Under the action of Gia, the four
quadrants of the grid each represent an orbit of lines. Each of these four orbits
of lines carries orbits of planes distinct from FÂ£ and Vg. In the next chapter, we
will study how the planes in an orbit carried by an orbit of lines in a quadrant
of the grid meet Â£n. We will see that it is sufficient to study planes in orbits of
the type 022 and orbits of the type 0Â£q
3.4.3 The Square Points of H
Let = {(l,s2 t2,s,t) : s2 t2 Â£ }, the subset of J~C analogous to
Â£a C Â£. That is, !Ka is the set of points of Oi corresponding to the set of
all (a, b) such that a2 b2 is a nonzero square. The group Hin stabilizing !KD
is generated by {zoa^ : a2 b2 Â£ }, M and N and [Hi : Hia\ = 2. It is
straightforward to show that, under the action of Hia, the points on l are in
three orbits:
{(1,0,0,0), (0,1, 0, 0)}, {(1, s2, 0, 0) : s Â£ F;} and {(1, r/s2, 0.0) : s Â£ F*}.
Similarly, there are three orbits of points on
{(0,0,1,1), (0,0,1,1)}, {(0,0, a, 6) : a2b2 Â£ } and {(0,0, a, b) : a2b2 Â£ $}.
The orbit under Hh splits into two orbits under Hia: Â£}} and which
contain the points of !H0 and respectively. Similarly, the orbit of lines
splits into the two orbits Â£^n and under the action of Hia.
32
Theorem 3.4.4 The orbits of planes under the action of Hia are as follows:
1. The tangent planes [0,1, 0, 0]r and [1, 0, 0. 0]T to the points (1, 0, 0, 0) and
(0,1,0,0), respectively form an orbit of size 2.
2. The tangent planes [0, 0,1, \}T and [0, 0,1, l]r to the points (0, 0,1,1) and
(0, 0,1, 1), respectively form an orbit of size 2.
3. The planes Vkg = {7rcD} = {[1, c_1, 0, 0]T : c Â£ }, form an orbit of size
q~ I
4 The planes Vx0 = {717^} = {[1, c_1,0, 0]T : c Â£ 0}, form an orbit of size
q~ I
5. The planes F^a = {7a b = [0,0,a,6]r} on the line ((1, 0, 0, 0), (0,1, 0, 0))
such that aab n TCa = {(1, s2 t2, s, t) : s2 t2 Â£ and sa + tb = 0} U
{(1,0, 0,0), (0,1,0,0)} form an orbit of size
6. The planes F^o = {7a,b = [0,0,a,fe]T} on the line ((1,0, 0,0), (0,1,0,0))
such that aa
{(1, 0,0, 0), (0,1,0, 0)} form an orbit of size
7. The planes F^0 = {7^ = [0,0,a,6]T} on the line ((1, 0, 0, 0), (0,1, 0, 0))
such that aa,b n TCa = {(1, s2 t2, s,t) : s2 t2 Â£ 0 and sa + tb = 0} U
{(1,0, 0,0), (0,1,0,0)} form an orbit of size
8. The planes meeting exactly one of (1,0, 0,0) or (0,1,0,0) and one point
on ((0, 0,1,1), (0, 0,1, 1)) \ {(0, 0,1,1), (0, 0,1, 1)} for some ^ Â£
F'ku form a single orbit of size (q l)2.
33
9. The planes meeting exactly one of (1,0, 0,0) or (0,1,0,0) and one point
on ((0, 0,1,1), (0, 0,1, 1)) \ {(0, 0,1,1), (0,0,1, 1)} n^,6 for some
Fxc0 form a single orbit of size (q l)2.
10. The planes on exactly one of the lines ((1, 0, 0, 0), (0, 0,1,1)),
((0,1,0, 0), (0, 0,1,1)), ((0,1, 0, 0), (0,0,1, 1)), or ((1, 0,0, 0), (0, 0,1, 1))
form an orbit of size 4(q 1).
11. The orbits 0^, uj G F* of size such that [1,1,0, w] e 0^
This includes a single orbit of tangent planes to points of CK not on
[1,0,0,0]T or [0,1,0,0]r. The tangent planes are the orbit and are
tangent to points of TCa when q = 3 mod 4 and are tangent to points of
JCgi when q = 1 mod 4.
12. The yy orbits u G F* of size . These are orbits of planes
meeting a single line of This includes a single orbit of tangent planes
to points of TC not on [l,0,0,0]r or [0,1,0, 0]r. The tangent planes are
tangent to points of Ka when q = 1 mod 4 and are tangent to points of
TC0 when q = 1 mod 4.
13. The yy orbits uj F* of size ' such that [1, 7?1, 0, uj\ 0^.
These are orbits of planes containing a single line of
14. The orbits 0^, u F* of size . These are orbits of planes
containing a single line o/Â£^.
Proof: We can check that Hia is transitive on the four lines in the statement
of number 9, via the action of (A/, N). Consider the action of {zua0 : a2 b2 =
34
1} Q %D on planes on any of these lines, say ,3 = [0,1, ui, uj]t for some
u / 0 on ((1,0,0, 0), (0, 0,1,1)). (3 C\ [0,0,0,1]. If zua,b(3 = 'OJc.dP for some
a2 b2 = c2 d2 = 1 then we must have a b = c d, whence a + b = c + d
and hence a = c and b = d. As there are q 1 representations a2 b2 of 1, the
orbit of (3 includes all planes on ((1,0, 0, 0), (0, 0,1,1)) which are not tangent to
Oi. Thus the orbit of [3 has size 4(q 1).
It is straightforward, if occasionally tedious, to verify the remaining state
ments.
Vji : q1 planes meeting in
s k h
Fk : q1 planes
meeting
in l
\ %
Vk0 V^ca
*
/
> F'Ka
Figure 3.2: A diagram of the hyperbolic quadric from the point of view of the
stabilizer of two points not on any line of the quadric.
The action of the stabilizer of two points of 0~C not on a line of the quadric is
similar to the action of Gi on Â£. Figure 3.2 is a diagram of K from the point of
view of the stabilizer Hi of {(1, 0, 0, 0), (0,1, 0, 0)}. The planes V
35
((0, 0,1,1), (0, 0,1. 1)) and the planes F'x meet in l = ((1, 0, 0.0), (0,1, 0, 0)).
The points of K not contained in either of the planes [1, 0, 0, 0]T or [0,1, 0, 0]T
are on the lines in orbits = Vxu fl Fxu and = Vx.0 fl Fx.0 under the
action of Hia. Each dot represents a line of containing a pair of points
(l,a2 b2,a,b), (l,a2 b2, a, b), a2 ^ b2. The intersections without dots
represent the lines in the orbit of m' = ((1,77,1, 0), (1,77, 1, 0)), under Hi.
Under the action of Hia, the four quadrants of the grid each represent an orbit
of lines. Each of these four orbits of lines carries orbits of planes distinct from
F^c and Vx We will see that, for our purposes, it is sufficient to study planes
in the ^ orbits of the type 0^, and in the ^ orbits of the type 0~xD.
36
4. Truncated Quadrics and Elliptic Curves
4.1 Point Counts on Planes Meeting Â£a
In view of Theorem 3.4.2, it is a simple matter to count the points of Â£D on
planes in certain orbits under Gm The planes in FgD each meet Â£D in q 1
points and the planes in Fg^ do not meet Â£n. The planes in VgQ each meet Â£n
in q T 1 points and the planes in Vg^ miss Â£n. Each plane of 0g;a lies on a line
n tangent to either (1,0, 0,0) or (0,1,0,0) and in a single plane of FgD. Thus
the q 1 planes on n each meet of the remaining 3~ (q 1) points of
Â£d. Similarly, the planes in 0g/^ each meet Â£D in ~ points. The tangents to Â£
meet Â£n in either 0 or 1 point. Counting the number of points of Â£a on planes
in orbits of type 0^, OgQ, Og^ and Og^ will take considerably more work.
In our discussion of the orbits of planes under the action of Gi, we found
that for the ^ orbits of type 0g[], the planes on the line m include 2 repre
sentatives of each of these orbits. Similarly, the planes on the line m' include 2
representatives of the orbits of type Og^.
In order to get a feel for the shape of Â£n, we checked, for small prime values
q, the intersection numbers of some planes with Â£n. Consider the planes on
((1,1,1, 0), (1,1, 1, 0)) distinct from [0,0, 0, l]r E F and [1,1, 0, 0]T E V, i.e.,
the set
T2 = K = [E1,0,cc]t:^GF;}.
Note that CPg contains two representatives, aw and a_w, from each orbit Og^.
As an example, in table 4.1 we have the intersection numbers of this family
37
Table 4.1: Intersection numbers of Â£n with planes in CPJ fr Q = 263
N
Planes meeting Â£n in N points
116
118
120
122
124
126
128
130
132
134
136
138
140
142
144
146
148
**
**
of planes for q = 263. We notice that Table 4.1 is symmetric with respect to
^=2 = 132, that each plane meets Â£D in an even number of points and that the
number of points per plane is relatively near 132. For odd primes p < 300, it
was observed that these characteristics hold in general.
Lemma 4.1.1 Let Â£n and tP^ be described as above. Then
1 Each plane in CPg meets Â£D in an even number of points.
2 For any N, there are an even number of planes of CP^ meeting Â£D in N
points.
Proof: Suppose a point of Â£D is incident with a plane = [1, 1, 0, oj\t of CPg,
that is
(l,s2 rjt2,s,t)[ 1, 1, 0, u]T = 1 s2 + rjt2 + cut = 0. (4.1)
38
If s ^ 0 then (1, s2 qt2, s, t) is also on aw. If s = 0, solve the quadratic in
t and note that the discriminant uj2 4q is nonzero and the roots are distinct.
Check also that these roots can never be zero. This proves 1. For 2, simply note
that (1, s2 rjt2, s, t) G au if and only if (1, s2 qt2, s, t) G a_w, and that each
aw meets the oval {(1, s2, s,0):sÂ£F}U (0,1, 0,0) in the points (1,1,1,0) and
(1,1,1,0).
The remainder of this chapter will be concerned primarily with the proof of the
following statements, and corresponding statements for the families of orbits of
planes 0^, 0^nQ, and 0^fQ.
1. If a plane of meets Â£D in N points, then ^ ^q < N < ^ + y/q.
2. If there is a plane of Tg meeting Â£D in ^ t points for some integer t,
then there exists a plane of Tg which meets Â£n in ^ +1 points.
We will see that the second statement, and the corresponding statements for
planes meeting the hyperbolic quadric J~C do not hold for all q.
Consider the plane au = [1, l,0,u/]T, uj G F* secant to Â£ and containing
the points (1,1,1,0) and (1,1, 1,0). Asking if a point of Â£n is on qw, there
arises the system of equations
(1, s2 r)t2, s, f)[l, 1, 0, cj]t = 1 s2 + r/t2 + ut = 0 (4.2)
s2 rjt2 a2 = 0. (43)
We may convert these to the homogeneous polynomials
f(x, s, t, a) = x2 s2 + r]t2 + ujxt (4.4)
39
g(x,s,t,a) = s2 jjt2 a2. (4.5)
This allows us to interpret solutions X = (x,s,t,a) to f{X) = g(X) = 0 as
points in projective space PG(3, q). Because 77 is a nonsquare, s2 gt2 = 0 has
no solution, so /(0, s. t, a) = 0 has no solution and a point P satisfying f(P) =
g(P) = 0 may be assumed to have the form P = (1, s, t,a). If P = (1, s, t, a) is a
solution to this system, then so is (1, s, t, a), and these two points correspond
to the single point (1, s2 gt2, s, t) on Â£n with s2 gt2 = a2. That is, if there
are N points satisfying f(X) = g(X) = 0, there are y points of Â£n on the
plane [1, 1, 0, ou]T. Each of / and g is a degenerate quadratic form over Â¥q in
variables x, s, t, a with respective matrices Mf and Mg,
1 0 UJ 2 0 \ O 0 0 1 0
0 1 0 0 Mg = 0 1 0 0
LO 2 0 V 0 0 0 v 0
! O 0 0 0 0 0 0 1
Let 6/ and Gg be the corresponding quadrics in PG(3, q). Then 6/ is a quadratic
cone with vertex (0,0,0,1) and a convenient carrier plane is [0,0,0,1]T. The
quadric Qg is a quadratic cone with vertex (1, 0,0, 0) and carrier plane [1,0,0,0].
Note that neither of the vertices is a point on the other cone, so the cones do
not share a linear component. With this choice of carrier planes, the vertex of
each cone is on the carrier plane of the other.
We postpone the continuation of this discussion in order to outline some
relevant results from algebraic geometry and the general theory of elliptic curves.
4.2 Some Background on Elliptic Curves
40
This section collects basic results from algebraic geometry from [21] and [15]
and on elliptic curves in particular from [22], [16], and [25] which is necessary to
justify the manipulations in coming sections. The first chapter of Shafarevich
[21] is particularly illuminating.
Let /i and /2 be homogeneous polynomials over an algebraically closed field
K and let Ci and C2 be projective plane curves defined by = {x e PG(2, q) :
/j(x) = 0} for = 1,2. A rational map from Cfi to C2 is a collection of rational
functions
The map $ is a birational equivalence if the functions
there exist functions ipj, j = 1, 2, 3 such that 0j o yj and o are the identity
on the points where the maps are defined. The following theorem is central to
our investigations. A proof and discussion may be found in Hartshorne [15],
chapter 6.
Theorem 4.2.1 Every curve is birationally equivalent to a nonsingular projec
tive curve which is unique up to isomorphism.
The genus of an algebraic curve is a nonnegative integer associated with the
curve which is invariant under birational transformation. Conics, for example,
are curves of genus g = 0. For curves over arbitrary fields, genus may be defined
algebraically via the RiemannRoch theorem. We again refer to Chapter 2 of
[22] . An elliptic curve is an algebraic curve of genus 1.
It can be shown (Silverman [22], Chapter 2) that any elliptic curve is iso
morphic to an affine curve E, together with a point (oo) whose points (x, y)
41
satisfy an equation
y2 + a\xy + a^y = x3 + CI2X2 + 04a: + a&, (46)
an equation in Weierstrass form, with coefficients in a field K. Any such curve
is isomorphic to a curve in the projective plane with an equation
y2z + a\xyz + a^yz2 x3 + a,2X2z + a^xz2 + ciqz3
via the map
(x,y) >[l,x.y], (00)  [0,1.0]
and conversely that any nonsingular curve given by a Weierstrass equation is an
elliptic curve.
Any isomorphism between curves with equations of the form 4.6 is given by
a change of variables
X = t2x' + u (4.7)
y = t3y' + vx + s (4.8)
with s, t, u, v G K and t 7^ 0, where K is the algebraic closure of K. See
[16], Chapter 3 for proof. We call this an admissible change of variables. If the
characteristic of K is not 2 or 3, equation 4.6 can always be transformed to an
equation of the form
y2 = x3 + Ax2 + Bx + C (4.9)
via an admissible change of variables. An elliptic curve is necessarily nonsin
gular, that is, it has no points (x.y) at which both partial derivatives vanish.
42
For a curve with equation y2 = f(x) as in equation 4.9, this is equivalent to
f(x) having distinct roots. We define an elliptic curve to be a nonsingular affine
curve whose points satisfy an equation of the form 4.9 together with a point oo.
We now state the HasseWeil Theorem, which is essential to our progress.
For a curve E, let #E(Fq) denote the number of Fgrational points of E.
Theorem 4.2.2 (HasseWeil) Let E be a projective curve of genus g defined
over a finite field Fg. Then
q + 1 2g^/q < N < q + 1 + 2g^q. (4.10)
We refer to [16] or [22] for a proof when g = 1 and [15] for the more difficult
result when g is arbitrary.
4.3 Birational Transformations Between Quartics and Cubics
In this section, we carry out algebraic transformations on equations of curves
whose points correspond to the points on the intersections of planes with the
truncated quadrics. The resulting equations are of the form 4.9, and we apply
the HasseWeil Theorem with g = 1.
4.3.1 Elliptic Curves FFom Orbits 0Â£D
We return to the equations 4.4 and 4.5 from section 4.1. These equations
arose from considering the intersection of a plane 7rw = [1, 1,0,lv]t with Â£.
From g(x, s,t, a) = 0 we have s2 gt2 = a2, which we substitute into equation
4.4 to get x2 a2 + cut = 0 We may assume that r / 0, since no point on
Gf Pi Qg is of the form (0, s, t, a). We solve for t
2 2
a
t =
UJX
43
and substitute into g(x, s,t,a) = 0 to obtain, after simplification
F(x, s, a) = 7jx2 + (u2 2rf)x2a2 + r/a2 ujx2s2 (411)
and we let 6p = {(a:, s, a) G PG(2,q)\F(x,s,a) = 0} be the corresponding
plane curve. When x = 0, there is the unique solution (0,1,0). Put x = 1 in
F(x, s, a) = 0 to obtain the equation
u2s2 = r]a4 + (cu2 2r])a2 + 7], (4.12)
the affine part of Gp.
We will show that equation 4.12 is birationally equivalent to an equation for
an elliptic curve in Weierstrass form, that is, an equation of the form 4.9. Our
hypothesis that Fg is an arbitrary finite field of odd order does not change. In
particular, these manipulations are justified when Â¥q has characteristic 3.
Our method follows one outlined in chapter 8 of Cassels [8]. Note that the
point (a, s) = (1,1) satisfies equation 4.12. Put1 u = to obtain (after
dividing through by u2)
^ = 4(1 + 1)*
u
27?), 1 1N2 T]
LO2 KU LO2
which leads to
,4 2 4 0.3 i ^ i 47?^
V?SA = u4 + 2 u* +
U!
u + ~u H
LO1
Put v = su2 and write the right hand side as G(u)2 + H(u), where
G(u) = u2 + g\u + g0 and
1The first step has the effect of moving the rational point (1,1) to oc on the resulting
elliptic curve and is not absolutely essential.
44
H(u) h\u + ho.
solving for coefficients we find cq = 1, go = h\ = 0 and ho = ~ Now
u2 = G(u)2 + H(u), so
(v + G(u))(v G(u)) = H(u).
Put i> + G(u) = f, whence v G(u) =  and then 2G(u) = t
Multiply through by 4f2 to obtain
H(u)
.2+2
^t2 = 2t3 2 1 (JL 1 1 ^ 3 1 to
U/'2 ' \.UJ2 ca4
t.
Let d = ut, so that
4d2 + 4dt + ^lt2 2t2 2 ( 2L iL ] t
cu2 w4
and then let r = 2d + t, so that d = Gr and after simplification
( 7] 4 ^
LO2) ' V LU2 UJA
Finally put y = 2r and x = 2f to put our equation into a rather nice form:
V \ _2 ^
uj J \ a/1 w
2
x
2 i ( 4r?tu
(4.13)
and let Egp, together with oo, be the elliptic curve whose points satisfy this
equation. Tracing through the various transformations, we find that the rational
maps
2x
a =
1
yx
(4.14)
45
2x3 + (1 8^)x2 y2
(:y ~ x)2
(4.15)
with inverse maps
4 ya2 + 2(u!2 Arj)a + 4r/ + 2uj2s
u2(a l)2
(4.16)
y =
Arja3 + 2(uj2 2rj)a2 + 2(lo2(s + 1) 2 y)a + 2(2y + oj2s)
(a l)3w2
(4.17)
map the curve with equation 4.12 to the affine part of E = E^a. The functions
from E to Cp are undefined only when a = 1, and the inverse functions are
undefined only when x = y. If we put y x in equation 4.13. we find that for
in Fg. Thus the only point on 4.13 with x = y is (0, 0) and so the only rational
points of E on which the maps 4.16 and 4.17 are undefined are (0,0) and oo.
The two points of Cf for which a = 1 are (1,1) and (1,1). We extend the
rational maps between CF and E by (1,1) oo and (1,1) (0,0). Thus
augmented, the rational maps 4.14 and 4.15 and their inverses 4.16 and 4.17 give
a bijection between the points of E (including oo) and the nonsingular points
of the projective curve CF that is, the points of GF different from (0,1,0).
Two nonsingular points (s, a), (s, a) on correspond to the single point
(1 ,s2 r]t2,s, t) with s2 yt2 = a2 on au O Â£a, where t = Thus the map
from the points of the elliptic curve E to the points on n Â£a is 2:1, and by
the HasseWeil Theorem we have the following result.
Theorem 4.3.2 Let a^, w 6 F* be a plane in the orbit 0^ and let =
W D Â£n. Then Nw is even and
x 7^ 0, the discriminant of the resulting quadratic is which is never a square
(4.18)
46
Let Egp denote the elliptic curve with equation 4.13 and Egn = {Eg^j^gF*.
In our example q = 263 in Figure 4.1, we find _264 + 2\/263j = 296 = 2(148)
which shows that the bounds of 4.18 are as close as possible wfithout some further
restriction.
4.3.3 Elliptic Curves From Orbits 0gn
Let IPg = {[1, p_1,0,o/r : u G Â¥q \ {0,2, 2}}. We choose a representa
tive plane op = [1, rj, 0, lu]t in the orbit Ogj^. Note that [1, 771, 0, 2]T are
tangent planes to Â£ at the points (1, 77,0, 1), hence the restriction. From the
equations
(1, s2 rjt2, s, t)[l, rj, 0, u]T = 1 ?/(s2 rjt2) + cut = 0
we obtain the equation
f]u2s2 = a4 + r](u 2)a2 + rj2.
(4.19)
The substitution
y/VX
(4.20)
y
v/r)(4x3 + x2(2 uj2) 2 y2
(4.21)
transforms 4.19 to
(4.22)
47
which is nonsingular whenever u> ^ 2. The inverse rational maps are
x =
y =
2 2
2a2 + 2r] + 2T]u)s + a; a
4a2
and
2dZy/fj + 77 2 + 2?7cjs + y/rjuj2a2
4 a3
(4.23)
(4.24)
Again we apply the HasseWeil theorem and find that for = aw fl Â£n,
uj 7^ 2
 ^9 < JV < ^ + A (4.25)
When uj = 2. the equation 4.22 is singular and the curves correspond to
the planes tangent at the points (1, 77,0,1). When q = 3 mod 4, 77 is a
square and the points (1, 77, 0, 1) are on Â£n, while when q = 1 mod 4 they are
on 8,0. The total number of points of Â£n on the planes a^, uj G F9 \ {0, 2, 2}
is therefore when q = 1 mod 4 and 9 ~l9 when q = 3 mod 4.
4.3.4 Elliptic Curves from Orbits 0^
Now consider !Kn, a subset of the hyperbolic quadric TC defined in Section
3.5 and stabilized by the group We look first at planes in the orbits 0^.
Similar to the situation with planes in orbits 0Â£Q when uj = 2, the planes
are tangent to 0~C at the points (1,1,0, 1). These points are on J{D when
q = 3 mod 4 and on when q = 1 mod 4. From the system of equations
(1, s2 t2, s. f)[l, 1)0, 1 s2 + t2 + cut 0
s2 t2 a2 = 0
we obtain the quartic
(4.26)
uj2s2 = a4 + (to2 2 )a2 + 1
(4.27)
48
which is equal to 4.19 if we set 77 = 1 in that equation. We adapt the birational
maps from the discussion of planes in Og. The rational maps
take 4.27 to
x
a =
y
(4x3 + x2 (2 u)2) 2 y2
S= 2unp
y=x6 +
2uj2
x
= X I x
UJ
U) U)
16 7
a>2
X
which is identical to equation 4.22. The inverse maps are
x =
y =
2ci2 + 2 T 2cos f ct.2ci2
T2
2a2 + 1  2cos 7 lo2cl2
4 a3
(4.28)
(4.29)
(4.30)
(4.31)
(4.32)
Although the curves that arise here are isomorphic to those from orbits 0^, the
point counts on planes are slightly different. Note that the plane [1, 1,0, w]T
meets TC in the four points (1, 0, iw'1, to~l) and (0,1, u;1, u^1) which are
not in IKn. These pairs of points lie on the planes [0,1,0,0]T and [1,0,0,0]T
tangent to TC at (1, 0, 0,0) and (0,1, 0,0), respectively.
Let j3w be a plane in the orbit 0^,, and Nw = /A, Pi . Then
+ (433)
by the HasseWeil Theorem.
4.3.5 Elliptic Curves from Orbits 0^D
49
Consider an orbit 0^, which contains the planes [1, 77,0, cu]T on the line
((1, rj~l, 1, 0), (1,7/1, 1, 0)). Note that for all uj E F*, these planes are secant
to IK. From the system
(1, s2 t2, s, i)[l, 77, 0, co]T = 1 7)(s2 t2) + ut = 0
s2 t2 a2 = 0
(4.34)
we obtain the quartic
uis\2 4 u2 2rj 2 1
 = aT +
V )
7]z
which we transform to the cubic
y2 = x3 +
2rj u2 2 to4 At)u/
2t]2 I6774
U,2 4:7]
X
CO
X'X 47]2) \ 47]2
(4.35)
(4.36)
using the technique from Section 4.3.1. Let E^a be the curve with equation
4.36 and E^n = {Â£^p : cu E F*}. The plane = [1, 77,0, co]T meets the
plane [0,1,0, 0]r at the points (1, 0, u;_1, cu1) and (0,1, cu~1, u~4) of K,
which are not points of
(s.t.a) (a;1, w_1, 0) to the system 4.34. Thus the number of solutions
(s, t, a) to 4.34 is two more than \aw D fKa.
To reiterate, let aw be a plane in the orbit 0^, and Nu = \a^ fllKn. Then
by the HasseWeil Theorem,
+ (437)
4.3.6 Families of Curves
50
Return now to equation 4.36 and make the substitution x i> x + ~4rj2V
Multiply both sides of the result by (^)6 and make the substitutions y i
y and x ^ x, to obtain
4 p
y =x\x~i \x~ ,.,2
4 p a/
U)*
(4.38)
which is identical to equation 4.13. That is, the families of curves EgD and E^D
are the same.
To reiterate the relationship between curves from TC0 and from Â£n, we state
the following theorem.
Theorem 4.3.7 The set of elliptic curves Eg that arise from considering the
intersections of planes in Tg with Â£n is identical with the set of elliptic curves
E^ which arise from considering the intersections of planes in 7^ with 7Ca. The
set of elliptic curves Eg that arise from considering the intersections of planes
in CPg with Â£n is identical with the set of elliptic curves E^ which arise from
considering the intersections of planes in 7^ with 7(a.
Thus we have two families of elliptic curves to consider, and they take sur
prisingly simple forms. Make the substitution a = y into equation 4.13, and
write
Eg = {y2 = x (x pa2) (x {pa2 l)) : a E F*}.
(4.39)
From equation 4.22, we obtain, via the substitution a = j, the set of curves
Eg = {y2 = x (x a2) (x {a2 l)) : a E Fq\ {0,1, 1}}. (4.40)
51
4.4 Summary of Plane Intersections with Truncated Quadrics
In this section, we give a complete summary of plane intersections with Â£D
and in terms of exact counts or bounds given by the HasseWeil Theorem.
We first explain how the bounds we have found give bounds on orbits we have
not yet considered.
4.4.1 Other Plane Orbit Types
We turn now to the orbits of planes OgQ, and 0^ and their
intersections with Â£n and TCD. These orbits have the same relationship with 80
and !H0 as the orbits we have already considered have with Â£D and TCD, and
the point counts of the intersections are easily calculated.
Choose a, 6 e Fq such that a2 ijb2 = rj. Then Lpa^ G Gi interchanges
Â£d with 80 and in particular, (1,1,1,0)ipa0 = (l.r], a, b) and (1,1, 1, 0)
(1,77, a, b), so the set IPgQ is exchanged with a set of representatives for orbits
0^0. As each nontangent plane of PG(3, q) meets Â£ in q +1 points, the number
N0 of points of Â£D on the plane
[1, 771, ujbrj1 ,uaiqlY G Ogg
is equal to the number of points of 80 on [1, 1,0, w]T. By our application of
the HasseWeil Theorem to the number of points on planes in we have
y/q < + y/q
We obtain the same bounds on the number of points of Â£n on planes in orbits
of type by counting the number of points of Â£$ on planes in orbits 0^
and mapping 0^ to 0 with ipa.b, which simultaneously interchanges Â£n and
52
8,0. Thus if a plane tt of PG(3, q) meets 8 in q + 1 points of which ~ + t are
points of Â£n, then an element of Gi taking Â£n to (such as cpa.b as defined
previously) takes it to a plane (pa,b^ meeting 80 in + t points, so Pa.b^ meets
Â£d in ^ t points. We call this a complementarity property.
Similarly, if we choose c and d G Fq such that c2 d2 = 77, then wc0 G Hi
interchanges the points of CKn with the points of TC^, and takes orbits 0^ and
0^ to orbits 0^ and 0^, respectively. We may choose as a representative
plane in an orbit of type 0^D the plane 7iy, = [1, 1, 0, uj\ for some u ^ 0. Then
7iy, contains the points (1, 0, uW, To;1) and (0,1, cu1, cu_1) of H and so meets
TCn Uin q 3 points. Thus a complementarity property holds for TC as well.
4.4.2 Summary of Plane Intersections for Truncated Quadrics
We summarize our results on the plane intersection numbers for Â£n and for
iHd in two theorems.
Theorem 4.4.3 Let 8a = {(l.s2 r/t2,s,t) : s2 r]t2 G } and let the orbits
of planes under Gia be as stated in Theorem 3.42.
1. The tangent planes to 80 each meet Â£Q in a single point.
2. The tangent planes to 8 \8a do not meet Â£n.
3. The planes do not meet Â£n.
4. The planes Vgn each meet Â£n in q + 1 points.
5. The planes in each meet Â£c in q 1 points.
53
6. The planes in F^ do not meet Â£n.
7. The planes in 0g/n each meet Â£D in ^ points.
8. The planes in Oei0 each meet Â£D in ^ points.
9. Each plane in an orbit weFJ each meets Â£n in N points, where N
is even and satisfies N\ < yfiq.
10. Each plane in an orbit uj Â£ F* meets Â£D in N points, where N is
even and satisfies Ar < y/q.
11. Each plane in an orbit 0^, u Â£ F* meets Â£c in N points, where N is
even and satisfies A^ < ^[q.
12. Each plane in an orbit uo Â£ F* meets Â£a in N points, where N is
even and satisfies \^ N\ < sjq.
Proof: When we are able to give an exact value, the number is at most a
consequence of the orbitstabilizer theorem. The bounds given in statements
9 and 10 are consequences of our application of the HasseWeil Theorem, and
the bounds in statements 11 and 12 are arrived at via the complementarity
property discussed prior to the theorem statement.
The structure of TCa is slightly more complicated than that of Â£n, and there are
several more orbit types.
Theorem 4.4.4 Let TC = {l,s2 t2,s,t) : s2 t2 Â£ } and let the orbits of
planes under Hia be as stated in Theorem S.f.f.
1. The ^~11 tangent planes to each meet in q 1 points.
54
2. The ^4p tangent planes to 04.0 each meet TCa in q 2 points.
3. The planes Okq each meet TCa in q 1 points.
4 The planes V^c0 do not meet lKn.
5. The planes F^a eac/i meet CKn in q 1 points.
6. The planes F^ do not meet lKa.
7. The planes meeting exactly one of (1,0, 0, 0) or (0,1,0, 0) and one point x
on {(0, 0,1.1), (0,0,1, 1)) \ {(0, 0,1,1), (0, 0. 1, 1)} such that x is on a
plane of F^a each meet CKn in points.
8. The planes meeting exactly one of (1, 0,0,0) or (0,1,0,0) and one point x
on {(0, 0,1,1), (0, 0,1, 1)} \ {(0, 0,1,1), (0, 0,1, 1)} such that x is on a
plane of F^ each meet !Ka in points.
9. The planes on exactly one of the lines ((1, 0, 0, 0), (0, 0,1,1)),
((0,1,0, 0), (0,0,1,1)}, {(0,1, 0, 0), (0,0,1, 1)), or {(1, 0,0, 0), (0, 0,1, 1)}
each meet tKa in ^ points.
10. Planes in an orbit 0^, a; Â£ F* \ {2, 2} each meet "Ka in N points for
some N satisfying Ar < q.
11. Planes in an orbit 0^, u Â£ F* each meet Jia in N points for some N
satisfying l2^ N\ < ^fq.
12. Planes in an orbit 0^, u Â£ F* each meet Ka in N points for some N
satisfying ^ N\ < ^fq.
55
13. Planes in an orbit O^c0, weFJ each meet dtD in N points for some N
satisfying ^ N\ < yfq.
14. Planes in the orbit of tangent planes to points ofK not on [1, 0,0, 0]T
or [0,1, 0,Of each meet Ka in q points when q = 1 mod 4 and in q 1
points when q = 3 mod 4.
15. Planes in the orbit of tangent planes to points of IK not on [1, 0, 0, 0]T
or [0,1,0, Of each meet !Kn in q 1 points when q = 1 mod 4 and in q
points when q = 3 mod 4.
Proof: Because H[a is transitive on the points of Ka, and hence on the tangent
planes to Ka, it is sufficient for 1 to consider the tangent plane [1,1, 2, Of to
the point (1,1,1, 0). This plane contains 2q + 1 points of K, including (1,1,1,0)
and the four points (1, 0,2_1, 2_1) and (0,1, 2_1, 2_1). Any element zua^ of
Hi such that a2 b2 E 0 interchanges Ka and and so the points of Ka
distinct from (1,1,1,0) on [1,1, 2, 0]T are equal in number to the points of
on [1,1, 2, 0]T The reasoning for 2 is similar.
The other exact counts are either immediate or are a consequence of the
orbitstabilizer theorem. The planes in orbits of type Ofn and OfD were han
dled in Sections 4.3 and 4.4. The bounds given in statements 11 and 13 follow by
our application of the HasseWeil Theorem and the complementarity property.
4.5 Sums of Point Counts
In this section, we use our knowledge of the number of incidences of sets
of representatives of plane orbits with the truncated quadrics to find the total
56
number of points in the elliptic curve families given in equations 4.39 and 4.40.
The planes
P? = {au} {[!> 15 0, lu]t : u: G F*}
all meet in the line ((1,1,1, 0), (1,1, 1, 0)) and partition the points of Â£ which
do not lie on either [0,0,0, l]r or [1, 1,0,0]T. Thus the number of incidences
of {op} H Â£n is seen to be As the map taking points of Â£n fi au to points
on the associated elliptic curve is 1 : 2, the total number of points on the family
of curves E = {E^} is q2 1, including multiplicities.
Now we have the problem of finding how the points of TCn are partitioned
by the set of planes on a single line in an orbit or On the line
((1,1,1, 0), (1,1, 1, 0)) G Tarn, each pair of planes [1, 1, 0, u;], to G F* are
in a distinct orbit 0^. We will write a^ = [1, 1,0, a/)2". When uj = 2, the
planes aw are the tangents to !K at the points (1,1,0,1) and (1, 1,0, 1).
In [1, 1, 0, 2}T, the plane tangent at (1, 1, 0,1), the tangent lines to Of may
be written n\ {(1,1 + 2A, 1 + A, A) : A G Fg} U {(0, 2,1, 1)} and ?r_i =
{(1,1 + 2A, 1 A, A) : A G Fg} U {(0, 2, 1, 1)}. The points (0, 2, 1, 1)
are on the plane [1, 0, 0, 0]T tangent at (0,1, 0, 0). The points (1,1 + 2A, 1 + A, A)
and (1,1 + 2A, 1 A, A) are the unique points of nx and n_i on the plane
[1, (1 + 2A)_1,0,0] G Vk. Thus [1, 1,0, 2)T meets TCn in q points when 1
is a square in Wq and in q 1 points when 1 is a nonsquare. In this way we find
the same counts for points of +CD on the plane [1, 1, 0, 2]T. 2 Let N be the set
of points of 9fa on one of the planes [0, 0,0,1]T, [1. 1,0, 2]r or [1,1, 0, 2]T.
The planes in the set TQ = {a^ = [1, 1,0, uj)t : uj ^ 0,2, 2} each contain
2It is possible that we could have made a more geometric argument here.
57
(1,1,1,0) and (1,1, 1, 0) and each of the g2 g+6~4j points of 3ia \ N are on
exactly one of these planes, where d = q 1 when q = 3 mod 4 and d = q
when q = 1 mod 4. Each plane in IP}} contains (1,1,1,0) and (1,1, 1,0) and
meets 2 points of CK fl [1,0,0,0]T. In particular, aw = [1, 1,0, a;] contains the
points (0,1, a;1, a;1) and (0,1, cj^1). Thus the planes of IP}} have a total
of q2~q+9 incidences with points of fK0D ([1, 0, 0, 0]TUd{), counting the 2(5 3)
incidences from {ay.,} meeting (1,1,1,0) and (1,1, 1,0). Each such incidence
yields 2 points on the curve with equation 4.29. Each curve also has 4 points
corresponding to the 2 points on the intersection ay D K fl [0,1,0,0]T. This
proves the following theorem.
Theorem 4.5.1 The family of elliptic curves
Eg = [y2 = Xs + (1 8f)x2 + fyjr5n*: Ef.l
U)
contains a total of q2 1 Fqrational points. The family of elliptic curves
E? = {y2 = x3
2lu2
, LO UJ
x
+ ()x:ioeFq\{072,2}}
contains a total of q2 2q 3 Fqrational points.
4.6 The Invariant j and Symmetry of Incidences
In this section, we show that for the family of planes = {a=
{[1, l,0,u;]T}, uj G F*, the planes meeting Â£n in ^ + t points are in one
toone correspondence with the planes in IPg meeting Â£n in f points. To
prove this result, we require more elementary properties of elliptic curves. Proofs
and further discussion of these results may be found in [16] and [22]. For our
58
results in Section 4.6.2 and beyond, we find it necessary to restrict ourselves to
finite fields Â¥q where q = pe and p > 3.
4.6.1 Admissible Changes of Variables
For the finite field Â¥q, let Â¥q denote the algebraic closure of Â¥q. As usual,
uj G F* and p is a nonsquare in F9. Two elliptic curves with Weierstrass equations
y\ = f(xi) and vl d(x2) with coefficients in Fg are isomorphic over F? if and
only if there exists a linear change of variables of the form
x\ u2x 2 + r
(4.41)
V\ = UZV2 + SU2X2 +1
where r, s, t e F9 and u G F*. The curves
E = {(x, y) : y2 = x3 + ax2 + bx + c} U {(oo)}
and
Ev {(a;, y) : py2 = x3 + ax2 + bx + c} U {(oo)}
are seen to be isomorphic over Fg, via the substitution y ^/rfxy. In this case
we say that En is a twist of E by rj. E and Ev are isomorphic over Fq2, and
since any nonsquare /? e Fg may be written (3 = pa2 for some a E Â¥q, the twist
of E is unique up to isomorphism.
Proposition 4.6.2 Let E and E71 be elliptic curves given by y2 = f(x) and
py2 = f(x), respectively, where f(x) = x3 + ax2 + bx + c and let #E be the
number of points on E over Fg. Then ffE + jfEn = 2(q + 1).
Proof: For each zero x of /, the point (x, 0) is on both E and Ev. Say / has
m zeros. Then for each of the remaining q m elements x of Â¥q, f(x) is either
59
square or nonsquare, yielding two points on either E or Ev. The two points at
infinity give the desired sum.
Recall that the curve Eu from the plane 7rw in orbit Og[J has k points if and
only if the plane au meets Â£n in exactly  points. To show that the symmetry
of incidences observed in Section 4.1 holds for all q. by Proposition 4.6.2 it will
be sufficient to show that for u 7^ 0, the elliptic curve Ew = {{x,y) : y2 =
x3 + (1 8^)x2 + ^)x} C (()} has as a twist the curve E$ for some
5 =Â£0. Consider the twist E^ 1
T) 1y2 = x3 + (1  ) x2 +
16?72 4T]
U!4 Ul2
X
and make the substitutions x 1> a 2y 1x + r and y 1> a 3rj 4y, yielding
3l
2 3 a2r](u2(3r + 1) 8y) 2
y = x \5x
a47]2(16rj2 4r]ui2(4r + 1) + ru/(3r + 2)
ar
a^ifrilQrf 4r)u2(2r + 1) + ru4(r + 1))
after clearing denominators. Put r = and a ^ to obtain, after simplifi
cation
2 3 2h^2 2 w44t7u;2
V =x +
xz +
x.
2i] ~ ' 16?72
Multiply the numerator and denominator of the coefficient on x2 by (^)2^,
and multiply the numerator and denominator of the coefficient on x by (^Â§)2,
whence
y2 = x3 +
1
(5)V
x.
60
So the curve Eg is a twist of Ew when S = Note that ^ = u implies rj = (f)2
contrary to the assumption that 77 is a nonsquare, so 8 ^ cu. Thus for each E^
in EÂ£ with q + 1 + t points, there is a different curve E$ in Eg with q + 1 t
points. This proves the following theorem.
Theorem 4.6.3 In the family of elliptic curves over Fg
if Eu is a curve in Eg then Ef is in EÂ£ as well. Further, if a particular curve
Eu is represented k times in Eg then Ef. is represented k times in Eg.
We can restate this in terms of the plane intersections awClÂ£c. If au meets Â£D in
of the relevant theorem for plane intersections is part of Theorem 4.6.6.
We might ask how often these complementary pairs of planes (and comple
mentary pairs of elliptic curves) occur in our various families. To find an answer
to this, we invoke more of the theory of elliptic curves. For the remainder of
this section, assume that we are working over a finite field F9, where q = pe f or
some positive integer e and for a prime p > 3.
4.6.4 The Invariant j
To an elliptic curve given by a Weierstrass equation with coefficients in a
field F, it is possible to associate a value j G F, the jinvariant of the curve that
is invariant under an admissible change of variables, 4.41. Thus any two curves
with the same jinvariant are isomorphic over the algebraic closure of F. For
curves with Weierstrass equation y2 = x3 + ax2 + b:r, the jinvariant is given by
^ I1 points, then as meets Â£n in
(7+1 __ t
2 2
points, where 5 = The statement
256(a2 36)3
^ b2(a2 46)
(4.42)
61
For details of the derivation and properties of j, see Chapter 3, section 3 of [16].
The curves given by equation 4.13 for uj G F* are the curves EÂ£ and were shown
to be isomorphic, as a set, to the curves E^. Further, we found that whenever
a curve FT is in the set
uji
UJ4 UJ2
x : u, 6 F} (4.43)
a quadratic twist E'% of that curve is also in Eg, and that a curve and its twist
occur the same number of times. From formula 4.42, the j invariants for this
family of curves are
16(16?72 4t]uj2 + o4)
] =
(4.44)
Tj2(jj4(Ar) uj2)2
for uj G F*. We have already seen that the curves E^ and E^ are proper twists
of one another, (and so are isomorphic over Fg2 but not necessarily isomorphic
over Â¥q) and that Eu and E_^ are isomorphic over Fq. Solving
16(16?y2 4rj52 + 84) 16(16?72 4 rju2 + uj4)
V254(4r] S2)2
for 8, the solution set is
4 7)
rj2u4(4r] u2)2
(4.45)
uj
rf
y/4r] uj2,
2i/rjLd 2\Jriuj2 4 if
UJ
\J uj2 4r]
OJ
and we find that there are at most 12 values of 8 such that Eu = E$ over Fq.
For the curves
2 uj2
Eg = {y2 = x3
the jmvariants are
x
uj4 uj2
+ (  X)1: w e Ft w ^ 2)
. 16(xj4 4xF + 16)3 . _ ,
J 4x 2 a\2 UJ & Fg \ \2, 2}
uj4(ujz 4 y
62
and in this case the values of 5 such that Ew = E$ are
5 G s Eluj. , 
4 (4.46)
U
uj \/4 uj2 \/ uj2 4
The curves E^ and E are easily seen to be isomorphic over Fg. When q =
LJ
1 mod 4, lu2 4 and 4 a;2 are either both square or both nonsquare. A simple
count shows that for some values of uj, both a;2 4 and 4 u2 are nonsquare.
Thus it is possible that the family of curves Eg does not contain a proper twist
of for every u G F*. When q = 3 mod 4, exactly one of uj2 4 or 4 uj is
a square, and it is straightforward to check that for each uo G Fg, at least one
value of 5 in 4.46 gives a twist of E^ by 1. We interpret this result in terms of
plane intersections with Â£D and 3La.
Theorem 4.6.5 Assume that q = 3 mod 4. Let = Tg = {[1, 1, 0, a;] : uj G
F*,u; 7^ 2} and fP^ = Tg = {[1, rj~l, 0,w] : uj G F*,a> 7^ 2}. Then for each
plane in meeting TCn in + t points for some integer t, there is a plane
in meeting 0ia in t points. If there is a plane in fPg meeting Â£a in
q~ + t points for some integer t, then there is a plane in fPg meeting Â£n in
t points.
We restate our theorems on the symmetry of families of elliptic curves in terms
of symmetries of families of orbits of planes under the actions of Gia and Hi0.
Theorem 4.6.6 Let q = pe be a prime power with p > 3, and let {Og^j^eF*,
{Ogw,. {0oW;. mi {OSS,} ^eF* be families of orbits of planes as de
scribed in Theorems 3.42 and 3.44
63
1. For each orbit in {O^qIwsf* whose planes meet Â£a in ^ + t points, there
is an orbit whose planes meet Â£D in ^ t points.
2. For each orbit in whose planes meet Ka in + t points, there
is an orbit whose planes meet !Ka in ^ t points.
3. When q = 3 mod 4, for each orbit in {Pzfyuev* whose planes meet Â£D in
+1 points, there is an orbit whose planes meet Â£n in ^ t points.
4 When q = 3 mod 4, for each orbit in {OgQj^eF; whose planes meet IKa in
~ +1 points, there is an orbit whose planes meet !Ka in t points.
Proof: These statements are an immediate consequence of Theorem 4.6.5 and
the correspondences between points of the truncated quadrics on planes of the
stated orbits and their associated elliptic curves.
Appendix B contains tables for the number of points on the two families of
elliptic curves which we have studied for primes q < 200. Table 4.2 is an
example and may be viewed as a refinement of Figure 4.1. The first column
states the number N of points on a curve and the second column states the
jinvariants of the curves which occur having N points, followed in parentheses
by the number of times that particular curve occurs. Referring to Table 4.2, we
see in this example that there are 2 curves with 232 points having jinvariant
44. This agrees with the first row of Figure 4.1, which shows 2 planes of Tg
meet Â£a in exactly 116 points.
64
Table 4.2: The number of Fgrational points on elliptic curves E = Ef1 for
u F263, their jinvariants and multiplicities in EÂ£.
E2
\E\ jinvariant (number of curves)
232 44(2)
236 153(6)
240 60(2) 93(2) 109(2) 143(2) 159(2) 163(2) 168(2) 189(2) 212(2) 245(2)
244 20(6)
248 23(2) 30(2) 35(2) 156(2) 166(2) 211(2) 218(2) 242(2) 261(2)
252 94(6) 103(6) 120(6) 162(6) 193(6)
256 129(2) 161(2) 175(2) 199(2) 239(2) 253(2)
260 15(6) 71(6) 187(6) 225(6)
264 31(4) 37(4) 85(4) 108(4) 110(4) 150(2) 184(4)
268 15(6) 71(6) 187(6) 225(6)
272 129(2) 161(2) 175(2) 199(2) 239(2) 253(2)
276 94(6) 103(6) 120(6) 162(6) 193(6)
280 23(2) 30(2) 35(2) 156(2) 166(2) 211(2) 218(2) 242(2) 261(2)
284 20(6)
288 60(2) 93(2) 109(2) 143(2) 159(2) 163(2) 168(2) 189(2) 212(2) 245(2)
292 153(6)
296 44(2)
65
Appendix A. Additional Results
The results given in this appendix are not used elsewhere in this thesis. The
logical place for Appendix A.l is after Section 3.1, and we adopt the notation
and perspective taken up through that point.
In Appendix A.2, we give equations for application of the chordtangent
group law on an elliptic curve from the family EÂ£.
A.l q 1 Elliptic Quadrics Intersecting in 2 Points
We take a moment to show that the orbits under Gi of the lines m =
((1.1,1, 0), (1,1, 1, 0)) and m! = ((1.77,1, 0), (1,77, 1, 0)) carry the points of
a number of elliptic quadrics, any two of which meet in the points (1,0,0,0)
and (0,1,0,0). The orbit of the point (1,77,1,0) under Gu together with
{(1,0, 0, 0), (0,1,0, 0)} form the elliptic quadric Â£1 = {x : xEiXT = 0} where
Â£1 =
0 2 0 0
Â§000
0 020
0 0 0 2/7
Choose 77 to be a primitive element of Fq, and for 1 < n < q 1, put
0 f 0 0
f 0 0 0
0 02 0
2/7
0
66
0 0
Then each Â£n = {x : xi?nxT = 0} is an elliptic quadric, and is a family
of q 1 elliptic quadrics pairwise intersecting only in {(1, 0, 0,0), (0,1, 0, 0)}. An
alternative description is
Â£n = {(1, rjn(s2 qt2), s, t) : s,te FJ U {(0,1,0, 0)}. (A.l)
Each Â£ is stabilized by G;, and Â£ = Â£*,, where qk = 2. Let D = diag[ 1, 77, 1,1].
Then (ZD) permutes {Â£} in a cycle. Each Â£n, l
matrices of the form
1 ?] n(a2 rjb2) a b
0 10 0
0 2a 1 0
0 2 r]b 0 1
and we find that the complete group of Â£n is generated by the set of all such
[r"b] and G/. Note that n here is not an exponent, merely a superscript on [rÂ£b].
Thus the group that stabilizes the set {Â£}of q 1 quadrics is generated by G;
and D.
A.2 Addition on the Curves
It is well known that the rational points of an elliptic curve, together with a
point oo form a group with identity oo under a law of composition derived from
the construction of chords and tangents to the curve. See Chapter 2 of [25] for
proof and a discussion. For the sake of completeness, we include the addition
formulae for the elliptic curve E^ whose equation is
W +(18X)x,+ (^_t2
V U)1 J \ LUq ur
X.
67
Let P = (xi,yi) and Q = (Â£2,2/2) points on C. If P ^ Q then P + Q = (x3,2/3)
is given by
Â£3
V2 yi
x2 X\
8(r 1 Xi X2
UJ
y 12/2 ,  x2yi
y3 =Â£3 +.
Â£2 Xi X2 X\
If P = (Â£1,0), then 2P = (00). Otherwise, to find 2P, let
m =
2^(3l? + 2(l8^)a:i+4(
At]2
0Jq
UJZ
Then
Â£3 = m2 + 8~r 1 2x\
u2
2/3 = m(x3 Â£1) + yx.
Each point (x, 0) on the curve is an element of order 2. so the order of the group
of the curve is divisible by 4.
We may use the birational transformations 4.16 and 4.17 to obtain formulae
for addition on the nonsingular points of the curve Gp with equation 4.12. The
resulting equations are quite messy and we will not reproduce them here.
68
Appendix B. Tables of Elliptic Curves
B.l Tables for Elliptic Curves
In Theorem 4.3.7, it was shown that in the analysis of certain plane intersec
tions with the point sets Â£n and !Ka, there arise two families of elliptic curves,
which we call EÂ£ and Eg. In keeping with the remainder of this thesis, q is a
power of an odd prime, and 77 is a fixed nonsquare element in Fg. We restate
equations 4.39 and 4.40. Our families of curves are
Eg = {y2 = x {x a2) (x (a2l)):oeF;\{1, 1}} (B.l)
and
Eg = {y2 = x (x rja2) [x [qa2 l)) : a F*}. (B.2)
In this appendix, we present tables for the families of curves which arise
from truncated quadrics when q < 200 is an odd prime. Recall from Section
4.6.4 that two elliptic curves are isomorphic over the algebraic closure of F^ if
and only if their jinvariants are equal. Each table lists in the first column N,
the number of points on a curve E in that family, and in the second column
are given the jinvariants of the curves that arise with N points, followed in
parenthesis by the number of curves in the family with that jinvariant and N
points. For example, when q = 7, the family of curves Eg contains 2 curves
with 4 points and jinvariant 0, 2 curves with 8 points and jinvariant 6, and 2
curves with 12 points and jinvariant 0. The family Eg contains 4 curves with
jinvariant 6. Recall that Eg contains 2 fewer curves because the line which
carries the corresponding planes is the intersection of two tangent planes.
69
Table B.l: Point counts and jinvariants for curves over F3
Eg
\E\ jinvariant (number of curves)
(0)
En
E jinvariant (number of curves)
4 0(2)
Table B.2: Point counts and jinvariants for curves over F5
En
\E\ jinvariant (number of curves)
4 3(2)
8 3(2)
Eg
\E\ jinvariant (number of curves)
8 3(2)
Table B.3: Point counts and jinvariants for curves over F7
E2
\E\ jinvariant (number of curves)
4 0(2)
8 6(2)
12 0(2)
Eg
\E\ jinvariant (number of curves)
8 6(4)
Table B.4: Point counts and ^invariants for curves over Fn
En
\E\ jinvariant (number of curves)
8 2(2)
12 1(6)
16 2(2)
Eg
\E\ jinvariant (number of curves)
8 2(4)
16 2(4)
70
Table B.5: Point counts and j'invariants for curves over F13
En
\E\ jinvariant (number of curves)
8 12(2)
12 11(4)
16 11(4)
20 12(2)
E?
\E\ jinvariant (number of curves)
8 12(2)
16 0(4) 11(4)
Table B.6: Point counts and jinvariants for curves over F17
En
\E\ jinvariant (number of curves)
12 10(4)
16 9(4)
20 9(4)
24 10(4)
EÂ£
\E\ jinvariant (number of curves)
16 9(4) 11(6)
24 10(4)
Table B.7: Point counts and jinvariants for curves over Fig
En
\E\ jinvariant (number of curves)
12 0(2)
16 5(2) 15(2)
20 18(6)
24 5(2) 15(2)
28 0(2)
E171
\E\ jinvariant (number of curves)
16 5(4) 15(4)
24 5(4) 15(4)
71
Table B.8: Point counts and jinvariants for curves over F23
EÂ£
\E\ jinvariant (number of curves)
16 6(2)
20 5(6)
24 3(2) 19(4)
28 5(6)
32 6(2)
E?
\E\ jinvariant (number of curves)
16 6(4)
24 3(4) 19(8)
32 6(4)
Table B.9: Point counts and jinvariants for curves over F29
E?
\E\ jinvariant (number of curves)
20 17(2)
24 12(4) 23(4)
28 16(4)
32 16(4)
36 12(4) 23(4)
40 17(2)
EÂ£
\E\ jinvariant (number of curves)
24 12(4) 23(4)
32 16(4) 18(12)
40 17(2)
Table B.10: Point counts and (/invariants for curves over F31
E?
\E\ jinvariant (number of curves)
24 11(2) 17(2)
28 0(2) 28(6)
32 2(4) 23(2)
36 0(2) 28(6)
40 11(2) 17(2)
Eg
\E\ jinvariant (number of curves)
24 11(4) 17(4)
32 2(8) 23(4)
40 11(4) 17(4)
72
Table B.ll: Point counts and jinvariants for curves over F37
Eg
\E\ jinvariant (number of curves)
28 17(4)
32 10(4)
36 15(4) 26(2) 30(4)
40 15(4) 26(2) 30(4)
44 10(4)
48 17(4)
\E\ jinvariant (number of curves)
32 10(4) 29(12)
40 15(4) 26(2) 30(4)
48 0(4) 17(4)
Table B.12: Point counts and /invariants for curves over F41
pn ^Â£
\E\ jinvariant (number of curves)
32 4(4)
36 22(4) 29(4)
40 11(4) 39(4)
44 11(4) 39(4)
48 22(4) 29(4)
52 4(4)
Eg
\E\ jinvariant (number of curves)
32 4(4) 6(6)
40 11(4) 39(4)
48 5(12) 22(4) 29(4)
Table B.13: Point counts and jinvariants for curves over F43
En ^Â£
\E\ jinvariant (number of curves)
32 22(2)
36 0(2) 24(6)
40 9(2) 12(2) 29(2) 31(2)
44 8(6)
48 9(2) 12(2) 29(2) 31(2)
52 0(2) 24(6)
56 22(2)
Eg
Â£! jinvariant (number of curves)
32 22(4)
40 9(4) 12(4) 29(4) 31(4)
48 9(4) 12(4) 29(4) 31(4)
56 22(4)
73
Table B.14: Point counts and jinvariants for curves over F47
EJ
\m jinvariant (number of curves)
36 38(6)
40 16(2) 25(2) 26(2)
44 37(6)
48 10(4) 36(2) 44(4)
52 37(6)
56 16(2) 25(2) 26(2)
60 38(6)
^E
\E \ jinvariant (number of curves)
40 16(4) 25(4) 26(4)
48 10(8) 36(4) 44(8)
56 16(4) 25(4) 26(4)
Table B.15: Point counts and jinvariants for curves over F53
E?
Â£ jinvariant (number of curves)
40 32(2)
44 7(4)
48 8(4) 22(4) 42(4)
52 25(4) 45(4)
56 25(4) 45(4)
60 8(4) 22(4) 42(4)
64 7(4)
68 32(2)
^E
\E\ jinvariant (number of curves)
40 32(2)
48 8(4) 22(4) 39(12) 42(4)
56 25(4) 45(4)
64 7(4) 17(12)
Table B.16: Point counts and jinvariants for curves over F59
E?
1 E\ jinvariant (number of curves)
48 20(2) 42(2) 44(2)
52 34(6)
56 7(2) 38(2) 43(2) 53(2)
60 15(12) 17(6)
64 7(2) 38(2) 43(2) 53(2)
68 34(6)
72 20(2) 42(2) 44(2)
Eg
\E\ jinvariant (number of curves)
48 20(4) 42(4) 44(4)
56 7(4) 38(4) 43(4) 53(4)
64 7(4) 38(4) 43(4) 53(4)
72 20(4) 42(4) 44(4)
74
Table B.17: Point counts and ^'invariants for curves over F6i
pH
Â£ jinvariant (number of curves)
48 15(4)
52 20(2) 35(4) 40(4)
56 1(4) 33(4)
60 4(4) 6(4)
64 4(4) 6(4)
68 1(4) 33(4)
72 20(2) 35(4) 40(4)
76 15(4)
Ef
\E\ jinvariant (number of curves)
48 0(4) 15(4)
56 1(4) 33(4)
64 4(4) 6(4) 32(12) 56(12)
72 20(2) 35(4) 40(4)
Table B.18: Point counts and jinvariants for curves over F67
E?
E jinvariant (number of curves)
52 0(2)
56 3(2) 9(2) 22(2)
60 12(6) 23(6)
64 33(2) 35(2) 42(2) 51(2) 57(2)
68 53(6)
72 33(2) 35(2) 42(2) 51(2) 57(2)
76 12(6) 23(6)
80 3(2) 9(2) 22(2)
84 0(2)
\E\ jinvariant (number of curves)
56 3(4) 9(4) 22(4)
64 33(4) 35(4) 42(4) 51(4) 57(4)
72 33(4) 35(4) 42(4) 51(4) 57(4)
80 3(4) 9(4) 22(4)
75
Table B.19: Point counts and jinvariants for curves over F71
\E\ jinvariant (number of curves)
56 33(2)
60 25(6) 42(6)
64 7(2) 11(2) 56(2) 60(2)
68 32(6)
72 17(4) 24(2) 40(4) 48(4)
76 32(6)
80 7(2) 11(2) 56(2) 60(2)
84 25(6) 42(6)
88 33(2)
Eg
\E\ jinvariant (number of curves)
56 33(4)
64 7(4) 11(4) 56(4) 60(4)
72 17(8) 24(4) 40(8) 48(8)
80 7(4) 11(4) 56(4) 60(4)
88 33(4)
Table B.20: Point counts and jinvariants for curves over F73
En ^8
!Ef jinvariant (number of curves)
60 26(4) 55(4)
64 47(4) 50(4)
68 52(4) 72(4)
72 25(4) 41(4) 43(4)
76 25(4) 41(4) 43(4)
80 52(4) 72(4)
84 47(4) 50(4)
88 26(4) 55(4)
Eg
\E\ jinvariant (number of curves)
64 0(4) 47(4) 50(4) 53(12)
72 25(4) 41(4) 43(4)
80 22(12) 49(6) 52(4) 72(4)
88 26(4) 55(4)
76
Table B.21: Point counts and jinvariants for curves over F79
Fn
\e\ jinvariant (number of curves)
64 3(2) 73(2)
68 10(6)
72 22(2) 26(2) 42(2) 74(2) 77(2)
76 0(2) 34(6) 63(6)
80 15(4) 21(4) 69(2)
84 0(2) 34(6) 63(6)
88 22(2) 26(2) 42(2) 74(2) 77(2)
92 10(6)
96 3(2) 73(2)
Eg
\E\ jinvariant (number of curves)
64 3(4) 73(4)
72 22(4) 26(4) 42(4) 74(4) 77(4)
80 15(8) 21(8) 69(4)
88 22(4) 26(4) 42(4) 74(4) 77(4)
96 3(4) 73(4)
Table B.22: Point counts and jinvariants for curves over F83
E?
\E\ jinvariant (number of curves)
68 44(6)
72 2(2) 11(2) 49(2) 53(2) 69(2)
76 66(6)
80 8(2) 14(2) 16(2) 78(2) 80(2)
84 50(12) 68(6)
88 8(2) 14(2) 16(2) 78(2) 80(2)
92 66(6)
96 2(2) 11(2) 49(2) 53(2) 69(2)
100 44(6)
Eg
E jinvariant (number of curves)
72 2(4) 11(4) 49(4) 53(4) 69(4)
80 8(4) 14(4) 16(4) 78(4) 80(4)
88 8(4) 14(4) 16(4) 78(4) 80(4)
96 2(4) 11(4) 49(4) 53(4) 69(4)
77
Table B.23: Point counts and jinvariants for curves over FÂ§9
Eg
\E\ jinvariant (number of curves)
72 79(4)
76 42(4) 62(4)
80 17(4) 46(4)
84 41(4) 54(4) 55(4) 59(4)
88 29(4) 39(4)
92 29(4) 39(4)
96 41(4) 54(4) 55(4) 59(4)
100 17(4) 46(4)
104 42(4) 62(4)
108 79(4)
Eg
E jinvariant (number of curves)
72 79(4)
80 17(4) 26(12) 37(6) 46(4)
88 29(4) 39(4)
96 1(12) 21(12) 41(4) 54(4) 55(4) 59(4)
104 42(4) 62(4)
Table B.24: Point counts and ^'invariants for curves over F97
E?
\E\ jinvariant (number of curves)
80 85(4)
84 76(4) 83(4)
88 44(4) 46(4) 80(4)
92 63(4) 78(4)
96 6(4) 31(4) 36(4) 48(4)
100 6(4) 31(4) 36(4) 48(4)
104 63(4) 78(4)
108 44(4) 46(4) 80(4)
112 76(4) 83(4)
116 85(4)
Eg
\E\ jinvariant (number of curves)
80 79(6) 85(4)
88 44(4) 46(4) 80(4)
96 6(4) 15(12) 31(4) 36(4) 48(4) 61(12)
104 63(4) 78(4)
112 0(4) 68(12) 76(4) 83(4)
78
Table B.25: Point counts and jinvariants for curves over Fi0i
\E\ jinvariant (number of curves)
84 7(4) 79(4)
88 6(4) 54(4)
92 41(4) 81(4) 88(4)
96 27(4) 60(4) 98(4)
100 11(2) 42(4) 69(4)
104 11(2) 42(4) 69(4)
108 27(4) 60(4) 98(4)
112 41(4) 81(4) 88(4)
116 6(4) 54(4)
120 7(4) 79(4)
Eg
\E\ jinvariant (number of curves)
88 6(4) 54(4)
96 27(4) 28(12) 30(12) 60(4) 65(12) 98(4)
104 11(2) 42(4) 69(4)
112 24(12) 41(4) 81(4) 88(4)
120 7(4) 79(4)
Table B.26: Point counts and jinvariants for curves over F103
E2
\E\ jinvariant (number of curves)
84 0(2)
88 40(2) 49(2) 84(2) 99(2)
92 93(6)
96 5(2) 29(2) 32(2) 43(2) 60(2) 70(2)
100 58(6) 89(6) 97(6)
104 23(4) 69(4) 80(2)
108 58(6) 89(6) 97(6)
112 5(2) 29(2) 32(2) 43(2) 60(2) 70(2)
116 93(6)
120 40(2) 49(2) 84(2) 99(2)
124 0(2)
Eg
\E\ jinvariant (number of curves)
88 40(4) 49(4) 84(4) 99(4)
96 5(4) 29(4) 32(4) 43(4) 60(4) 70(4)
104 23(8) 69(8) 80(4)
112 5(4) 29(4) 32(4) 43(4) 60(4) 70(4)
120 40(4) 49(4) 84(4) 99(4)
79
Table B.27: Point counts and jinvariants for curves over F107
Eg
\E\ jinvariant (number of curves)
88 49(2)
92 27(6)
96 19(2) 30(2) 46(2) 57(2) 63(2) 64(2) 77(2)
100 32(6) 103(6)
104 26(2) 39(2) 43(2) 69(2) 97(2)
108 16(6) 72(12)
112 26(2) 39(2) 43(2) 69(2) 97(2)
116 32(6) 103(6)
120 19(2) 30(2) 46(2) 57(2) 63(2) 64(2) 77(2)
124 27(6)
128 49(2)
E?
Â£ jinvariant (number of curves)
88 49(4)
96 19(4) 30(4) 46(4) 57(4) 63(4) 64(4) 77(4)
104 26(4) 39(4) 43(4) 69(4) 97(4)
112 26(4) 39(4) 43(4) 69(4) 97(4)
120 19(4) 30(4) 46(4) 57(4) 63(4) 64(4) 77(4)
128 49(4)
Table B.28: Point counts and jinvariants for curves over F^
Eg
\E\ jinvariant (number of curves)
92 77(4)
96 29(4) 67(4)
100 10(4) 15(4) 58(4) 84(4)
104 19(4) 86(4) 93(2)
108 22(4) 45(4) 65(4) 94(4)
112 22(4) 45(4) 65(4) 94(4)
116 19(4) 86(4) 93(2)
120 10(4) 15(4) 58(4) 84(4)
124 29(4) 67(4)
128 77(4)
E1^
\E\ jinvariant (number of curves)
96 29(4) 67(4) 72(12) 89(12)
104 19(4) 86(4) 93(2)
112 0(4) 6(12) 22(4) 45(4) 65(4) 94(4)
120 10(4) 15(4) 58(4) 84(4)
128 4(12) 77(4)
80
Table B.29:
\)int counts and jinvariants for curves over F113
Eg
\E\ jinvariant (number of curves)
96 1(4) 42(4)
100 5(4) 40(4)
104 20(4) 25(4)
108 11(4) 29(4) 49(4) 64(4) 95(4) 97(4)
112 41(4) 59(4)
116 41(4) 59(4)
120 11(4) 29(4) 49(4) 64(4) 95(4) 97(4)
124 20(4) 25(4)
128 5(4) 40(4)
132 1(4) 42(4)
Eg
\E\ jinvariant (number of curves)
96 1(4) 42(4) 90(12)
104 20(4) 25(4)
112 15(12) 41(4) 59(4) 94(12)
120 11(4) 29(4) 49(4) 64(4) 95(4) 97(4)
128 5(4) 24(12) 33(6) 40(4)
81
Table B.30: Point counts and jinvariants for curves over F127
Â£ jinvariant (number of curves)
108 0(2) 12(6)
112 18(2) 54(2) 71(2) 78(2) 81(2)
116 19(6) 63(6)
120 14(2) 37(2) 67(2) 72(2) 90(2) 98(2) 113(2) 124(2)
124 85(6) 103(6)
128 77(2), 95(4) 126(4)
132 85(6) 103(6)
136 14(2) 37(2) 67(2) 72(2) 90(2) 98(2) 113(2) 124(2)
140 19(6) 63(6)
144 18(2) 54(2) 71(2) 78(2) 81(2)
148 0(2) 12(6)
Eg
\E\ jinvariant (number of curves)
112 18(4) 54(4) 71(4) 78(4) 81(4)
120 14(4) 37(4) 67(4) 72(4) 90(4) 98(4) 113(4) 124(4)
128 77(4) 95(8) 126(8)
136 14(4) 37(4) 67(4) 72(4) 90(4) 98(4) 113(4) 124(4)
144 18(4) 54(4) 71(4) 78(4) 81(4)
82
Table B.31: Point counts and jinvariants for curves over F131
Eg
\E\ jinvariant (number of curves)
112 32(2) 74(2) 109(2)
116 59(6)
120 3(2) 6(2) 8(2) 29(2) 53(2) 66(2) 69(2) 83(2)
124 2(6) 73(6)
128 1(2) 15(2) 34(2) 52(2) 130(2)
132 25(6) 28(12) 50(12)
136 1(2) 15(2) 34(2) 52(2) 130(2)
140 2(6) 73(6)
144 3(2) 6(2) 8(2) 29(2) 53(2) 66(2) 69(2) 83(2)
148 59(6)
152 32(2) 74(2) 109(2)
Eg
Â£ jinvariant (number of curves)
112 32(4) 74(4) 109(4)
120 3(4) 6(4) 8(4) 29(4) 53(4) 66(4) 69(4) 83(4)
128 1(4) 15(4) 34(4) 52(4) 130(4)
136 1(4) 15(4) 34(4) 52(4) 130(4)
144 3(4) 6(4) 8(4) 29(4) 53(4) 66(4) 69(4) 83(4)
152 32(4) 74(4) 109(4)
83
Table B.32: Point counts and jinvariants for curves over F137
E?
\E\ jinvariant (number of curves)
116 70(4)
120 34(4) 68(4) 82(4) 123(4)
124 86(4) 99(4)
128 16(4) 47(4)
132 3(4) 12(4) 67(4) 85(4)
136 19(4) 88(4) 116(4) 118(4)
140 19(4) 88(4) 116(4) 118(4)
144 3(4) 12(4) 67(4) 85(4)
148 16(4) 47(4)
152 86(4) 99(4)
156 34(4) 68(4) 82(4) 123(4)
160 70(4)
Eg
Â£ jinvariant (number of curves)
120 34(4) 68(4) 82(4) 123(4)
128 16(4) 47(4) 50(12) 128(12)
136 19(4) 88(4) 116(4) 118(4)
144 3(4) 12(4) 45(12) 54(12) 67(4) 73(12) 85(4)
152 86(4) 99(4)
160 70(4) 84(6)
84
Table B.33: Point counts and jinvariants for curves over Fi39
Eg
\E\ jinvariant (number of curves)
120 41(2) 90(2) 107(2) 124(2)
124 0(2) 114(6) 123(6)
128 27(2) 34(2) 39(2) 91(2) 106(2)
132 115(6) 137(6)
136 12(2) 37(2) 38(2) 48(2) 52(2) 73(2) 85(2) 102(2)
140 60(6) 65(12)
144 12(2) 37(2) 38(2) 48(2) 52(2) 73(2) 85(2) 102(2)
148 115(6) 137(6)
152 27(2) 34(2) 39(2) 91(2) 106(2)
156 0(2) 114(6) 123(6)
160 41(2) 90(2) 107(2) 124(2)
E1^
\E\ jinvariant (number of curves)
120 41(4) 90(4) 107(4) 124(4)
128 27(4) 34(4) 39(4) 91(4) 106(4)
136 12(4) 37(4) 38(4) 48(4) 52(4) 73(4) 85(4) 102(4)
144 12(4) 37(4) 38(4) 48(4) 52(4) 73(4) 85(4) 102(4)
152 27(4) 34(4) 39(4) 91(4) 106(4)
160 41(4) 90(4) 107(4) 124(4)
85
Table B.34: Point counts and jinvariants for curves over F149
\E\ jinvariant (number of curves)
128 59(4)
132 22(4) 85(4) 111(4) 128(4)
136 89(2) 98(4) 124(4)
140 53(4) 72(4) 138(4)
144 7(4) 35(4) 36(4) 84(4) 114(4) 118(4)
148 42(4) 94(4)
152 42(4) 94(4)
156 7(4) 35(4) 36(4) 84(4) 114(4) 118(4)
160 53(4) 72(4) 138(4)
164 89(2) 98(4) 124(4)
168 22(4) 85(4) 111(4) 128(4)
172 59(4)
E f
\E\ jinvariant (number of curves)
128 52(12) 59(4)
136 89(2) 98(4) 124(4)
144 7(4) 35(4) 36(4) 56(12) 83(12) 84(4) 114(4) 118(4)
152 42(4) 94(4)
160 53(4) 63(12) 64(12) 72(4) 122(12) 138(4)
168 22(4) 85(4) 111(4) 128(4)
86
Table B.35: Point counts and jinvariants for curves over F^i
1^1 jinvariant (number of curves)
128 98(2)
132 34(6) 136(6)
136 6(2) 21(2) 32(2) 71(2) 114(2) 129(2)
140 87(6) 107(6)
144 10(2) 19(2) 37(2) 69(2) 85(2) 88(2) 122(2) 126(2)
148 0(2) 13(6) 104(6)
152 67(2) 101(4) 143(4) 148(4)
156 0(2) 13(6) 104(6)
160 10(2) 19(2) 37(2) 69(2) 85(2) 88(2) 122(2) 126(2)
164 87(6) 107(6)
168 6(2) 21(2) 32(2) 71(2) 114(2) 129(2)
172 34(6) 136(6)
176 98(2)
Eg
\E\ jinvariant (number of curves)
128 98(4)
136 6(4) 21(4) 32(4) 71(4) 114(4) 129(4)
144 10(4) 19(4) 37(4) 69(4) 85(4) 88(4) 122(4) 126(4)
152 67(4) 101(8) 143(8) 148(8)
160 10(4) 19(4) 37(4) 69(4) 85(4) 88(4) 122(4) 126(4)
168 6(4) 21(4) 32(4) 71(4) 114(4) 129(4)
176 98(4)
87
Table B.36: Point counts and j'invariants for curves over F
E?
\E\ jinvariant (number of curves)
136 1(2) 48(4) 50(4)
140 2(4) 49(4) 128(4)
144 37(4) 72(4) 100(4) 149(4)
148 10(4) 31(4) 119(4) 148(4)
152 41(4) 90(4)
156 5(4) 47(4) 123(4) 142(4)
160 5(4) 47(4) 123(4) 142(4)
164 41(4) 90(4)
168 10(4) 31(4) 119(4) 148(4)
172 37(4) 72(4) 100(4) 149(4)
176 2(4) 49(4) 128(4)
180 1(2) 48(4) 50(4)
E1^
E jinvariant (number of curves)
136 1(2) 48(4) 50(4)
144 0(4) 37(4) 72(4) 76(12) 100(4) 149(4)
152 41(4) 90(4)
160 5(4) 36(12) 47(4) 68(12) 69(12) 107(12) 123(4) 142(4)
168 10(4) 31(4) 119(4) 148(4)
176 2(4) 49(4) 116(12) 128(4)

