Computing mo dular curves via quaternions

David R Kohel

National University of Singap ore

January

Based on a talk given at the Fourth CANT Conference

Number Theory and Cryptography

University of Sydney December

x1 Intro duction

Mo dular curves are of central interest for b oth the theoretical and compu

tational investigation of elliptic curves In the course of proving Fermats Last

Theorem Wiles and TaylorWiles established that a large class of

elliptic curves are parameterized by a mo dular curve Cremona has develop ed

eective algorithms and p erformed extensive computations of these parametriza

tions In a dierent direction Elkies has used explicit mo dels for mo dular

curves to make signicant practical improvements to a theoretical p olynomial

time algorithm of Scho of for computing the trace of Frob enius on an elliptic

curve over a nite eld This has made it p ossible to compute the numb er of

p oints on elliptic curves over nite elds whose cardinality measures hundreds

of decimal digits In order to apply this algorithm one must precompute a

large numb er of explicit mo dels for mo dular curves

One approach to the problem of computing mo dels for mo dular curves is to

pro duce a basis for the space of weight two cusp forms Such forms corresp ond

to dierentials on the curve by which one can construct the canonical mor

phism to pro jective space When a curve is of genus greater than two and not

hyp erelliptic the canonical morphism is an emb edding and gives a nonsingular

mo del for the curve

For the purp oses of computation it serves to have the additional informa

tion of the Hecke mo dule structure on dierentials This gives information

on the decomp osition of the Jacobian and on curves covered by the mo dular

curve In particular we exploit the explicit action of the canonical involution

to decomp ose the cusp forms into invariant and antiinvariant eigenspaces The

investigation of parametrizations of curves of higher genus is aided by a Hecke

mo dule decomp osition of the space of dierentials

It will b e the purp ose of this article to discuss certain isomorphisms of Hecke

mo dules dened in very dierent contexts and to describ e isomorphisms among

them The relation b etween sup ersingular elliptic curves and the ideal theory

in a quaternion algebra app ears in the classical work of Deuring which in

the mo dern theory is prop erly stated as an equivalence of categories The basis

problem of Eichler provides the means of relating the ideal theory to mo d

ular forms Using this theory Pizer describ es an algorithm for computing

mo dular forms The metho d of graphs of Oesterle and Mestre rephrases the

theory of quaternion ideals in terms of sup ersingular elliptic curves This gives

an intuitive metho d for relating the Hecke mo dule dened as a subgroup of the

divisor group of a mo dular curve with the space of mo dular forms of weight two

In this work they express the Hecke op erator T n in terms of the adjacency

op erator of a graph of sup ersingular elliptic curves Via the ab ove mentioned

equivalence of categories the ideas of Oesterle and Mestre translate into the

computationally simpler world of the ideal theory of a quaternion algebra Us

ing a metho d which is in essence that of Pizer one can compute an array

of quadratic forms determining the Brandt morphism For any n the Hecke

op erator T n can b e extracted as the Brandt matrix of nth representation

numb ers of these quadratic forms

In section two we discuss quaternion algebras and their ideal theory and

follow in section three with a discussion of the equivalence b etween sup ersingular

elliptic curves and certain ideals over a maximal order In section four we recall

the main ideas of the metho d of graphs of Mestre and Oesterle Section ve

intro duces the Brandt morphism given in terms of the Brandt matrix of theta

functions for quadratic forms asso ciated to the mo dule of homomorphisms of a

basis of ideals We conclude with a discussion in section six of the computational

asp ects of computing mo dular curves using the ideal theory of quaternions

As an app endix to this article we give a table of characteristic p olynomi

als of the Hecke op erators which suce to determine the decomp osition of the

Jacobian of corresp onding mo dular curve We further give examples of com

putations of the ring of mo dular functions combining several ideas from the

article of Elkies For any given level one can make improvements to this

approach A signicant advantage however is that this approach is systematic

thus suitable for implementation or for proving b ounds for the computational

complexity

x2 Quaternion algebras over Q

A quaternion algebra A over Q is a central simple algebra of dimension four

over Q The numb er theory of these algebras is analogous to that of numb er

elds In particular we have an noncommutative theory for each of the following

ob jects and concepts from commutative numb er theory

Maximal orders There exist innitely many maximal orders of any quater

nion algebra however they fall in nitely many isomorphism classes

Ideal theory We can study the onesided and twosided ideals of a given

maximal order in a quaternion algebra Again these fall into nitely many

classes

Ramication and splitting The quaternion algebra A is said to split at

the rational prime l if A A Q is isomorphic to M Q Otherwise

l l l

A is said to ramify at l and A is a division algebra Likewise A is said to

l

split or ramify at innity if A R is a matrix algebra or a division algebra

Quaternion algebras are analogous to quadratic extensions of Q In fact

the analogy go es further every element x of A not in the center generates a

quadratic extension of Q

Example The matrix algebra M Q is a quaternion algebra which we call

the split quaternion algebra over Q Let x b e the element

and set K Q x Then K is isomorphic to the ring Q X X X Every

maximal order is conjugate to the order M Z

Example Let A Q Q i Q j Q k b e the quaternion algebra dened by

the relations

i j k ij j i

Then A ramies at and at innity and O Z Zi Zj Z where

i j k is the unique maximal order up to isomorphism

x3 Sup ersingular elliptic curves

Let k b e an algebraically closed eld of characteristic l An elliptic curve E

is sup ersingular if and only if its endomorphism ring O EndE is an order

in a quaternion algebra Moreover A O Q is ramied at l and at innity

and O is a maximal order in A

Let E b e a xed sup ersingular elliptic curves over k Then the map F

HomE F determines a bijection of the set of isomorphism classes of sup ersin

gular elliptic curves with the isomorphism classes of lo cally free rank one right

O mo dules This is prop erly stated as an equivalence of categories as follows

Theorem Let k be an algebraic closure of a nite eld let S be the category

of supersingular el liptic curves over k and let E be an object in S Then the

functor Hom E to the category of local ly free rank one right modules over

S

O EndE is an equivalence of categories

Consequences We note a few consequences of the theorem

Under the equivalence isogenies of elliptic curves corresp ond to nonzero

O mo dule homomorphisms Isomorphism of ob jects is functorial thus

the nite set of isomorphism classes in each category are in bijective cor

resp ondence

Given any right O mo dule of the form HomE F we can cho ose any

element Then the dual determines an emb edding

b HomE F O EndE

as an ideal of O By the equivalence of categories every lo cally free rank

one right mo dule over O is isomorphic to one of the form HomE F and

all of its emb eddings in O are determined in this way

The degree of a morphism I J of right O mo dules is dened which

we refer to as the norm N in the category I The norm may b e dened

lo cally or as the squarero ot of jJ I j

For nite extensions k F the functor F HomE F where

l

is the Frob enius morphism gives an equivalence of sup ersingular elliptic

curves over k with an appropriately dened category of pairs

One can dene the j invariant of an ideal I To make the latter well

dened we must sp ecify an orientation O k as describ ed in Rib et

An orientation is a homorphism to k with the kernel equal to the unique

prime ideal containing p The image is a quadratic extension over the

prime eld in which the j invariant of I lies

In its full generality we take a category of sup ersingular elliptic curves

with level N structure and ideals of an Eichler order of index N in the

maximal order

In terms of computations the two categories are quite dierent The j

invariant of an elliptic curve is trivial to compute while the endomorphism ring

and isogenies are generally dicult In contrast determining homomorphisms

and the endomorphism ring is easy for ideals and determining the j invariant

of an O ideal is presumably of comparable diculty to that of determining the

j invariant mo dulo l of a binary quadratic lattice

x4 Metho d of graphs

Following Mestre we asso ciate a graph to a set S of representatives of the

isomorphism classes of S Fix an integer n Let S b e the set of vertices and let

the edges E b e the isogenies E F of degree n with cyclic kernel up to

isomophism of F Dene

i t

-

E S S

-

i t E F

This denes a directed multigraph For an edge E F the curve E i

is called the initial vertex and F t is called the terminal vertex

For a prime n p the numb er of edges with initial vertex E are p in

numb er in bijection with the p cyclic subgroups of E p ZpZ ZpZ

Due to automorphisms of E fewer edges may terminate at E

Example Let l There are three sup ersingular elliptic curves over the

algebraic closure of F Since none of these curves has automorphisms group

larger than fg we can view the graph as undirected For p we have the

graph

r r

r

Up to isomorphism there is exactly one sup ersingular elliptic curve dened over

the prime eld and two curves one conjugate to the other dened over a

quadratic extension The necessary regularity of the graph and the auto

morphism induced by the Frob enius morphism completely determine the ab ove

graph of isogenies The adjacency matrix of the graph is the matrix

A

T

with characteristic p olynomial X X X We will see that T can b e

interpretted as a Hecke op erator on the space M Q of mo dular forms

of weight two for The rational ro ots of the characteristic p olynomial for

T imply that the Jacobian of the mo dular curve X splits as a pro duct

of elliptic curves over Q

We construct a Hecke mo dule asso ciated to a graph of nisogenies as follows

Let M b e the free ab elian group with basis S Dene T n M M to b e

the adjacency op erator on M

X

T nE t

1

i E

We dene T n to b e the nth Hecke operator on M Dene an inner pro duct

on M by

j AutE j if E F

hE F i

otherwise

extending by linearity Then T n is selfadjoint with resp ect to the inner pro d

uct

hE T nF i hT nE F i

and this numb er equals the count of cyclic isogenies of degree p from E to F

Dene an op erator An by

X

T m An

2

r mn

the adjacency op erator of the graph of all nisogenies

In terms of the basis S fE g for M the op erators T n and An have

i

matrix representations where j Aut E jAni j is the numb er of isogenies of

j

degree n from E to E

i j

For all relatively prime integers n and m we obtain T nT m T nm

and T nD is a symmetric matrix where D is the diagonal matrix with entries

D i i j AutE j The op erators An satisfy the relations

i

Anp ApAnp pAn

The Hecke algebra T is dened to b e the algebra over Q generated by the

op erators T n

x5 Brandt morphism

Let M M Q We dene the Brandt morphism

Q

M M M l Q

Q Q

P P

deg n

by E F q hAnE F iq where the rst sum is over all ele

ments of HomE F then extending linearly to M That the images lies

Q

in M l Q is a wellknown result for theta functions Eichler proved

as part of his work on the basis problem that M and M l Q are

Q

isomorphic as Hecke mo dules We state this result in the form of the following

theorem

Theorem The map T n T n of Hecke operators denes an isomor

phism of Hecke algebras on M and M l Q such that the Brandt morphism

Q

is a nondegenerate Hecke bilinear map

T nE F E T nF T nE F

and such that the traces of T n on M and T n on M l Q agree

Q

We compute the Brandt morphism as follows The degree map

deg Hom F F Z

S

is a quadratic map deg n n deg and by means of a choice of ba

sis gives a quadratic form By the equivalence of categories of Theorem

we can identify the mo dule Hom F F with a mo dule Hom I I of O

S O

homomorphisms of ideals For a choice of basis we call the asso ciated quadratic

form the norm form

Example Let O Z Zi Zj Z dened ab ove O is a right principal

ideal ring thus any right ideal I is isomorphic to O itself and the ring of O

endomorphisms of I is isomorphic to O acting by left multiplication In the

ab ove basis the norm form is given by

N x x i x j x f x x x x

x x x x x x x x

We represent a quadratic form f by its Gram matrix M In this case we write

f as the pro duct

t t

f x x x x X XMX X

P

N

q is equal to the where X x x x x Therefore the series

O

theta series

X X

f x x x x n

1 2 3 4

q a q

n

n x x

1 4

where a is the nth representation numb er of f For the ab ove example we

n

obtain the Eisenstein series

q q q q q q q q q q q

which generates the mo dule M Q

In the previous example the Hecke op erator T acted on M Z

with characteristic p olynomial X X X The eigenspace of is that

generated by the Eisenstein series and the eigenspaces of the eigenvalues

and are rational cusp forms each dening an isogeny class of mo dular elliptic

curves over Q

x6 Computational asp ects of mo dular curves

The mo dular curve X l has a singular mo del j j based on the map

l l

X l X X taking a mo duli p oint for the isogeny E F to the

pair j E j F There exists a Fricke or canonical involution w X l

l

X l which takes to its dual b determining the involution j j j j

l l

on the singular mo del

It is often convenient to compute rst X l X l w a curve of genus

l

at most one half that of X l The mo dular functions on X l are just those

functions on X l invariant under w Moreover by means of a decomp osition

l

of mo dular functions into invariant and antiinvariant spaces under w the order

l

of the p oles at each cusp coincide so relations b etween functions can b e reduced

to linear algebra on the Fourier expansions around the single cusp at

l has degree and is ramied precisely at The morphism X l X

the p oints of complex multiplication by an order of discriminant l l or

l Thus there are precisely R hl hl hl ramication p oints

where hD is the class numb er of an order of discriminant D when such an

order exists or zero otherwise By the RiemannHurwitz formula we obtain

g l R

g l

where g l is the genus of X l and g l is the genus of X l Moreover

R is the numb er of sup ersingular elliptic curves which can b e dened over F

l

and dim M l Q g l is the total numb er

For the nitely many curves for which g l equals we can compute a

l then obtain the function eld of X l as a quadratic Hauptmo dul for X

extension by a function antiinvariant under w We will thus fo cus on metho ds

l

applicable as g l and g l grow large

Let ff f g b e a basis for the space S l Q of cusp forms Then

g

we dene the canonical morphism to pro jective space by

g

-

X l P

-

Q f Q f Q

g

When X l is nonhyp erelliptic of genus greater than two the canonical mor

phism is an emb edding In practice we will take a sp ecial subset of a basis for

S l Q consisting of forms with prescrib ed zeros at We treat some

sp ecic examples in App endix I I

In order to eciently compute S l Q and its subspaces of invariant and

antiinvariant forms we will make a detailed study of the Brandt morphism and

the decomp osition of M We b egin with the following corollaries of Theorem

Q

Corollary Let U be a Hecke submodule of M Then the orthogonal decom

Q

position M U V is a decomposition of Hecke modules Moreover v lies in

Q

V if and only if u v for al l u in U

Pro of Let u U and v V and let T lie in T Since T u U we have

hu T v i hT u v i so T v lies in V Since the T n span T as a Q vector space

the latter statement is clear from examination of the co ecients of u v

P

n

hT nu v iq 

n

Corollary v M M l Q is an isomorphism of Hecke mod

Q

ules if and only if v is not contained in any proper Tsubmodule of M

Q

Pro of By Theorem the map v is a homomorphism of Hecke mo dules

and by the last statement of Corollary it follows that v is injective if

and only if v lies in no prop er Hecke submo dule It remains only to show that

v is surjective when Tv M By the trace condition of Theorem

Q

dim M TrT TrT dim M l Q

Q

so is surjective Let u and w lie in M and write w T v Then u w

Q

u T v T u v so the image of v is all of M l Q 

M has a decomp osition as E S where E is the Eisenstein space gen

Q Q Q Q

erated by

X

E j Aut E j E

E S

P P

and the cusp space S is the orthogonal complement f a E a g

Q E E

From Pizer we know that the canonical involution w acts as T l on

l

l M thus we also have a decomp osition M M M where dim M g

Q

Q Q Q

g l g and dim M l The canonical involution acts by sending E to

Q

E where is the Frob enius automorphism and E is the representative in

S of the curve conjugate to E Thus the spaces M and M are spanned by

Q Q

fE E E S g and fE E E S g resp ectively

By taking the intersection with the previous decomp osition we obtain an

orthogonal decomp osition M E S S where S M Note that

Q Q

Q Q Q Q

S is the kernel of T l and S is the kernel of T l on S Moreover

Q

Q Q

T p has eigenvalue p on E for all primes p l and eigenvalue for p l

Q

Since each such space is dened as the kernel of certain Hecke op erators so is

the image We thus have the following corollary

Corollary Let v M Then E v S v and S v are con

Q Q

Q Q

tained in the space of Eisenstein series invariant cusp forms and antiinvariant

cusp forms respectively

It is clear from Corollary that for general v equality will hold with the

resp ective image space We can thus exploit the structure of M to decomp ose

Q

the Hecke mo dule b efore mapping to the resp ective submo dules of M l Q

Remark A sup ersingular elliptic curve E is S lies in M whenever E can b e

Q

dened over the prime eld This is the basis of the observation of Pizer

Remark that in the matrix of with resp ect to the basis S not every row

or column can span M l Q

Using the equivalence of categories of Theorem we nd a basis for M in

terms of right ideal classes for a xed maximal order O in the quaternion algebra

ramied at l and For each pair I J we determine the reduced quadratic

norm form of the mo dule Hom I J JI For the resulting p ositive denite

O

quaternary quadratic forms over Z there exits a unique reduced form Beginning

with the ideal O we construct a basis for M by cho osing neighb oring ideals in

the graph of homormorphisms of small degree in analogy with elliptic curves

For two ideals ideals I and J we test for identity via the reduced Gram matrix

of Hom I J

O

By means of an implementation of the arithmetic of quaternion algebras

using the computer algebra package Magma V we have computed the

Hecke mo dule of mo dular forms of weight two and small level For instance

for prime level the array of reduced Gram matrices of the ideal forms is the

following

From the representation numb ers of the ab ove quadratic forms we nd the rst

few Hecke op erators act on this basis by the matrices

T T T

which have resp ective characteristic p olynomials

g t t t t t t t

g t t tt t t t

g t t t t t t t

In each case the eigenvalue p of T p corresp onds to the one dimensional

space of Eisenstein series and the second linear factor to a one dimensional factor

of J By calculating suciently many co ecients of the corresp onding

normalized eigenform

f q q q q q q q q q q q q

q q q q q q q q q

we can verify that this corresp onds to the single isogeny class of conductor

of Cremonas tables

App endix I

We collect in the following tables the characteristic p olynomials T p

and T p of the Hecke op erator T p on S and S for p and and

Q Q

all primes l up to p X X T X X X X X X X X X X X X X X X ) l ( X X X X X X on X X X X X X X X X X X X X X X X X X X X p X X X X X X X X X T X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X op erators X X X X X X X X X X X e X X X X k X X X X X Hec l of g l g p l p p olynomials T p X X X X T X X X X X X X X X X X X X X X l g Characteristic l g p l

X X X X X X X X X X X X X X X X p X X X X X X X X X X X X T X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X p X X X X X X X X X X T X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X l g l g p l

App endix II

Computations of mo dular curves

The mo dular curve X l is dened in terms of generators and relations

for the eld of meromorphic mo dular functions of weight zero for l We

dene Y l to b e the ane op en subset H l of X l The space of weight

zero mo dular functions holomorphic on H can b e identied with the ring R of

functions on Y l In order to have a working mo del for computing with the

l

curve we require expressions for j q and j q in R Moreover if we are to

apply the ideas of Elkies for eciently computing isogenies of curves we

require a description of the ve Eisenstein series

l

l l

q E q E q E q and E q E

L

in the ring M M l Q of mo dular forms for l

k

We thus collect here the denitions of standard mo dular functions as well

as some calculations of the graded rings of mo dular forms for l For the

present purp oses we will restrict to prime level l

Standard mo dular forms

The function E q denotes the normalized Eisenstein series with Fourier series

k

expansion

k n

X X

n q k k

n

nq E q

k k

n

B q B

k k

n n

where n denotes the sum of the r th p ower of the divisors of n and B is

r k

the k th Bernoulli numb er We write

l

k l

E q l E q E q

k k

k

k l

E q l E q E q

k k

k

l

k l

Note that w E q l E q so that E q is antiinvariant and E q

l k k

k k

l

is invariant under w While neither E q nor E q is a mo dular form the

l

l

function E q is generating the Eisenstein space of weight two for l

The delta function q is dened in terms of Eisenstein series as E q

E q The Dedekind eta function is a th ro ot of q having q

expansion

Y

n

q q q

n

It is wellknown that q hence also q has no zeros on the upp er half

plane H Thus we will b e able to invert q to pro duce new functions without

intro ducing p oles on H

Genus zero curves

For l and the mo dular curve X l has genus zero hence we can

nd a degree one function on X l For this purp ose we take the Hauptmodul

l

q

h

l

q

dened as a degree one function having neither zeros nor p oles on H The func

tion h is holomorphic outside of the cusps and transforms under the involution

l n

w as w h l h We set u equal to hl for appropriate n and nd

l

the following expressions for the j invariant in terms of u

q u

l u w u u j

l

q u

q u u

l u w u u j

l

q u

q u u

l u w u u j

l

q u

q

w u u l u

l

q

u u u u

j

u

q

l u w u u

l

q

u u u u u u

j

u

Similar functions can b e found for curves X l where l is one of the ten primes

and for which X l is of genus zero

Graded rings of mo dular forms

In order to pro duce the graded ring of mo dular forms we must nd mo du

lar forms of low weight such that for all k monomials in the forms span the

spaces M l Q over Q The ring of mo dular forms of level is M

k

Q E q E q The dimension of the space M l Q is

k

k k k

From Theorem in Shimura we nd that for l and all even k

greater than that

k k

dim M l k g l k

k

k k

l k R dim M l k g

k

where g l g l R with R hl hl and where

and

l l

Recall that hD denotes the class numb er of a binary quadratic order of dis

criminant D where such exists and is zero otherwise Let K Q X l b e the

function eld of X l and note that KM K for any in M l Q

Level

The graded ring of mo dular functions for is Q X Y with

u

X X E and Y E

u

where u is the Hauptmo dul dened ab ove for By means of an analysis

of the ramication of X X we nd the dimensions of the spaces of

mo dular forms of even weight are given by

dim M Q k

k

dim M Q k k k

k

and verify that this agrees with the dimensions of the graded comp onents of

Q X Y We express the standard Eisenstein series in terms of X and Y as

follows

E q Y E q X

E q X E q X Y

Level

The graded ring of mo dular functions for is Q X Y Z I where

X E q Y E q and Z

and where I is the ideal X XZ Y The forms Y and Z can b e

expressed in the ring Q uX as

u u

Y X X and Z

u u

where u is the Hauptmo dul dened ab ove for As ab ove we deduce from

the ramication of X X and Theorem of Shimura that

the dimensions of the spaces of mo dular forms of even weight are as follows

dim M Q k

k

dim M Q k k k

k

and verify that this coincides with the dimensions of the graded comp onents of

Q X Y Z I We express the Eisenstein series for in terms of X Y and

Z by the following relations

E q X E q Y E q X

E q X Z E q X Y

Mapping forms to functions

Determining the full ring of mo dular forms of a given level is an impractical

undertaking in general as the size of a basis for M l Q grows in prop ortion

k

to b oth k and l Rather following Elkies we nd a mo dular form of

weight with no p oles on H then map the spaces M l to meromorphic

m

m

functions on X l all of whose p oles lie at the cusps by f f

For the purp ose of constructing such a we intro duce certain generalized

theta function of weight one Let a b e an ideal in the quadratic imaginary order

R of discriminant D Then

X

Nx

q q

a

xHomRa

where Nx is the cardinality of the ideal quotient axR is a mo dular form

of weight one and level jD j The pro duct of any two such forms is a mo dular

form of weight two and level jD j which is antiinvariant under the canonical

involution

For l mo d we can set D l to obtain mo dular forms of level l As

in Elkies manuscript we can mo dify the construction for forms asso ciated

to discriminant D l to obtain forms with characters and level l when

l mo d

Since ramied in R the ring R R is isomorphic to F where

aa and let R R f g b e the Fix an isomorphism R R

unique character on R R Then

X

Nx

xq q

a

xHomRa

where is dened on a via the reduction to aa and isomorphism with R R

We may take to b e the following function according to the congruence

class of l mo dulo

l mo d Set where is a linear combination of functions

l a

having maximal zero at the cusps and with each a in the correct genus such

that and have the same character There exists such an ideal with

a l

for all primes l mo d except for the three primes and for

a

p

l has class numb er two which Z

l mo d Set for the linear combination of functions

l a

having the highest order zero at the cusps

l mo d Set where the unique antiinvariant cusp

l

form with a zero of maximal order at the cusps

In each case is a mo dular function of weight holomorphic on H which is

antiinvariant under the canonical involution

Level

Using a decomp osition of S Q into invariant and antiinvariant

forms we set ff f g equal to the echelon basis

f q q q q q q q q q

f q q q q q q q q q

for the space S Q of invariant cusp forms and let fg g g g b e the

basis

g q q q q q q q q q q

g q q q q q q q q q

g q q q q q q q q q

for the antiinvariant forms such that g g f f Note that b oth f and g

are uniquely determined as the invariant and antiinvariant forms having zeros

of maximum order at the cusps By Chap er IV x of Hartshorne the forms

f and f dene the degree canonical morphism of X to a genus zero

curve C X where is the hyp erelliptic involution of X Set t

e e

and u equal to the invariant functions

t f f q q q q q q q

u g g q q q q q q

satisfying the equation

u t t u tt t

Then t is a generator for the function eld of C having p oles only at the

cusp of X and its image under Since the twist of u by the hyp erelliptic

e

involution

u t t u q q q q q q

e

has value at the cusp we conclude that u is holomorphic on H We eliminate

the p ole of t to nd a degree function v u t on X

Let R b e the imaginary quadratic order of discriminant and set a R

and b equal to a prime ideal of norm We obtain two mo died theta functions

q q q q q q q q q

a

q q q q q q q q q q

b

set and nd a degree function

b

uu uu u

w g tu v

t v

holomorphic on H and conclude that u v and w generate the ring of functions

on X with p oles only at the cusp

To complete the calculation of mo dular functions for X we set x equal

to the degree function f g and y equal to the degree function

f f v w x

y

u u

We verify that x and y are holomorphic on H by expressing their squares in

terms of u v and w

x u v u

y u v u uw uv u w v u

By the RiemannRo ch theorem we see that these functions generate the ring

of meromorphic functions on X with p oles only at the cusps Each of the

functions E E E E and E can thus b e expressed in

terms of u v w x and y

Level

By computing the Hecke mo dule of theta function we nd the echelon basis

ff f f g

f q q q q q q q q q

f q q q q q q q q q

f q q q q q q q q q

for the space S of cusp forms invariant under the canonical involution

Likewise we compute an echelon basis for the space of antiinvariant cusp forms

in S and take the four forms

g q q q q q q q

g q q q q q q q

g q q q q q q q

g q q q q q q q

of highest zeros at the cusps Necessarily g is the form with no zeros

on the upp er half plane H

We set X f Y f and Z f and nd a nonsingular quartic relation

X X Y Z X Z XZ Y Z Y Y Y Z Z

dening the canonical emb edding of X as a plane curve We set x y

and z to b e the mo dular functions f g f g and f g resp ectively

Next we set u v and w equal to the functions on X

u g g q q q q q q q q

v g g q q q q q q q q

w g g q q q q q q q q

v u v v uv u u uv u v u

of degree and resp ectively By eliminating co ecients we nd a function

r uw v uv u of degree and the ring of mo dular functions for X

is generated by u v and w satisfying the relations

w u v w v uw uv w v u

w uv u v u v u v v uv u u

By the relation z uy v x y x we nd that u v w x and y generate

the ring of mo dular functions on X with p oles only at the cusps We

verify that the gap sequence consists exactly of the g l gaps accounted for by

the RiemannRo ch theorem so that these functions indeed generate the full ring

of mo dular functions holomorphic on H

To complete a working mo del for X l we solve for x xy and y in terms

of u v and w and nd expressions for the ve functions

E q E q E q E q and E q

with g in Q u v w x y

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