Involutory elliptic curves over $\mathbb {F}_q(T)$

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

A

NDREAS

S

CHWEIZER

Involutory elliptic curves over F

q

(T )

Journal de Théorie des Nombres de Bordeaux, tome

10, n

o

_{1 (1998),}

p. 107-123

<http://www.numdam.org/item?id=JTNB_1998__10_1_107_0>

© Université Bordeaux 1, 1998, tous droits réservés.

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

Involutory elliptic

curves over

Fq (T)

par ANDREAS SCHWEIZER

RESUME. Pour n ~

_Fq

_[T],

G

_désigne

un _sous-groupe d’involutions d’Atkin-Lehner de la courbe modulaire

_{Xo (n)}

de Drinfeld. On determine tous les n et G tels que la courbe

GBXo (n)

est rationnelle

ou

_elliptique.

ABSTRACT. For n ~

_Fq

_[T]

let G be a

_subgroup

of the

Atkin-Lehner involutions of the Drinfeld modular curve

_{Xo (n) .}

We

de-termine all n and G for which the

_quotient

curve

_{GBX0 (n)}

is rational or

_elliptic.

0. INTRODUCTION

In

_[M&#x26;SD]

Mazur and

_{Swinnerton-Dyer}

called a modular

elliptic

curve

involutory

if its Weil uniformization is

_{given by}

a

_subgroup

of the

Atkin-Lehner involutions.

_They

listed several

_examples.

Later Kenku

_[Ke]

showed that there exist

_{only finitely}

many.

Using

a

_slightly

more

_general

definition for

elliptic

curves over

_{Fq (T)}

we

ask: For which n E and which

_subgroups

G of the Atkin-Lehner involutions of the Drinfeld modular curve

Xo (n)

is the

quotient

curve

GBXo(n)

elliptic?

In this paper we will show that there are

_{only finitely}

_many

(for

_{all q}

together)

and list them.

Two

_strategies

are used to restrict the

_possible

n. The first

_method,

due

to

_Ogg,

consists in

_counting

rational

_points

on a suitable reduction. The

second one transfers the

problem

from n to divisors I of n

_{by exploiting}

the

knowledge

how

_automorphic

forms for lift to

_automorphic

forms for

Fo (n) .

Combined,

these two

_{strategies yield surprisingly good}

bounds for

deg(n).

In the

_remaining

cases the genus of

G~Xo(n)

can be calculated or

estimated

_using

the

_explicit

formulas of Section 2.

The results are

gathered

in

_Propositions

_{3.1, 3.2, 5.6,}

and 5.7 for

_elliptic

1. BASIC FACTS

Let

_Fq

be a finite field

with q elements,

A :_

Fq [T]

the

polynomial ring,

K := the rational function

field,

_{:=1~q ( (T -1 ) )}

its

_completion

at

oo

(=

T ),

and C the

completion

of an

algebraic

closure

of K .

The letters

n, m,

C, ...

will denote monic elements of A and p will be a

_prime

(i.e.,

a

monic irreducible

_element)

of A.

_Throughout

this _paper we will assume

s

n i with different

_primes

_pi E

A,

and d will denote the

_degree

of n.

i=i

The group GL2 (.K~ ) acts on on and on the Drinfeld

upper

halfplane

Q = C -

Koo

by

fractional linear transformations. So does

the Hecke congruence

subgroup

The

_quotient

space can be

compactified

by

adding

the finite set of

cusps

ro (n)BPl (K).

The

resulting

rigid analytic

space is called the Drinfeld

modular curve

Xo(n).

As an

_algebraic

curve

Xo(n)

is defined

already

over K and has

_good

reduction at all finite

_{primes p}

with

_{p In.}

Its _genus is

_([Gel

_p.79~

or

[G&#x26;N]):

where

For _every

_m(n

with

_{(m, ~)}

= 1 there exists the

(partial)

Atkin-Lehner

involution on

Xo (n),

sometimes

simply

denoted

by

Wm

if the n is

clear. It is

_given

on

(resp.

by multiplication

from

the left with any y matrix

( ::

(nc md)

b

with a ’ ’

b

c ’d e A and determinant

rm for some _q E

_F:.

From this it is obvious that

_{Wml Wm2}

= with

m3 =

(ml-.

Consequently

the Atkin-Lehner involutions

_{of Xo (n)}

form a 1,m2

2-elementary-abelian

group

W(n)

of order 2.

In

_particular,

_represented

for

_{example b}

_bY

(1), is

called the

(f )

&#x3E; iS th

Theorem 1.1

_([Sch3]).

Let n E

_{Fq [T]}

be

_{of degree}

d. Then

a) Xo(n) is

rational

_if

and

_{only if d}

2.

b) Xo (n) is

elliptic if

and

_only

_if

q =

2,

d

= 3,

and _n has _a

multiple

_root.

c) Xo(n) is

hyperelliptic if

and

_{only if}

d = 3

(except

the

elliptic

_cases

listed under

_b)

or

_if

_q = 2 and n =

(T2

+ T ₊

1)2.

In all these cases the curve

X+(n)

is rational. In

particular,

for

hyperelliptic

Xo(n)

the

_{hyperelliptic}

involution is

_always

the

_full

Atkin-Lehner involution.

Throughout

the _paper we will

freely

use the fact that we can

change

co-ordinates

_by

an affine transformation T N with

_1F9

and _y E

_Fix

This reduces

_drastically

the amount of

_explicit

calculation we will have to

do.

Let T be the Bruhat-Tits tree of

_PGL2(K~).

Its oriented

_edges

are the

cosets where

_J

is the so-called Iwahori

_subgroup.

For

precise

definitions and the connection between Q and T see

[G&#x26;R].

We

only

need some functorial

properties.

The _group

_GL2(K~)

acts from the

_right

on the functions on the oriented

edges

of T

_{by acting}

from the left on the

_argument.

If _{cp is} such a function

and M E

_GL2(K~)

we write _cp o _M for the function which on the oriented

edge

e takes the value

By

we denote the

(finite-dimensional)

R-vector _space of

R-valued, alternating,

harmonic,

ro(n)-invariant

functions on the

edges

of

T, having

finite

_support

modulo

_ro(n).

Then the Atkin-Lehner involutions

act from the

_right

on

_H,

(T,

~"~ .

The

_importance

of for our _purposes lies in the fact that

for _every

_subgroup

G

_{of W(n) the genus}

of the curve

GBXo(n)

equals

the

dimension of the

_subspace

of G-invariant elements

1 -1

If tin and

then is an

injective

linear

map-ping

from

_{H, {T,}

([) into

_{H, (7,}

Defining

the _space of newforms the

_{orthogonal complement}

of the _space of oldforms relative to the so-called Petersson scalar

_product,

one obtains

([G&#x26;N]):

Thus,

for _every

_specific

n the dimension of the _space of newforms can be

calculated

_recursively.

In the

_analogy

between

_Xo(n)

and classical modular curves

(over

C or

a Hecke congruence

_subgroup.

For more

information,

as for

_example

the

(proven!)

analogue

of the famous

_conjecture

of

_Shimura,

_Taniyama,

and

Weil,

see

[G&#x26;R].

The

_following

results are more or less

analogous

to Lemmata

_{25, 26,}

and 27 in

_[A&#x26;L].

non-trivial

_eigenform

_for

wàn)

with

_eigenvalue

I

_represents

W,(3)

then

One

_important

consequence of Lemma 1.2 is:

If S is a

_{subspace of ;}

is stable

under

_{W (n) .}

For I _{= p} a

prime

this results from the fact that

W(n)

can be

generated

by

involutions

_Wm

with

_{~p ~m.}

The

_general

case then is

proved by

iteration.

Especially

we see that the _space of oldforms in

H, (T,

is stable

under

_W(n).

Since the Atkin-Lehner involutions are

self-adjoint

with

re-spect

to the Petersson scalar

_product,

the _space of

_newforms,

_too,

must be stable under

_{W (n) .}

_Taking

S =

Hrew(’T,

_we obtain

Proposition

1.3. The

_subspaces

If

_g(GBXo(n))

= 1 for _some

subgroup

G of

W(n),

then E =

GBXo(n)

is

an

_elliptic

curve defined over K because

Xo(n)

and the Atkin-Lehner

invo-lutions are

_already

defined over K and

Xo (n)

_always

has K-rational

_points.

Generalizing

the definition of

_[M&#x26;SD]

_slightly

we call E an

involutory

elliptic

curve.

E,

being

a

_quotient

of

Xo(n),

has

_{split multiplicative}

reduction at oo.

Moreover,

the _space of G-invariant elements of

_{H, (T,}

is one-dimen-sional.

_{By Proposition}

1.3 its basis vector _’P is contained in one of the

subspaces

As the

_splitting

of into

ain

" °

new and old

_parts

of different level and under the action of the Hecke

_algebra

and the Atkin-Lehner involutions reflects the

_{corresponding decomposition}

of the Jacobian of

_{Xo (n),}

the conductor of E is oo - L

Obviously,

an

involutory elliptic

curve E =

G B Xo (n)

is _a

_strong

Weil

curve if and

_only

if

cond(E)

= 00. n

(i.e.

the

_{corresponding}

_’P is a

_newform)

and

_g(UBXo(n))

&#x3E; 1 for all _proper

_subgroups

U of G.

For more information on how data of

(not

_necessarily

involutory)

_strong

Weil curves are encoded in the

corresponding

newforms see

[Ge3].

Now we

decompose

H,(T,

into its simultaneous

eigenspaces

under

the action of

_W(f).

If V is such an

_eigenspace,

then

by

Lemma 1.2 for

every

possible

choice of a and À the

lifting

(ia

r + iQ )(V)

is contained in

,

a( a simultaneous

W(n)-eigenspace

ofH!(7,JR)ro(n).

From this we see that the _cp E

_H!(7,

_belonging

to an

involutory

elliptic

curve E = _{BJ ,} is contained in a

_subspace

(ia

C)(V)

where

, _{at ’}

-

is a one-dimensional simultaneous

W(f)-eigenspace

of

H!

(T,

IR)rO

(t),

Let G be the _group of all Atkin-Lehner involutions which have

eigenvalue

+1 on V. Then E is

isogenous

over K to the

_involutory

_elliptic

curve E =

GBXo(C).

Since the

_strong

Weil curve in the

_isogeny

class of E

is

_{necessarily involutory,}

we have

proved

Proposition

1.4.

_{Every involutory}

_elliptic

curve

G B Xo (n) is

_isogenous

over K to an

_involutory

_strong

Weil curve

of

conductor oo - I

for

some

[In.

2. CALCULATING THE GENUS OF

_GBXo(n)

We obtain almost the same formula as in the classical

_setting

([Kl]).

where

_#(Wm)

denotes the number

_of

_fixed

_points

_{of Wm}

on

Xo(n),

and 6 is 1

_if

_{q is}

even and 0

if

q is

odd.

Proof.

This is

_essentially

the Hurwitz formula. If q is

odd,

all ramification

of the

_covering

_{Xo(n) --+ GBXo(n)}

is tame. If _q is _even, all ramification is wild but

_by

a theorem of

Gekeler,

based on results of R. Crew and

M.

_Raynaud,

the second ramification _groups are trivial

(compare

[Sch3],

Proposition

7).

0

There are two

_types

of fixed

_points

of

Wm

on

Xo(n),

those which are

cusps and those which are interior

points

(i.e.

in We denote

their

_respective

numbers

by

and

Proposition

2.2. Let

_Wm

be a non-trivial Atkin-Lehner involution

of

Xo(n).

For y E A define

the number- theoretic

_function

a)

If q is

odd we have

Here r denotes the number

of different

prime

divisors

of

~.

b)

If

q is even zve have

otherwise. The

_polynomials

_yl and y2 are understood to be monic.

Proof.

Combine

_Propositions

1 and 9 of

_[Sch3].

0

Proposition

2.3

_([Sch3]).

_{If q}

is

_odd,

the number

_of

interior

_(i.e.,

non-cusp)

fixed

points

of

Wm

on

Xo(n) is

Here a E a

fixed

_non-square, the

_{Legendre symbol,}

and

Lemma 2.4

_([Sch3]).

_Suppose

that _q is even and n = [m with

(1, m)

= 1

and deg(m)

&#x3E; 1.

_{Write m = m2m1}

_{with m1, m2}

_{E Fq [T],}

_{where ml}

_is

square-free.

There exist

_uniquely

determined _{.5, t} E not

_necessarily

_monic,

with m1 = Let

a)

The numbers

_of

interior

_fixed

_points

_of

Wm

on

Xo(m) is

b)

More

_generally,

we have

.

1-In

_particular,

_#t(W~’)

= 0

if m

is _a

square.

c)

If

there exists a _p with

ordp ( C)

&#x3E; +

_1,

then = 0.

d)

If

C is

_square-free

and m is not a _square, then

Example

2.5. Let _q = 2 and _n =

T (T 3

+ T 2

+ 1 ) .

Then

g (X o (n) )

= 6 and

has 5 fixed

_points

and hence =1.

T3+T2+1 + +

We claim that the

_{WT3+T2+1-invaxiant}

form in

_{H, (T,}

is a

new-form. This can be seen as follows:

Since acts as -1

_(and

Wi

as

+1),

&#x3E; it

follows from Lemma 1.2 that

W, (n)

and have

_opposite

_signs

on

follows from Lemma 1.2 that

_Wn

and

_WT

have

_pp

_g

on

So acts as -1 on the oldforms. We conclude that the

involutory

T3+T2+1

elliptic

curve

has conductor oo - +

T~

+

_1).

It is even a

_strong

Weil _curve, for

Xo (T(T

+

T 2

+

₁₎₎

is not

_elliptic.

Furthermore one

_easily

calculates that

has genus 0.

Example

2.6. Now let _q = 2 and n =

T 2 (T 3 +T 2 + 1 );

then

g (X o (n) )

= 13.

Moreover,

WT2

has 2 fixed

_points

_(which

are

cusps)

and

Wn

has 4 but

Lemma 2.4

_{gives only}

a crude upper bound for the number of fixed

points

of

_WT3+T2+1.

_Proposition

2.1 tells us that

_WT3+T2+1

has

_2,

6 or 10 fixed

Using

Lemma 1.2 and the fact that

_Hfew(T,

is

2-dimen-sional,

we see that

H, (7-, R) ro (,)

contains a 6-dimensional

_{subspace of}

_WT2

-invariant oldforms. As

_{g (WT2 X o (n) )}

= 6 we conclude that

_{every WT2}

-invariant element

_(and

a fortiori _every

W(n)-invariant element)

of

H~ (T,

R) ro (n)

must be an oldform.

Now _suppose that + is a

YV(n)-invariant

oldform. Then

, _a(

I cannot be

T3

+

T2 + 1

_(same

reason as in

_Example

_2.5),

so

Moreover,

p must be

WT3+T2+1-invariant,

for otherwise the

lifting

would

have

_opposite

_eigenvalues

under

_WT2

and

_Wn.

_Finally,

the fact that the

WT3+T2+1-invariant

form in

_{H, (T,}

has

_eigenvalue

-1

un-der

_WT

_(compare

_Example

_2.5)

forces À = -1.

Summarizing:

the

_subspace

of

_{W(n)-invariant}

forms in

_{H, (T,}

_{R) ro (n)}

is one-dimensional with basis

_{2T T T3+TZ+1 (SP)}

where _{cp is} the

_{WT3+T2+1-invariant}

form in

H, (T, ~)ro (T(T3+T2+1)) )

discussed in the

previous example.

So

is an

_elliptic

curve with conductor 00. +

T~ +

₁₎

and

_isogenous

over

F2(T)

to the

_strong

Weil curve +

T 2

+

_1)).

Now that we know the number of fixed

_points

of

W 32T2 +T 2+1,

_’ we

T3+T2+1 7

easily

check that none of the curves +

T2

+

₁₎₎

is

_elliptic.

3. THE CASE

_deg(n)

3

According

to Theorem

_1.1,

_deg(n)

= 3 is the smallest

possible

case for

which an

_elliptic

G,Xo(n)

might

exist.

If

_{deg(n) =}

_3,

_{then the genus of}

_{Xo (n)}

is _q if n is

square-free,

and

q -1

if not. We will also

_freely

use the fact that then the full Atkin-Lehner

involution

Wn

acts as -1 on

H, (T,

R) ro (n)

(compare

Theorem 1.1 or

[Ge2]).

Moreover,

since there are no

_oldforms,

_every

_elliptic

_quotient

of

Xo (n)

has

conductor oo . n.

Proposition

3.1. Let

_deg(n)

3.

_Up

to

_{af jirce transformations}

T e

yT + ~3

(’Y, (3

E

_IE’q,

_’Y

₅₄

_0),

the curve

Wm, Xo (n)

is

_{elliptic exactly}

in the

For q

= 2 the _curve

2-isogenous

to the

elliptic

curve

XO(T2(T

+

_1)).

All other curves are

_strong

Weil _curves, and in

fact

these are all

_of

the

_strong

Weil curves

_for

the

_given

_q and n.

The

_equations

_of

the

_strong

Weil curves

for

_q = 2 _are

Proof.

If

_deg(m)

=

1,

then _{gm :=}

g(Wm BXo(n))

=

2 (q

_{+ 1-}

f),

where

f

E

_{~0,1, 2}}

_depends

on _{q, n,} and m

(see

[Ge2],

8.5 and

8.8).

This can also

be derived from the formulas in Section 2. Since = _{g - gm,} n’

there remains to determine all m of

_degree

one with _{g. =1} or _gm =

_{g -1.}

For odd q this

implies

I

q - 1 - 2 (q + 1),

whence

_q

3 or

_{q 5,}

respectively. Similarly

for even

q. The

remaining

possible

values for q and n are then checked

_by

direct

calculation.

The remark on

_strong

Weil curves follows from the table in

[Ge2, p.142~

and the

_equations

for _q = 2 _are

_given

in

[Ge3]

and

(G&#x26;R~.

D

Proposition

3.2.

_Up

to

_{transformations}

T ~

_’YT+(3

with _y,

₍₃

E

_Fq,

-y

~

0,

the 12 curves listed below are the

_{only elliptic}

curves

of

the

form

Here, of

course,

F4

=

F2(~).

The _curves

for

q = 3 _are

_2-isogenous

_to the

elliptic

curves

given

in

Proposition

~. ~. All others are

_strong

Weil curves.

Proof.

If

_GBXo(n)

is

_elliptic

and

_IGI

=

_4,

then _{n must} be _a

product

of 3

different linear

_factors,

because G cannot contain

_Wn.

Fix a simultaneous

eigenbasis

B = for all

and

_{similarly B_+}

and B_ _ .

For

_{odd q}

we know from the

_proof

of

_Proposition

3.1 that

?’

1,

and

_fih

_(because

is

a basis for the elements

having

_{eigenvalue -1}

under

_Wml

IB++I = 1

this leads to _{q -} 3 +

2 ,

and hence

_q

7. For even q a

similar

_argument

_yields

_{q 4.}

D

Proposition

3.3. For

_deg(n)

3 the curve rational in and

only

in the

_following

cases:

a)

deg(n)

2, q

arbitrary,

G any

subgroup

of W (n).

b)

deg(n)

= 3,

q

arbztrary,

G a

subgroups

_of

W(n)

_contaznzng

the

_full

Atkin-Lehner involution

_Wn.

c)

For q

= 2 in addition _{to cases}

a)

and

b),

I),- - -1 ..,. , _ _ . , ,

d)

For q

= 3 in addition _{to cases}

a)

and

b),

Proof.

From Theorem 1.1 it is clear that the curves

_GBXo(n)

in case

_a)

and

b)

are rational.

If

_deg(n)

= 3 and G doesn’t contain

Wn,

by

the _same

theorem,

Xo (n)

must be

_elliptic

or

G

(

=

4,

since the

hyperelliptic

involution is

_unique.

Then

_by

the same

_argument

as in the

_{preceding proofs}

we obtain

q 3.

0

4. DIVIDING BY A SINGLE INVOLUTION

Proposition

4.1. The

_only

rational curve

of

the

form

with

deg(n) &#x3E;

4 is the one with _q = 2 and _n = m =

(T 2

₊ T ₊

1)2.

Proof.

This follows

_immediately

from Theorem 1.1. D

Lemma 4.2. Let G be a

subgroups

of W(n).

a)

If

there exists a

_Q

E

_Fq

with

_{(T - 0) In,}

then

b)

If

there exists a

prime

p E A of degree

2 with p

1 n,

then

Proof.

a)

The curve

Xo (n)

has

_good

reduction mod

(T - {3)

and the

Atkin-Lehner involutions induce non-trivial

_{automorphisms}

of the reduced curve.

Thus

_{Xo (n)}

mod

_{(T -}

_Q)

is a

covering

of

degree

G

(

of a curve of genus

g(GBXo(n)).

Therefore the

_right

side of the

_inequality

is an _upper bound

for the number of rational

_points

of

_{Xo (n)}

rnod

_{(T - /3)}

over the

quadratic

extension of

_{A/(T -}

_{3).

On the other

_hand,

we know from the

proof

of

Lemma 18 in

_[Sch3]

that 2’

+

is a lower bound for the number of these

q+l

rational

_{points, namely:}

there are at least 21 rational _cusps

and $#

interior

q+l

points

coming

from a

special supersingular

Drinfeld module.

b)

is

_{proved similarly.}

D

Lemma 4.3.

a)

If WmBXo(n) is

elliptic

and

_tim

with

_1,

then 1.

b)

If WmBXo(n) is

elliptic

and

_{mi (In}

with

_deg([)

_deg(n),

then

_{WmBXo (1)}

is rational.

Proof.

a)

Suppose

CPl, CP2 E

H,(T,

are

_{linearly independent.}

If

_they

are

_{Wm -invariant,}

then

_by

_taking

a = 1 in Lemma 1.2 we obtain I

two

_{linearly independent}

_Wm-invariant

elements of

_{H, (T,}

n) .

A similar construction works if _cpl and _p2 have

_{eigenvalue -1}

under Wm .

So far we have

proved

2.

By

Theorem 1.1 this

implies

3 or _q =

2, r

=

(T2

₊ T +

1)2.

Hence _~p1 and _w2 are

necessarily

newforms. Therefore the 4 forms

are also

_{linearly independent}

and we obtain a 2-dimensional

_Wm-invariant

subspace

of

_H,

_(T,

(n) .

b)

₎

Assume

_that

_! E

_{r )}

is invariant under m . Then _n,

or both as

space.

... ,

So acts as

identity

on this 2-dimensional

0

Corollary

4.4. and

_{Wm B Xo (n)}

is

_elliptic,

then s 3 and

Proof.

We _may assume Then

_{g(Xo(~1)) G}

1

by

_part

a)

of the

pre-ceding lemma,

and Theorem 1.1

_{implies s -}

1 2 and

_(E

_{ei) -}

₁

3. 0 Now assume that

WmBXo(n)

is

elliptic

and

deg(n) &#x3E;

4.

If q &#x3E; 4,

then

_{by Corollary}

4.4 there exists

_{a /3}

E

_IF9

with

_{(T-(3) In.}

Thus

15 q3

a contradiction.

If _q =

3,

then n =

T 2 (T -1 ) (T

₊

1) (according

to

Corollary

4.4 _up to

translation the

_{only possibility}

with

_{(T3 -}

is excluded

_by

Lemma 4.3

a).

Hence

_{(T3 -}

_{T) In,}

and Lemma 4.2

_implies

_~(n)

120

_(and

conse-quently

d

_4).

The few cases

holding

this condition are excluded

_by

direct

calculation.

So we are left with _{q = 2.}

Applying

4.2, 4.3,

and 4.4

imposes

the

restriction that d = 4 or n is irreducible of

degree

5.

Using

the

_explicit

formulas from Section 2 we obtain

Proposition

4.5.

_Elliptic

curves

of

the

f orm

Wm 1 Xo (n)

with

deg(n) &#x3E;

4

exist

_{only for}

_q = 2. There are 10 such curves

_{given by}

the

_following

values

and their translates under T H T + 1:

The

_first

three curves are

_strong

Weil curves. Their

_equations

over

I~2

(T)

are:

The curves

E4

and

E5

are

_isogenous

over

F2 (T)

to

_respectively

Xo(T(T + 1)2).

To find the

_equations

of the

_strong

Weil curves one

_proceeds

as in Section

4 of

_[Ge3].

This involves the calculation of some Hecke

_operators

on the

quotient

graph

ro (n) ~ T,

which is carried out in

_Chapter

4 of

_~Schl~.

The results are also listed in

[Sch2].

5. DIVIDING BY ALL INVOLUTIONS

is a

W(n)-invariant

_subspace

is invariant under

_W(1)

and i

are two

_{linearly independent}

_{W(n)-invariant}

forms in

Corollary

5.2.

_Suppose

Proo f .

Lemma 5.1 combined with

_Propositions

4.1 and 4.5. 0

Proposition

5.3. For _q &#x3E; 7 there don’t _{exist arcy}

_involutory

_elliptic

curves over

_1Fq(T).

Proof. Suppose

inequality

of Lemma 4.2.

_a)

can be weakened to

and

provided

there exists a

₃

E

_IE9

with

_{(T - ,0) In.}

First we show that for

_{q &#x3E;}

16 there exist no

_involutory

curves. If

T)Xn,

inequality

(*)

shows s 3. If

_applying

Lemma 5.1

_a)

and then

_inequality

_(*)

we obtain s - 1 3. As

_{q &#x3E;}

_{16, again}

(Tq -

and

_{consequently s}

3 must hold for

_general

n.

_Furthermore,

_equation

(**)

implies

qd

q4,

and hence d 3. But this was

_already

16 16

-excluded in

_Propositions

3.1 and 3.2.

For _q E

_{9,11,13}

the same

_argument

yields

d = _s = 4. After aSine

transformation we _may assume n =

T (T -1 ) (T - ,Q) (T - y)

where

0, 1, (3,

~y

are different

and q

is a _square in

Using

Proposition 2.1,

one sees that

g(W(n)BXo(n))

&#x3E; 1.

_(For

the number of fixed

_{points of Wm}

with

_deg(m)

= 3

it suffices to estimate and

_h()

_by

q + 1 +

Likewise,

for _q = 8 the

decomposition

types

(1,1, 1, )

and

(2,1,1)

must

be treated

_by

direct

_computation.

0

Lemma 5.4. Let be such that

_g(W(n)~Xo(n)

1. Then

_(with

d =

deg(n))

_one

of

the

following

assertions must hold:

Proof.

We treat

_only

the case _q = 3 in detail.

First we assume

(T3 - T) ,.rn.

Then s 5

by inequality

(*)

and d 6

by

inequality

(**).

But

_{if d 6,}

then s cannot be

_greater

than 4.

Moreover,

the

_{only possible decomposition}

_type

for n with

_{s = 4,}

namely

(1,1, 2, 2),

is excluded

_by

Lemma 4.2

_a).

Hence s 3 and

_reentering

_(**)

_gives

d 5.

The above discussion

_{already implies s _}

4 for

_general

n.

Thus,

if

(T3

-we can

apply

Lemma 4.2

b)

and obtain

43 -

3d-3

£(n)

1600,

and

consequently

d 5.

The

_proofs

in the other cases are

quite

similar.

For q

= 7 the _cases

3 s d =

4,

which resist Lemma

4.2,

have to be excluded

by

direct

computation.

D

It is even

possible

to

_improve

on some of the

_bounds,

but since we want to

determine all

_{involutory elliptic}

curves _anyway, it’s less work to start from below

_(i.e.

first settle the case d =

4)

and make _use of Lemma 5.1.

Carrying

out all the calculations one obtains the

following

three

propositions.

Proposition

5.5. There are

exactly

9 cases with

deg(n) &#x3E;

4

for

which the curve

_rational,

_namety

(up

to translations T

_{+ ~3):}

The

_only

rational curve

of

the

form

GBXo(n)

where

deg(n) &#x3E;

4 and G is a _proper

_subgroup

of W(n)

is the one with _q =

2,

n =

T(T+1)(T2+T+1)

Proposition

5.6. There are 72

elliptic

curves

of

the

form

with

_{deg(n) &#x3E;}

_{4, namely}

_(up

to

_affine

_{transformations}

T f-t

_-yT

+ ,Q):

As usual

_{F4 =1F2}

_(1?).

The number I denotes the

_length

_of

the orbit

_of

n

under

_affine

_{transformations.}

Proposition

5.7. A

_complete

list

_of

all

_elliptic

curves E = where

deg(n) &#x3E;

4 and G is a _proper

subgroup of

W(n)

is

given

below. Each

horizontal box consists

_of

one orbit under

affine trans f ormations

T H

_’YT

+

0.

One sees that all the

_elliptic

curves with conductor 00 . n listed in

Propo-sitions 5.6 and 5.7 are

_strong

Weil curves.

The calculations

_(compare

_Example

_2.6)

also show that

_{for q}

= 2 and

n =

T(T

₊

1) (T2

₊ T ₊

1)

the curve

(WT, Wn) B Xo (n)

is

_isogenous

over

F2 (T)

to the

_strong

Weil curve

WT B Xo (T (T2 + T +

1)).

The other three

involutory

curves over

F2(T)

with conductor are

_isogenous

For _{q =} 3 and n =

T 2 (T -1 ) (T

+

1)

the curve

W(n) ) Xo(n)

is

isoge-nous to +

_1)).

The last curve in the table of

Proposition

5.7 is

_isogenous

to

_(WT(T-1),

_19)).

For all other curves listed it is clear to which

_strong

Weil curve

_they

are

isogenous,

either

_{by translating}

T or because there is

only

one

_involutory

strong

Weil curve of the

_given

conductor.

Acknowledgements.

The

_author,

_being

_{supported by}

CICMA in form of

a

_{post-doc position}

at Concordia

University

and McGill

University,

wishes

to _express his

_gratitude

to all three institutions.

REFERENCES

[A&#x26;L] A. O. Atkin and J. _Lehner, Hecke Operators on _03930(m), Math. Annalen 185 _(1970),

[Ge1] E.-U. Gekeler, Drinfeld-Moduln und modulare Formen über rationalen

Funktio-nenkörpem, Bonner Mathematische Schriften 119 _(1980).

[Ge2] E.-U. _{Gekeler, Automorphe} Formen über _{Fq (T)} mit kleinem Führer, Abh. Math. Sem. Univ. _Hamburg 55 _(1985), 111-146.

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Springer LNM _601, Berlin _Heidelberg New York 1977, 239-246.

[M&#x26;SD] B. Mazur and P. _{Swinnerton-Dyer,} Arithmetic of Weil Curves, Invent. Math. 25 _(1974), 1-61.

[Sch1] A. Schweizer, Zur Arithmetik der _{Drinfeld’schen} Modulkurven _{X0(n), Dissertation,} Saarbrücken 1996

[Sch2] A. Schweizer, Modular automorphisms of the _Drinfeld modular curves X0(n), Collect.

Math. 48 _(1997), 209-216.

[Sch3] A. Schweizer, Hyperelliptic Drinfeld Modular _Curves, in: Drinfeld _modules, modular schemes and applications, Proceedings of a _workshop at Alden _{Biesen, September 9-14,}

1996, (E.-U. Gekeler, M. van der _Put, M. _Reversat, J. Van _{Geel, eds.),} World Scientific, Singapore, 1997, pp. 330-343

Andreas SCHWEIZER

Dept. of Mathematics Concordia _University

1455 Boulevard de Maisonneuve Ouest

Montreal, Quebec H3G 1M8 Canada

E-mail : schWeizmcicma. concordia. ca

Dept. of Mathematics McGill _University 805 Sherbrooke West

Montreal, Quebec H3A 2K6 Canada