Analytical construction of Weil curves over function fields

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

E

RNST

-U

LRICH

G

EKELER

Analytical construction of Weil curves over function fields

Journal de Théorie des Nombres de Bordeaux, tome

7, n

o

_{1 (1995),}

p. 27-49

<http://www.numdam.org/item?id=JTNB_1995__7_1_27_0>

© Université Bordeaux 1, 1995, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Analytical

Construction of Weil

Curves Over Function Fields

par Ernst-Ulrich GEKELER (*)

Let

_{f :}

H -7 C be _a

cuspidal

modular form of

weight

two for the Hecke

congruence

subgroup

ro(N)

of the modular group

SL (2, Z),

and suppose it is a new

eigenform

with rational

eigenvalues

for the Hecke

algebra.

With

f

one can associate an

elliptic

curve E

_{= E f}

defined

over Q

with conductor

cond(E) =

N and a

Q-morphism

_p from the modular curve

Xo (N)

-H U (cusps)

to E such that

_{f (z)dz}

=

p* (w),

where ú) is an invariant

differential on ~E. Such a curve E is called a Weil _curve, and a

_strong

Weil curve if _p is maximal.

In different levels of

_clarity

and

_{concreteness,}

_Y.Taniyama,

G. Shimura and A. Weil

_conjectured

that each

_elliptic

curve

E/Q

_appears as a Weil curve in an

_essentially

_unique

_way. Modulo known

_facts,

the truth of the

conjecture

yields

canonical

_bijections

between the sets of

a)

normalized _cusp forms

_f

as

above;

b)

one-dimensional

_isogeny

factors of new

_part

of the

Jacobian of

_{Xc (N);}

and

c)

isogeny

classes of

_elliptic

curves

E/Q

with

cond(E)

= N.

Andrew Wiles

_perhaps

(**) has _proven the above

_conjecture

in the case

where N is

_squarefree,

which would have remarkable arithmetic

conse-quences.

In the

_present

_article,

I want to describe a similar

relationship

between

elliptic

curves and

_{modular/automorphic}

_forms,

but where the base field

_Q

is

_{replaced by}

a rational function field K =

Fq(T)

over a finite field

_Fq.

In

contrast with the number theoretical

_situation,

the assertion

_{(see (1.13))}

corresponding

to

_{Shimura-Taniyama-Weil’s}

_conjecture

STW is _proven,

in-cluding bijections

between the sets that over K substitute the sets

_a),

_b),

Manuscrit _reçu le 23 Novembre 1994.

(*) _{Supported by DFG, Schwerpunkt "Algorithmische Zahlentheorie und Algebra"}

(**)After some turns, Wiles’ _proof seems now to work _(time of _proofreading Nov. _94).

c)

above. Here

_c)

deals with

elliptic

curves

E/K

with

_{split multiplicative}

reduction at

_infinity.

The assertion comes out

_{by combining deep}

results of

_{Grothendieck, Jacquet-Langlands,}

and

_Drinfeld,

and is treated in detail

in

_[10].

_Actually,

the function field

_analogue

of STW is valid for

_arbitrary

global

function fields. But in order to avoid

_{complications}

_coming

from class

_numbers,

and since some more concrete results on the

relationship

between _cusp forms and

_elliptic

curves are established

_only

over rational

function

_fields,

I will restrict to this case.

We will have need for some

_ingredients

that

_replace

_e.g. the

_complex

upper

half-plane

~,

the modular curves

Xo (N),

and the modular forms for

subgroups

of

_SL(2,

_Z).

In

_fact,

there are different

_{substitutes, leading}

to

-

a

_{rigid analytic}

_theory

(Drinfeld

modular curves and

_forms,

values

in characteristic _p =

char(Fq),

_see

[8]),

and

-

an

_{automorphic theory}

(as

in

_{Jacquet-Langlands}

[15],

values in

characteristic

_zero),

the two

_being

related

_through

congruences.

I will

_give

a brief sketch of these theories in section one. In section

_two,

I

will discuss theta functions for

_subgroups

r of

_{GL(2, A),}

where A =

Fq [T]

is

the

_ring

of

_integers

in K.

_{They provide}

the link between the characteristic _p and characteristic zero

theories,

which

technically

is

_given

by

diagram

(2.7).

Now we can

analytically

construct the Jacobian

Jr

of a Drinfeld modular

curve as a torus divided

_by

the

_period

lattice of theta functions. There results a

_description

of the _group

o(r) =

~(Jr)

of connected

_components

of the Néron model of

Jr

through automorphic

data.

In section

_3,

the

_strong

Weil curve

_E~

associated with an

_automorphic

Hecke

_eigenform

_{cp is} described. It is a Tate curve at the

_place

o0 of

_K,

whose

_period

can be calculated

by

means of theta functions. In

_particular,

its Néron

_type

at

_infinity

is determined.

Except

for Theorems 3.2 and 3.17 and their

_corollaries,

this is carried out in detail in the

_joint

_paper

_[10]

with M.

_Reversat,

to which I refer for

_proofs.

So

_{far, everything}

can be

generalized

to

_arbitrary

function fields

_(of

transcendence

_degree

one over

Fq).

In the case of a rational function

field,

however,

one can do much bet-ter.

_Among

_others,

we calculate the

degree

_n~ of the Weil uniformization

p~ onto

E~,

and show that its

prime

divisors are

precisely

the _congruence

primes

of _cp.

There are two reasons

why

the

proofs

of these last-mentioned results

presently

fail to work for

_general

function fields:

_First,

it is not known whether the _map

_{j:Ï’ ~}

of

_(1.9)

is

_{always bijective}

_(see

also the discussion in

_[10]).

_Second,

in the case of a class number &#x3E;

_1,

the

argument

(3.17)

showing

the

_{perfect duality}

_(after

_tensoring

with

_Z[p-1])

of the Hecke

algebra

5l with

H,

(T, Z)

must

_apparently

be

_{replaced by}

some

adelic

_argument,

which at the least will

_require

additional efforts.

In the fourth and last

_section,

some

examples

are

_given

that illustrate

how

"automorphic"

(and

easily

calculable)

properties

of _cp

_correspond

to

properties

of the Weil curve

_E~.

I am convinced that

evaluating

that

relationship

will

produce

_important

progress in

questions

like the

Birch-Swinnerton-Dyer

conjecture

(or

rather

the Artin-Tate

_conjecture)

for

_elliptic

curves over function fields. While

completing

this _paper, I

_enjoyed

the

_hospitality

of the Institute for Ad-vanced

_Study

in

_Princeton,

to whose staff and members I would like to

express my sincere

gratitude.

Notation

1FQ

= finite field of characteristic

p with q elements

A =

Fq [T],

T indeterminate

K =

Fq (T),

with

completion

_at

infinity

-

~q~~~))~ ~ =

T’ 1

.

Ooo

=

IF q

oo-adic

_integers

C =

completed

algebraic

closure of _w.r.t.

its normalized =

q)

absolute

value 1.1 [

G =

group scheme

GL(2)

with center Z

r(1)

=

G(A)

=

GL(2,

A)

r = _congruence

subgroup

of

r(1),

in _{most cases}

r =

ro(n): =

{( ~~)

_c

c -

0

(mod

n)},

where _n c A

non-zero ideal =

_positive

divisor of K

_coprime

with _o0

r =

rab

modulo torsion

X =

G(Ooo)

=

GL(2,Ooo)

J

_{= {( ~ ~) E X Ie == 0}

_(mod

_7r)}

Iwahori

_subgroup

T = Bruhat-Tits _tree of

PGL(2,

with sets

of vertices and of oriented

_edges

Tm

= Hecke

_operator

(m

positive

divisor)

x = Hecke

algebra

_over

Z, acting

_on

1. Modular and

automorphic

forms over K

(1.1)

The

_K-analogue

of the

_complex

_upper

_half-plane

H is the

_Drinfeld

upper

half-plane

It has a natural structure of one-dimensional

_analytic

_space over C and even over

(~12~, ~5~, [10]).

The _group

G(K~)

acts on il

_through

fractional

linear transformations: =

az+b

The

_subgroup

_r(l)

and thus

C

c

d )

cz+d

*

each _congruence

_subgroup

r C

_r(l)

acts with finite

_stabilizers,

whence the

quotient

r B

S2 exists as an

analytic

_space over In

fact,

we have the

following

result due to Drinfeld

_[4].

THEOREM 1.2. There exists a smooth

affine algebraic

curve

_Mr

_defined

over a

finite

abelian extension K’

of

K such that

r B

S2 is

isorrtorphic

as an

analytic

space with the

analytification

Mrn

of

Mr.

Let

_Mr

be the smooth

_projective

model of

_Mr.

The curves

_Mr

are called

Drinfeld

modular _curves; their

_study

is the content of

_[8],

to which we refer

for the

_following.

On

_Mr,

there are two kinds of

_{distinguished}

_points:

the

Mr(C),

which are in canonical

bijection

with the finite set

r B

of orbits of r on and the

_elliptic

_points,

which are the

classes mod r of those z E 9 whose stabilizers

r.,

are

strictly larger

than

r

_{fl Z(K).}

Since

Mr

is

_uniquely

determined

_by

the closed

_analytic

_space

r B

f2 U

_{cusps},

we henceforth will make no difference between _e.g. the

point

set

_{r B}

_S2,

the associated

_analytic

_{space -}

_Mrn,

and

_Mr.

(1.3)

A

_Drinfeld

modular

_form

of

_weight

k and

_type

m for r

(where

k is a

non-negative

integer

and m a class modulo

d(r)

=

cardinality

of

det(r)

_c

lFq)

is a function that satisfies:

(ii)

f

is

_holomorphic

_(in

the

_rigid

_sense);

(iii)

f

is

_holomorphic

at the _cusps.

Condition

_(iii)

means that for each _{cusp s} of

_{r, ,}

_f

has a _{power series}

expan-sion with _respect to a

uniformizer ts

of s.

Clearly,

_ts,

or rather

_too,

_plays

the role of the classical uniformizer

_{q(z) -}

_e2"2z;

it is

_specified

in loc. cit.

V

_2,

3. We let be the C-vector space of i times

cuspidal

modular forms of

_weight

k and

_type

m, where

f

is i times

cuspidal

if its order w.r.t.

ts

is &#x3E; i for each _{cusp s} of r. Then

dime

_Mk,m

is

_always

_finite,

and

{holomorphic

differential forms on

Mr}.

Note that

_{f (z)dz}

is

_holomorphic

at a _cusp s if and

only

if

ords, f (z)

_&#x3E; 2

since

9 =

const.

_t2

whereas

_classically,

=

2Jriq.

This

explains

the

dz .8) y dz q p

"double

_cuspidal"

condition.

EXAMPLE 1.5

_(cf.

_{[14], [6], ~9~).}

Let r =

r(l)

=

GL(2, A).

Then there

are three

distinguished

modular

forms g,

h,

A of

(weight, type)

equal

to

(q - 1, 0),

(q

+

_1,1),

_{(q~ -1, 0),}

_{respectively,}

such that

Moreover,

= -A,

and j

:= g9+1/0

identifies with There is

_precisely

one _cusp

( j

=

oo )

and _one

elliptic

point

( j

=

0).

Formulas for the numbers of _cusps and

_elliptic

_points

and for the

gen-era

g(r)

=

genus(Mr),

when r =

ro(n)

or r =

F(n) (=

full

congruence

subgroup

of divisor

_n),

_may be looked _{up in}

_[6]

or

[8].

(1.6)

Let T be the Bruhat-Tits tree of

_PGL(2,

_(cf.

_[21]).

It is a

(q

+

_I)-regular

tree with set of vertices

_X(T)

=

G(Koo)jX Z(K~)

and

set of oriented

_edges

_Y(T)

=

Z(K~),

where the canonical _map

from

_{Y (T)}

to

_{X (T)}

associates with each

_edge

e its terminus

t(e).

The _group

acts from the left on T. If r C

_r(I)

is a _congruence

_subgroup,

the

quotient

graph

rB ’T

is the

_{edge-disjoint}

union of a finite

graph

(rB7)0

and a

finite number of half-lines

hs

(i.e.,

h,

is

_isomorphic

with

_{o o a}

labelled

_by

the _{cusps s} E

_{r B}

Pl (K)

of r. There is a canonical

_surjective

G(Kc,,c,)-equivariant

map A from S2 to

₍₌

_points

of the realization

T(R)

of ~’ with rational

_{coordinates),}

which _may be described as follows: Recall

_[13]

that

_{corresponds bijectively}

to the set of

_similarity

classes of non-archimedean norms on the

_Koo-vector

_space

K~.

Then

_A(z)

is the class of the norm

~~ I

.

liz,

where

lI(u, v)lIz

:= The

map A

and the

derived _maps

_Ar:

_{r B}

S2 -&#x3E;

_{(r B 7)(Q)}

_may be used to describe admissible

coverings

of n and of

Mr

=

rB0.

E.g.

for v _E

X(T)

and _{e E}

Y(T),

is

_isomorphic

with minus

_(q

+

₁₎

discs and

_{a-1 (e)}

with an annulus

Iz

E C

_{Izl [}

_1}.

The orientation of e

_corresponds

to an orientation

(1.7)

For _any abelian group

B,

let be the group of maps p: - B

subject

to

-We also

_put

_.B)r

c

B)r

c

B)r

C

_H(7,

_B)

for the

sub-groups of those p that also

satisfy

(iii)

+

(iv’), (iii)

+

(iv), (iii),

respectively,

where

(iii) cp( Ie) = cp(e)

(1’

E

_r);

]

(iv) cp

has

_compact

_{(= finite)}

_support

modulo

_r;

(iv’) p

comes from

_Hi

(1’,

B --+

_a!

(1’,

B)r.

Intuitively,

(iv)

means that _~p vanishes

eventually

on each of the half-lines

hs,

and

_(iv’),

that it vanishes on the whole of

h.

Here and in the

_sequel,

we consider _{cp E}

B)r

as a B-valued function on the

edges

Y (F B T)

of

the

_graph

_{r 1}

T.

(1.8)

These _groups, for B

_equal

to one of

are

_closely

related to the

_geometry

of

_Mr,.

_Namely,

let

r

be the _group

rab

(r

made

_abelian)

modulo its torsion

_subgroup.

It is a

finitely generated

free abelian group of

rank g

=

g(r),

where

a) g

= rank

H, (r B

T,

Z)

= rank

H!

(’1,

Z) ,

and

b) q

= _genus

Mr

(see

( I .12 ) ) .

The first relation comes out as follows: Let F* be the factor _group of r

modulo the

_subgroup

_generated

_by

the elements of finite order. Then r* is

canonically isomorphic

with the fundamental _group of

_{r B}

T

_([21]

I Thm.

13,

Cor.

_1),

which

_yields

_(F*)

~

_{~’, 7~) .}

_{Let ~p}

E

_{T, Z),}

regarded

as a r-invariant function _p: Z. For e E

_put

_{n(e) :}

=

index of in the stabilizer _group

_r~

of e in r. Then

_{~p*: e}

H

n(e)cp(e)

is a well-defined element of

Z) ,

and _cp H

p* :

HI

(r 1 ~’, z)

1Jj (J’,

Z) r

is

_injective

with finite cokernel.

_Together,

we have a _map

THEOREM 1.9.

_If

r is a Hecke congruence

subgroup

ro(n)

then j is

an

isomorphism.

The

_proof

is

_given

in

_[17],

_using

an

explicit description

of the

graph

rBT.

This method neither works for more

general

_groups r nor for more

general

function

_rings

A than

_Fq[T].

_However,

the statement remains true for

congruence

subgroups

r which are

p’-torsion

free. In

general,

the cokernel

of j

is finite with order

_prime

to p

(see

[10]

for

details).

(1.10)

From now _on, we assume that r =

ro(n)

with some divisor n of

A. In

_[7]

and

_[10]

it is shown how

H,

(7,

C)r

may be

interpreted

as a

space of

automorphic

forms in the sense of

Jacquet

and

Langlands

[15].

In

particular,

that _{space is}

_equipped

with

a)

a "Petersson" scalar

product

( . , .

);

b)

Hecke

_operators

_Tm

for each divisor m of

A;

c)

a canonical

integral

structure

H~(’J’,7G)r.

The scalar

_product

comes from the L2-norm on the discrete set

_{Y(r B ~),}

where the volume of an

edge

i E

_{Y(r B}

_T)

is

_{given by}

=

2 n(e)-1

(e

E

_{Y (7)}

above

_e,

see

(1.8)).

The Hecke

_operator

_Tm

is derived from a

correspondence

on

Y(r BT)

=

r B

Z(Koo).

Regarding

the latter as an adelic double coset

_([7]

3.3), Tp

for m = _p

prime

and

coprime

with n _may be defined as in

[24]

Ch.

VI. For _p a divisor of _n, the definition has to be

slightly

modified. One

possible

definition

_{(compatibilities}

are checked in

[10]

Sec.

9)

is as follows:

Consider p

as a function on

G(Koo) .

Then

where the sum is over _a,

_b,

d E A such that _a, d are

_monic,

_(ad)

= _m,

(a,

n)

=

1,

and

_deg

b

_deg

d.

These Hecke

_{operators Tm}

have the usual

_properties:

_They

commute

mutually, satisfy

Tm.m’

=

Tm

Tm

for _m and m’

coprime,

for a

_{prime p,}

_Tpi

is a

_polynomial

with

_integral

coefficients in and

Tm

is Hermitian w.r.t.

the Petersson

_product

if m is

_coprime

with n.

_Furthermore,

the

_integral

structure is stable under the

_Z-algebra

J£

_generated

_by

the

_Tm,

and is

_integral

w.r.t. the Petersson

_product.

(1.11)

Now the

_{correspondence}

_Tm

_may also be defined on

Mr,

_using

the

If _e.g. m and n are

_{coprime, Tm}

associates with each

_{point x}

on

Mr

(=

rank-two Drinfeld module with a certain level

structure)

the collection of

all x’ E

Mr

for which there exists a

_{cyclic m-isogeny}

from x to x’

(see [8]

for

_{details) .}

Let

_Jr

=

Jo (n)

be the Jacobian of

Mr

(recall

that r =

ro(n)).

Thus

_Tm

induces an

endomorphism

of

Jr,

and of the .~-adic

cohomology

HI

_(Mr,

_Ql).

These

_operators

have

_essentially

the same

_properties

as those

listed for

_Tm,

and are therefore labelled

by

the same

_symbol

"Tm" .

The

following

deep

result is due to Drinfeld

_(Thm.

2 of

_[4],

_specialized

to our

situation).

THEOREM 1.12. There is a canonical

_{2somorphisrrt}

between

and

_H!

_(~’,

_Ql)r

_{Q9 sp.} The

_isomorphism

is

_compatible

with the actions

_of

_a)

the Hecke

_{operators Tm}

and

_b)

the local Galois _group

_of

Koo.

Here _{sp is} the two-dimensional

_special

£-adic

_{(.~ ~ p)}

_{representation}

of

_([2]

_3.1).

It acts on since

Mr

is defined over H. We will _{not go into} details of

(1.12)

and its

_{interpretation}

as a

reciprocity

law

_{([4], [3], [10]),}

but

_{will just}

use the

_principle

that the

_splitting

of

_Hl(Mr,

_Qe)

and thus of

_Jr

under the action of the Hecke

_algebra

X is

encoded in = which is

effectively computable.

(1.13)

Now let

_{Q) r}

C

Q) r

be the new

_part,

_i.e.,

the

orthogo-nal

_complement

of the different

_embeddings

of Bj

_(0’,

into

H.

(0’,

Q) ,

where m runs

_through

the strict divisors of n.

_{Correspondingly,}

let

be the new

_part

of

Je(n)

=

Jr,

which is well-defined _{up to}

_isogeny.

Then

also

_Qt)

_{Q~)~ 0}

_sp.

We will consider

elliptic

curves

E/K

with conductor and with

_split

multiplicative

reduction at oo.

By combining

results of Grothendieck and

Jacquet-Langlands, E gives

rise to an

automorphic

_{representation,}

whose new _{vector cp} = _{pE appears in}

Brew(’J,

By

the

above,

E _appears

as an

isogeny

factor of

conversely,

_every factor of dimension one

of is an

elliptic

curve defined over K with the

_properties

stated

above. For a detailed

_discussion,

see

[10],

Section 8.

Applying

well-known

facts from the

_theory

of

_automorphic

forms

_(notably

_{"multiplicity}

_one"),

there result canonical

_bijections

between the sets of

a)

normalized Hecke

_eigenforms

_{’P in}

_{Q) r}

with rational

eigen-values ;

b)

one-dimensional

_isogeny

factors of

c)

isogeny

classes of

_elliptic

curves

E/K

with

_cond(E)

= _{n ’}

oo, and

It is this statement which we label as the

_analogue

of STW over the

func-tion field .K. The relafunc-tion between an

eigenform

_cp and an associated

_elliptic

curve E is as follows:

_{If p}

is a

_prime

not

_dividing

n, then the

eigenvalue

_Ap

of

_Tp

on _cp satisfies

where

_Fp

:=

A/p

has qp elements and

E(JFp)

is the _group of

_Fp-valued

points

of the reduction. But note

_that,

so

_far,

_{STW /K}

is a sheer existence

statement. Whereas

_c)

~

a)

is easy _to obtain

(by

Grothendieck,

the

L-functions associated to E are

polynomials

in

_q-S,

so one catches _cpE

_by

computing

a finite number of reductions of

E)

and

_b)

~

c)

_{is more or}

less

_{tautological,}

_a)

~

b)

is the difficult

_part.

In the _next section I will

show how to construct E as

quotient

of

Jo(n), ’P

being given.

Note that

the condition

_{"split multiplicative}

reduction" is

_equivalent

to

_being

a Tate

curve; so E will be described

_through

its Tate

_period

at

_infinity.

2. Theta functions and the Jacobian

(2.1)

Let

_f

be a nowhere

vanishing holomorphic

function on

Q,

e =

(v, w)

an

edge

of

~’,

and .

where

₁₁

.

liu

is the

spectral

norm on the admissible subset U of S2.

Intu-itively,

r( f )(e)

measures the

growth

of

f along

e. Then

r( f ){e~

E Z and

r( f ): e

~

r( f )(e)

defines an element of as results from the

rigid

residue formula

_[5].

More

_precisely,

r:

_{f H}

r( f )

_yields

a

G(K~)-equivariant

exact sequence

( loc.

cit.)

which is basic for what follows.

_Next,

for each differential form w on Q and e =

(v, w),

_we let

e)

be the residue of _{w w.r.t.} the oriented annulus

,X-’(e).

Again

from the residue

_theorem,

_res(w):

e F-+

res(w, e)

is in

_C),

is

_commutative,

where d

_log:

_{f I-t 1f}

and red is reduction

_modp.

f

(2.3)

A

_hotomorphic

theta

_fonction

for r C

_r(l)

is an invertible

holomor-phic

function u: Q -+ C such that for each _{y E} r there exists

cu(q)

E C* with

_{u(-yz) = cu(,)u(z),}

and which is

_holomorphic

and non-zero at the

cusps of r

(see

[10]

Sec.

5;

this makes sense in view of the functional

equations

of

_u).

Let

_8h(r)

C

_On(Q)*

be the _group of

_holomorphic

theta functions. For u E

_eh(r),

d

log u

= dz is

r-invariant,

which means

that

_{u’(z) /u(z)}

lies in

_M2,I(r),

and in

_fact,

in

_{m22,1 (r).}

_Next,

we construct

holomorphic

theta functions for r = Let a _E an

_arbitrary

base

_point,

and

_put

/ _...v

Products of this

_type

have been introduced and studied in a different con-text

_by

Manin-Drinfeld and

_by

Gerritzen-van der Put. The

_following

the-orem is

largely

due to Radtke

_{~18~ .}

THEOREM 2.4.

_(i)

The above

_product

converges

locally

uniformly

to an

invertible

_function

Uo on S~ that does not

depend

on -the choice

of

w E Q.

(ii)

ua is a

holomorphic

theta

function,

whose

multiplier

c, : ’Y ~-4

cu,

(q)

is a

homomorphism

r -~ C* that

only depends

on the class

of

a in

r :=

_{rab jtor(rab).}

(iii)

The association ~ is

_symmetric

and

defines

a

sym-metric bilinear _map

_{f rom}

r

x

r

to

_{K~ ~}

C* .

The

_relationship

between theta functions and

_automorphic

forms is de-scribed

_by

the

_following

_results,

whose

_proofs

are

_given

in

_[10].

_{Let j}

and

r be the _maps introduced

in ~ 1.8~

and

(2.1).

THEOREM 2.5. For a E

_r,

_{r(u,) = j (a) .}

whose arrows are induced from a Ua and r:

90(0)*

2013~ is

well-defined and commutative. Here we

wrote j(a)

for j

(class

of a in

F).

Similar notation will be used in what follows.

Since

_ker(r)

= C*

and j

is

bijective,

we have

COROLLARY 2.6. u and r are

_bijective.

Let

_M2,1(r,

be the

_Fp-subspace

of

_M2,1(r)

of forms

f

such that

f (z)dz

has its residues in

_Fp.

Dimension considerations show the

_properties

indicated in the commutative

_diagram

Here the _upper vertical arrows are

u(z)

H

u’(z)/u(z)

and reduction

_modp

(both surjective),

and the lower vertical arrows come from base extension

Fp

~ C.

THEOREM 2.8. For

_{a, (3}

E

_r,

we have

(Petersson

scalar

_product

in

_H!

_(,

_{Z)T ).}

COROLLARY 2.9. The

_symmetric

bilinear

_form

on

is

_positive

_definite,

and

-is

_injective.

We now

proceed

to construct the Jacobian

_Jr

=

Jo (n)

of

Mr

(r

=

ro(n)).

Let

_{Tr /Ooo}

be the

_split

torus with character _group

_P,

_{i.e., Tr}

=

We _may

_regard

r

via c as a

_subgroup

of

Hom(P,

K~).

Now the assertion of the last

_{corollary implies}

that the

quo-tient

_Tr/c(r)

_(which

_always

exists as an

analytic

_group

variety)

is in fact an abelian

_variety

(see

[16], [11]).

THEOREM 2.10. The Jacobian

Jr

of

Mr

is

_{canonically isomorphic}

with

Tr/c(f).

I. e., f or

each

_complete

extension L

_of

_Koo

in

_C,

there is an exact sequence

o f L-analytic

groups 0 -i

Hom(r,

L*) -+

0.

A

_proof

of the theorem is

_given

in

_[10]

in a more

_general

_context,

_along

with a

_precise

_description

of the Abel-Jacobi _map. In the next

_section,

we shall use our

description

to construct

_strong

Weil curves as

_quotients

of

_Jr.

_Presently,

we derive a _consequence about the _group of connected

components

of the N6ron model of

Jr .

Let 9

be the N6ron model of

_Jr

at oo,

go

C 0

the connected

component

of the

_identity,

_{and §(r)}

_{= ~ ( Jr ) :}

:=

8/ao

the _group of connected

_components

_(see

_[1]).

The Petersson

_product

( . , . )

on H := is

integral,

which

yields

a canonical

_injection

’

COROLLARY 2.11. The _group

_o(r)

_of

connected

components

of

the Neron model

_{of Jr is canonically isomorphic}

with

_Hom(Uj

_(’1,

_{Z) r,}

_{Z) ji(1Jr (7,}

Proo f .

Consider the commutative

_diagram

where v is derived from the valuation Since r is

_free,

the middle column

is exact.

_Identifying

r

with H =

Hi

(T,

Z)

by

_means

of j, v

o c becomes the

map i,

and the snake lemma

_yields

the exact _sequence

Recall that

_Hom(r,

_O§~)

= It follows from the construction

_given

in

_[16]

that =

§(Jr) =

~(T),

thus the

_right

hand term of the above _sequence

_equals

_4&#x3E;(f).

D

The

_meaning

of

_Corollary

2.11 is that

_0(r)

_(or

rather its

_size)

is the

"regulator"

of the lattice

H!

(7,

in the _space

at

of

_automorphic

forms, equipped

with the Petersson

_product.

A similar assertion on

mod-ular

_symbols

occurs in

[23],

Thm. 14.

3. The Weil Uniformization

In the

_present

_section,

we construct the

_strong

Weil curve associated to a

_primitive

Hecke

_{eigenform p}

E

Z)r

with rational

_eigenvalues.

_(cp

primitive:

Z)r

for n &#x3E;

l.)

We

_identify

r

with H =

Z)r

by

means

of j,

_writing

both _groups

additively.

Let A C

_K~

be the

_subgroup

fcw(a)

a

E

_r}.

As is shown in

_[10],

_{Prop. 9.5.1,}

A has a

subgroup tZ

of

finite

_index,

where &#x3E; 0.

_Choosing

_{v~ (t)}

_minimal,

we have

where d is a divisor

of q -1

and _jjd C

_K£

the _group of d-th roots of

_unity.

In our case A =

Fq [T]

however,

we can show:

THEOREM 3.2. In the above situation d = 1. In other

words,

the

group A

itself

has the

_form

t71

with some

uniquely

determined’t E

_Kg,,,

&#x3E; o.

Forget

the theorem for the

_moment,

and consider the commutative

dia-gram

where

_{ev: f ~--.}

is evaluation on _cp E

_{LI! (7,}

Z)

=

r.

Note that

rais-ing

to d-th _powers

_gives

_Tate(td)(C).

_{By GAGA,}

the

_right

hand arrow _pr, is a

morphism

of abelian

_varieties,

so the

el-liptic

curve

Tate(td)

is an

isogeny

factor of

Jr.

For each

prime

13 f

_n,

the

_p-th

Hecke

_operator

_Tp

acts on

Tate(td)

as

multiplication by

_{Ap =}

eigenvalue

of _cp under

_Tp

_(details

_being

worked out in

_[10]

Section

_9).

It follows

_{(loc. cit.)}

that _pr, and its

_target

_Ecp

:=

Tate(td)

are defined over

K. From

_(1.14)

we see that

_Ecp

belongs

to the

_isogeny

class determined

by

cp. In

fact,

_Ecp

is the

strong

Weil _curve, as is immediate from the

con-struction. Now let

Po

be a fixed K-rational

_point

on

Mr

=

ro (n) B

_S2,

say,

i:

_{Mr - Jr .}

Then the

_strong

Weil uniformization of

_Eep

is _{p_} := _prep

_{o i,}

described

_by

If w is a coordinate on C* and w

= dw

the associated invariant differential

w

on

_E,

we have

where

_{f (z)}

E is the modular form

_{f (z)}

=

(see

(1.4)).

But

,

Uli

z

note

_that,

unlike the classical case,

knowledge

of

f (z)

does not suffice to

determine

_E~,

due to the loss of information caused

_by

reduction mod _p

(compare

(2.7),

and Ex.

_(4.4)!).

(3.6)

We are left _{to prove} Theorem 3.2. In what

follows,

H

:= H~

(T,

Z)

=

and

_{c~

E

_{H ~}

_I

_{(cp, a)}

=

0}.

Consider the

following

numbers:

d as

given

by

(3.1),

i.e.,

d =

#(tor(A));

n :=

_degree

of the

_strong

Weil uniformization p,~;

m :=

a)

&#x3E; 0

_a

_{E H};}

congruence number of cp.

The theorem will come out

_{by comparing}

these numbers. Some of our

arguments

are

_{inspired by}

[19]

and

[25).

Let

_{p§ : E,}

=

Jac(E,) -

Jac(Mr)

=

Jr

be induced

by

Picard

functo-riality.

Then

_p*

is

_injective

_{because p} is a

_strong

Weil

_{uniformization,}

and

PROPOSITION 3.8. n = d . r.

Proof.

In what

_follows,

_working

with tori and abelian

_varieties,

we write

down the C-valued

_points

only.

Let E° =

im(p*)

C

_Jr.

It is the

subva-riety

of

Jr

that

_corresponds

to the

_integrable

subtorus

_Hom(r/cpl,

_C*)

of

C*)

=

Tr(C).

Hence

EW(C)

=

C*)/A,

where

The _{map pr~} E‘~ -&#x3E;

_E~

is therefore

_given

_by

where as

usual,

ev( f )

= Now ev

I

A is

_injective.

For let ca be in

the kernel. Then ca = 1 on

cp-1-,

so _ca has values in the _group of

r-th roots of

_unity.

But this means the theta function ua is bounded and

hence constant

_(i.e.,

ca =

1),

since Q is a Stein

_domain.

_Furthermore,

ev(A) _

1} =

=

t,dz

has

index d2 .

_r in A =

x

tz.

For _any natural number

_i,

let i* be its

p-free

_part.

We then obtain

r* for the order of

_ker(ev)

=

Hom(P/ZV

₊ and _{n -} n* for the

order of

_ker(pr,

_I

=

ker(mult.

by

n).

Chasing diagrams

in

(*)

yields

n - n* =

d2 ~

_{r .}

_r*,

which

implies n

= d - _{r as} stated. 0

Let 5l be the Hecke

_algebra

on H = as defined in

(1.10).

Note that it does not agree with the Hecke

algebra

considered in

(10~,

for

two reasons:

a)

5l acts on the whole of

H,

not

only

on its new

_part;

b)

it also contains

_operators

_{Tp with p}

_n.

In

_particular,

1C

_{Q9 Q}

is in

_general

not

_semisimple.

We also

_regard

5l as an

operator

algebra

on

Jr

=

Jo(n),

which

yields

inclusions

Let e E

_{M 0 Q}

be the

_idempotent

that

_corresponds

_{to ’P}

E

H.

Then

r = denominator of _e in End

H

(i.e.,

r = least

and

(3.10)

n = denominator of _e in End

.Ir.

Here

_(3.9)

is

_trivial,

and the

_(easy)

_proof

of

_(3.10)

_may be found in

_[25],

proof

of Thm. 3. Let s be the denominator of e in 9t. From the above

inclusions,

we have

Next,

consider the Fourier coefficients

_c(1)

=

for 0

_E

as introduced in

_[24]

Ch. III. The

_arguments

D are divisors on

_{K, i.e.,}

D

_{= f .}

_ooi,

where

_f

is a finite divisor

(=

fractional ideal of

A),

and

c(1)

vanishes if a fails to be

_{non-negative.}

The "constant" Fourier coefficient co

of o

vanishes

_by

virtue of the _cusp condition

_(1.7)(iv).

The

_c(a)

describe the restriction

_{of 0}

_(regarded

as a function on

GL(2,

to the

_subgroup

of

_{matrices ( § § )}

in

_GL(2,

For the convenience of the

_reader,

we

_give

(without proof)

the relevant formulas for

_c(a).

These _may be derived from the adelic formulas of

_[24]

p. 20

by

tedious but standard calculations.

Let 7r =

T-1 be the uniformizer

at oo, and write D =

f .

ooi,

where f

is

generated

by

the monic element

_f

of A. Let further _q: C* be the character

where

_{Tr: Fq - Fp}

is the trace and _{go is any} nontrivial character of

_{IFp .}

*

Then

where

Note that

_by

_(i),

the

_c(9)

_{determine 1/J}

since the matrices

( 71": : ~)

_represent

(ii)

reflects the flow condition

_(ii)

of

_(1.7).

Hecke

_operators

and Fourier coefficients are related as follows: If

1/;’

=

Tip

then

The fact

that 1b

is an

eigenform

for

Tp

with

eigenvalue

Asp

yields

(p, f

coprime,

d :=

_deg

_p)

where

_{the qj}

are

_given

_by

the _{power series}

identity

In the last _case,

_{Asp =}

if

_{p 11}

n and

_Ap

= 0 if

P2

n.

The sum in

(3.12)(i)

may be

simplified

to

with

_{v (y) = - 1 if y}

has a term of order 7r in its

_{-7r-expansion,}

and

_v(y)

=

q-1

if it has no term of order 7r.

Similarly,

the sum in

(3.12)(iii)

splits

(besides

the term

_{corresponding}

to u =

0)

into

partial

_sums of

_type

But the double class of

(

does not

_change

if u is

_{replaced by}

cu. Hence the value of that

partial

sum is

simply

and

_c(f)

is a linear combination with coefficients in of values of

_0.

COROLLARY 3.15. The Fourier

_coefficients

of1/;

E

Lii(7,Z)r

lie in

Z[p-1] -

Conversely,

if o

E

8!(T,C)r

has its Fourier

_coefficients

in

then o

also takes its values in

E

_Uj

_CJ,

(C)r.

Now consider the bilinear

pairing

(H

=

H~

(71

Z) r)

(Fourier

coefficients of

_T1jJ

w.r.t. the trivial divisor

_(1)).

THEOREM 3.17. The

_Pairing

is

_{non-degenerate}

and becomes a

perfect

pair-ing

after

tensoring

with

In other

_words,

the determinant of

_{( . , . ) with}

_respect

to _any Z-bases of X and H is a _power of _p.

Proof.

1)

Let 1/1

E H be in the

_right

kernel. Then 0 = =

qdeg

for

all

_positive

finite divisors _m, which

_{gives o}

= 0

by

(3.12)(i’).

2)

Let T E x be in the left kernel. Then for all

_q0

E

H

and all

_positive

finite divisors m, 0 = =

cTmT1b((l))

=

qdeg

_i.e.,

To

= 0

and therefore T = 0.

3)

Let _p be a

prime

number

and 1b

E

H

such that

_{(T, ~) -}

0

(mod t)

for all T E J4l. Then for all _m,

_{(Tm, ~)}

= =

"‘c~(m),

and all the

_cw(m)

are

_congruent

to 0

_{(mod ~).}

_By

_(3.12)(i’)

this

_implies

that 9

is divisible

_by

t.

The theorem follows from

1, 2

and 3. D

LEMMA 3.18. Then number

_{slr (see (3.11))}

divides the determinant

_of

the

pairing

( . , . )

of

(3.17).

Proof.

se E 5l is

_primitive

and E

_(s/r)H

for

_{all 9}

E

_H,

so

(se, V))

E

(s /r)Z.

~

Proof of

Theorem 3.2. The number d

_equals

_by

_(3.8),

so divides the p-power

s/r ((3.17)

+

_(3.18)).

On the other

hand,

d is a divisor

_{of q -}

1.

0

We state

_separately

what has been proven

along

with Theorem 3.2. Let cp be a

primitive

Hecke newform with rational

eigenvalues,

and let m and r =

(cp,

be the numbers defined in

(3.6).

COROLLARY 3.19. The number m is the

_pole

order

_of

the

j-invariant

_jE

_of

the

_strong

Weil curve E associated with _cp.

COROLLARY 3.20. The congruence number r

equals

the

degree n of

the

strong

Weil

_{uniformization of}

E.

Note that the last result is

_stronger

than the

_{corresponding}

result over

Q,

which seems to be established for

_prime

conductors

_only

_(~25~

Thm.

_3).

4.

_Examples

In the

_following

_examples

we

always

take _q =

_2,

thus r =

ro(n) =

{

GL(2, A)

with some n E A =

F2 [T] ,

n =

(n).

The

Peters-cd

I

son

product

on will be derived from the

L2-norm

on

Y(r B T),

where each

_pair

of inverse

_edges

contributes the volume 1

_{(compare (1.10)).}

Remark

_4.1.

Under our

_hypotheses,

two

_isogeneous

_elliptic

curves

_E,

E’ over K =

F2(T),

which _are Tate _{curves over} and whose

_j-invariants

j,

j’

satisfy

= Voo (j’)

are in fact

_isomorphic.

The

_{number 9}

=

genus(Mr)

= rank

Hi (77

is calculated in

(6],

_see

also

_[8].

If

_deg

n

_{27 g}

=

0,

_so there _{are no}

elliptic

_curves

E/K

with

conductor n. oo and E x = Tate _curve.

_Up

_to coordinate

_changes

_in

T,

there are

_precisely

two conductors n such

_{that g}

=

_1,

_given

_{by n}

=

T3

EXAMPLE 4.2. Let n =

T3.

The

graph

r B

T looks

where - - - indicates a _cusp

= infinite half-line

(see [7]

Sec.

5).

with discriminant A =

T8 and invariant j

=

T4

has

split multiplicative

reduction at oo and conductor

T3,

as a routine

application

of Tate’s

algo-rithm

_[22]

shows. Hence E is the

_strong

Weil curve associated with _p,

i.e.,

E =

Mr

in this case.

EXAMPLE 4.3. Let n =

T2(T -

1).

The

graph

r

B

T is

(loc. cit.)

If H =

Zp

then m =

(p, cp)

= 6. The

_strong

Weil _curve is

EXAMPLE 4.4. Let now n =

T(T2

-f-T ₊

1).

The

graph

r B

T looks

(loc.

cit.)

Let ₁₁ and ₁₂ be the

_cycles

of

_length

_4,

oriented

_{counterclockwise,}

and

wl, w2 the involutions attached to the

prime

divisors PI =

(T)

and _p2 =

(T2

+ T +

₁₎

of n. The Hecke

eigenforms

_cpl := _{’1’1 + "/2? W2} :=

_{+ "/2}

in

The Jacobian

_Jr

is

_isogeneous

to

_Ei

x

_{E2, where Ei}

is the

_elliptic

curve

The numbers mi are = 3 and _{m2 =}

( c~2 , ~y2 ~

=

5,

the congruence numbers ri

equal

2 for ~p1 and W2.

Equations

are

_given

by

.El

and

_E2

intersect in

_Jr

in their common

subgroup

scheme of two-division

points.

Since p = 2 itself is _{a congruence}

prime,

the two

weight

two modular

forms

_(Ui

= i =

1, 2)

_agree.

EXAMPLE 4.5. Put n =

T4

₊

T 3+1 ,

which is

prime

inF2 [T] .

Here

r B

T is

The canonical involution w acts as the reflexion at the middle

_axis,

and

it is easy to see that the _+-space of w in

Bj (T,

R)’

has dimension one with

basis vector _cp := _{/1 - 12,} which is

primitive

for

H!

(7,

Now

(’P, cp)

= 16

and m = =

8,

_{so r} =

deg

p~ = 2

Mr/w

= E is the

projection).

The curve E is

_given

by

The

_{Birch-Swinnerton-Dyer}

_conjecture

_predicts

the rank 1 for

_E(K);

in

A

_complete

list of

_strong

Weil curves over

F2(T)

for conductors n of

de-gree 4,

along

with the calculation of their relevant arithmetic

_invariants,

has been

_given

_by

A. Schweizer

_[20].

That list can be extended without

difficulty

to

_{higher degrees}

of n and

_larger

constant fields

_{Fq .}

The

_philosophy

behind these

_examples

is as follows: Calculations on H not

_{only yield}

the numbers m = and _r =

deg

PE for the

strong

Weil curve E in a

class,

but also

provide

insight

into the

position

of E

"on the modular curve

Mr,".

This means that _e.g. the

_image

of _cusps,

elliptic

points,

Heegner

points...

of

_Mr

in E can be

determined,

and also

the action of Atkin-Lehner involutions and any other

geometric

data that exist on

Mr.

_Obviously,

this enhances the toolbox of methods available for

a

deeper study

of

elliptic

curves over function fields.

REFERENCES

[1]

S. BOSCH, W. LÜTKEBOHMERT and M. _RAYNAUD, Néron models, Grundlehren. Math. _{Wiss., Springer,} Berlin-New York, 1990.

[2]

P. _DELIGNE, Formes modulaires et representations de _GL(2). In: Modular

Func-tions _of One Variable II, Lect. Notes Math., vol. _{349, Springer,} Berlin Heidelberg

New _{York, 1974.,} pp. 55-105.

[3]

P. DELIGNE and D.

_HUSEMÖLLER,

Survey of Drinfeld modules., Contemp. Math.

67 _(1987), 25-91.

[4] V. G. DRINFELD, Elliptic Modules, Math. Sbornik 94 _(1974), 594-627 _(Russian);

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[5]

J. FRESNEL et M. van der PUT, Géométrie Analytique Rigide et Applications, Progr. Math., vol. 18, Birkhauser, Basel Boston, 1981.

[6]

E.-U _{GEKELER, Drinfeld-Moduln} und modulare Formen über rationalen

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[7]

E.-U _{GEKELER, Automorphe} Formen über

_Fq(T)

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Berlin _Heidelberg New York, 1986.

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[10]

E.-U GEKELER et M. REVERSAT, Jacobians _{of Drinfeld} modular _curves, submit-ted.

[11]

L. _GERRITZEN, On non-archimedean representations of abelian varieties, Math. Ann. 196

_(1972),

323-346.

[12]

L. GERRITZEN and M. van der PUT, Schottky Groups and _{Mumford Curves,} Lect. Notes _Math., vol. _{817, Springer,} Berlin Heidelberg New _York, 1980.

[13]

O. GOLDMANN and N. IWAHORI, The space of p-adic norms, Acta Math. 109

(1963),137-177.

[14]

D. GOSS, 03C0-adic Eisenstein Series _for Function Fields, Comp. Math. 41 _(1980), 3-38.

[15]

H. JACQUET and R. P. LANGLANDS, Automorphic forms on _GL(2), Lect. Notes

Math., vol. 114, Springer, Berlin _Heidelberg New York, 1970.

[16]

D. MUMFORD, An analytic construction of degenerating abelian varieties over

complete rings, Comp. Math. 24

_(1972),

239-272.

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U. _NONNENGARDT, Arithmetisch _{definierte Graphen} über rationalen

Funktionen-körpern, Diplomarbeit, Saarbrücken

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W. _RADTKE, Diskontinuierliche _Gruppen im _{Funktionenkörperfall, Dissertation,}

Bochum

_(1984).

[19]

K. RIBET, Mod p Hecke operators and _congruences between modular _forms, Invent. Math. 71 _(1983), 193-205.

[20]

A. _{SCHWEIZER, Dissertation,} in _preparation.

[21] J.-P. _{SERRE, Trees, Springer, Berlin-Heidelberg-New York, 1980.}

[22]

J. TATE, Algorithm for determining the type of a singular fiber in an elliptic pencil. Modular Functions _of One Variable IV, Lect. Notes _Math., vol. _{476, Springer,} Berlin

Heidelberg New York, 1975, pp. 33-52.

[23]

J. TEITELBAUM, Modular symbols for

Fq(T),

Duke Math. J. 68 _(1992), 271-295.

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Ernst-Ulrich GEKELER FB 9 Mathematik

Universitiit des Saarlandes Postfach 15 11 50

D-66041 Saarbrfcken