On the order of vanishing of modular $L$-functions at the critical point

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

H

ENRYK

I

WANIEC

On the order of vanishing of modular L-functions

at the critical point

Journal de Théorie des Nombres de Bordeaux, tome

2, n

o

_{2 (1990),}

p. 365-376

<http://www.numdam.org/item?id=JTNB_1990__2_2_365_0>

© Université Bordeaux 1, 1990, tous droits réservés.

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

On

the order of

_vanishing

of

modular

L-functions

at

the

critical

_point

par HENRYK IWANIEC

1. Introduction

The

_nonvanishing

of L-functions at

_{special points}

is an attractive area

of research in

_contemporary

number

_theory,

see

[7]-[11].

One

_example

is the

_{Rankin-Selberg}

zeta-function

_{L( f ~}

associated with a

holo-morphic

cusp form

f

of

weight

2 and Maass cusp forms _gj of

eigenvalue

Aj

= In this case the

_nonvanishing

of

L( f

₀

_{at s = sj}

plays

a role in the work of R.

Phillips

and P.Sarnak

[6]

on deformations of

groups and was

proved

to be true for

_infinitely

_{many cusp}

_{forms 9j}

_by

J.-M.

Deshouillers and H. Iwaniec

_[3].

Another

_example

is the

Birch-Swinnerton-Dyer conjecture

which asserts _{that the rank of the group of rational}

_points

on an

elliptic

curve E defined

over Q

is

equal

to the order of

vanishing

of

the associated Hasse-Weil L-function

_{L(s, E~}

at s = 1

(the

_center of the

critical

_strip).

Recently

V.A.

_Kolyvagin

_[4]

has

_proved

that the _group of rational

_points

on a modular

elliptic

curve .E is finite 0 and that the L-function

L(s,

E,

~~)

twisted

_by

a suitable real character _x~ has

simple

zero at s = 1.

The latter condition was

subsequently proved

to hold true for

_infinitely

many discriminants d

by

D.

Bump,

S.

Friedberg

and J. Hoffstein

[2]

and

independently

by

K.

_Murty

and R.

_Murty

_[5].

In these notes we establish

(from scratch)

quantitative

results on

_{Kolyvagin’s}

condition.

2 - Statement of results

Let E be a modular

elliptic

curve defined

over Q

and

be the Hasse-Weil L-function associated with E. Thus

is a _cusp form of

_weight

2 which is a newform of level

N,

where N is the

conductor of E. The L-function is entire and it satisfies the functional

equation

where w = ~1. We are interested in curves E for which

L(1, E~ ~

_0,

so

the functional

_equation

holds with the

_sign

w = 1. The twisted L-function

where Xd

is a real

primitive

character to modulus d

_prime

to N is also entire and it satisfies the functional

_equation

with the

_sign

Wd =

wxd(-N).

In the

sequel

we let d range over the set

D =

{d :

0 d - for some v

prime

to

_4N}

and we let be the Kronecker

symbol.

Thus if d is

_squarefree

Xd is the

primitive

character to the modulus d which is associated with

the

_imaginary

_quadratic

field

_{Every prime}

_dividing

N

_splits

in

~(~~.

Moreover we have

_{w,~ _ -1,}

so

by

(1)

it follows that

Our aim is to _prove that

_{E, Xd)}

has a

_simple

zero at s =

1,

i.e.

0 for

_infinitely

many d in D. To this end we shall evaluate

two sums of

_type

and

where ¿b

means that the summation is restricted to

_squarefree

numbers and F is a smooth

function, compactly supported

in

R+

with

_positive

mean

THEOREM. For _any c &#x3E; 0 and ~’ &#x3E; 1 we have

and

with some constants

_{a ~}

0 and

_Q

which

_depend

on the curve E and the

test function F.

COROLLARY.

_Suppose

E _&#x3E; 0 and Y _&#x3E;

c(e).

Then 0 for at

Ieast Y2~‘~-f real

_primitive

characters _Xd to modulus d E

D,

d Y.

3. Estimates for the coefficients of

_f

The Fourier coefficients a~ of the cusp form

f

are

_{multiplicative.}

More

exactly,

for Re s &#x3E;

_3/2

we have the Euler

product

with _ap = = 0

if piN

and

lapl

=

1,3pi

=

pl /2

In the

latter case the result was

proved by

M. Eichler and P.

Deligne.

It

yields

the

following

bound for the coefficient an

(known

as the

Ramanujan

conjecture)

where denotes the divisor

_function,

_T(n)

« nF. This bound can be

slightly

improved

on _average.

Indeed,

arguing

as G.

Hardy

and E. Hecke

with Parseval’s formula and

_using

the boundedness of

_{y f (z~}

we

_get

Similarly

we

_get

for _any real a and M &#x3E;

2,

the

implied

constant

depending

on

f only.

In

LEMMA I. Let a be real

and V)

be a

periodic

function of

period

r. We then

have

where

Moreover,

if

_{11/’1 (}

1 and s is a

positive

integer

then we have

and

PROOF: The sum on the left-hand side of

₍₁₁₎

is

_equal

to

I I

whence the

_inequality

₍₁₁₎

follows

_by

_(10).

If 1 we obtain T

r’/2.

by Cauchy’s inequality.

For the

_proof

of

₍₁₂₎

we can assume that

(r, s)

= 1

by

changing 0

suitably.

Then we

_apply

₍₁₁₎

for in

_place

of where

xo is the

principal

character to the modulus s. We obtain

which

_gives

_(12).

_Finally

we derive

(13)

from

(12).

The sum on the

Hence

₍₁₃₎

follows

_by

_(8).

4.

_Approximate

formulas for

_{L’(1, E, Xd)}

We shall express

L’(1,

E,

Xd)

in terms of the

rapidly

convergent

sums

where V is the

_incomplete

_{gamma function defined}

_by

We have

Moving

the

_integration

to the line Re s =

-3~4

we _pass a

simple pole

at s = 0 with residuum

L’(1,

E,

Xd)

by

virtue of

(2).

On the other hand

the

_integral

over the line Res =

-3/4

is

equal

to

-.A(d2 NX -~ ,

Xd)

by

the

functional

_equation

_(1).

This

_gives

for _any r1’ &#x3E; 0 and d in D which is

_squarefree.

In

_particular

we have

By

(9)

we infer

trivially

that

A(X,

Xd)

«

~~ ~2

for

any X

&#x3E; 0 and

_inserting

this to

₍₁₄₎

we obtain

5. Estimation of the fourth moment of

_~’~l,

_E,

_Xd)

By

the

_large

sieve

_inequality

_(see

_[1])

_together

with

₍₈₎

we

_get

On the other hand

_by

₍₁₄₎

we have for _any

D,

d

Y,

d

squarefree

that

Combining

both results we infer the _upper bound

(5)

for

S4(Y).

6. An

_approximate

formula for the first moment of

_L’(1,

_E,

_X,~~

By

(15)

we obtain

Now we relax the condition that d is

squarefree by

introducing

the factor

then we

_split

the sum

_according

to whether a A or a &#x3E; A and in the latter case we return to

_squarefree

numbers

_by

_extracting

square

divisors of

a-2 d.

We obtain

_{51 (Y)}

=

S +

R,

_say, where

and

Here A is a

large

number to be chosen later. In the term

_{A(X, Xh2d)}

with X =

b2dVN

we return to

L’(1,

E,

by reversing

the

_arguments

as follows

the second line

_being

obtained

_by

₍₇₎

and the third line

_by

_(16).

_Finally

applying

(5)

and the H61der

_inequality

we conclude that

7. A transformation of S

It remains to evaluate S. For

_{(a, 4N)}

= 1 and

d E

D we have

Every

n can be written

uniquely

as the

product

n = where k has

prime

factors in

_4N,

£m is

_prime

to 4lV and m is

_squarefree.

For n written

this _way and d in D we have =

Xd(M)

subject

_to

(d,~) ==

1. The last

condition is detected

_by

the familiar formula of M6blus

_giving

Next, by

means of Gauss sums we write

where Em, = 1 if m =

_{1 (mod 4),}

_c~ = i if

m - -1 (mod 4)

and 4N4N -=

1(

mod

_m).

This

_gives

where

We

_{put A}

=

min(1/2,

and

split

S =

So+S, +S2,

where

SO,

Sl,

S2

denote the

_partial

sums restricted

_by

the conditions r =

_{0, 0}

irl

Am,

8. Estimates for

_S2

and

_S,

LEMMA 2.

_Suppose

_g(x)

is a smooth and

integrable

function on R with

derivatives

_g(j)(x)

«

_(Ixl

+

X)-J

for

_{all j &#x3E;}

1 the

_implied

constant

de-pending

on j only.

Suppose

a is real

and q is

a

positive integer

such that

aq is not an

integer.

We then have

for

_{any j &#x3E;}

_2,

the

_implied

constant

_depending

_{on j only.}

PROOF:

_By

Poisson’s formula the sum is

equal

to

where denotes the Fourier transform of

_9(x).

We have

g(y)

«

_by

the

_partial

_{integration j}

_times,

whence

₍₁₈₎

follows

by

trivial summation over u.

To estimate

_S2

we sum over d first

by

an

appeal

to

_(18).

For

_{any j &#x3E;}

2

we

_get

«

(n

₊

Y)-3,

whence

S2

« 1.

d

To estimate

₅₁

we sum over m first

using

(13)

and

partial

summation

together

with the relation

and then we sum over r

trivially

getting

9. Evaluation of

_So

Since r = 0 we have = 0 for all m &#x3E; 1 and the terms with

m = 1

yield

where

We

split

the summation over d into residue classes modulo 4N. Each class

contributes

for

_{any j &#x3E;}

_2,

and the number of relevant classes is

Hence

where

and

with

To evaluate the series we

appeal

to

analytic properties

of the

The

_required

_properties

are inherited from the

properties

of the

Rankin-Selberg

zeta-function

--The

_{Rankin-Selberg}

zeta-function is

_meromorphic

on

C,

holomorphic

on

Re s &#x3E; 1

_except

for a

simple pole

at s = 2 with residuum

and it satisfies a functional

equation

which connects

H(s)

with

H(2 -

s).

Moreover,

as shown

by

G. Shimura

[12]

the function

is entire.

_By

the

_{Phragmen-Lindelof}

_priuciple,

_using

the functional

equa-tion,

it follows that

Since

_L(s)

_{agrees with}

_{L(2s -}

_{1, sym2~/~(4’s -}

₂₎

=

H(2s)~~(2s - 1)

up to

an Euler

product

P(s),

_say, which converges

_absolutely

in Re s &#x3E;

_3/4

we

conclude that

_L(s)

is

_holomorphic

in Re s &#x3E;

_3/4,

it satisfies

and that

Now

_by

the contour

_integration

we

_get

by

the

_expansion

_{r(s) = s-l - ’/}

+ ...,

where y

is the Euler constant.

with

and

10. Evaluation of the first moment of

_L’(1,

_E,

_Xd).

Conclusion

Collecting

the established evaluations we infer that

which

_gives

₍₆₎

on

_taking

A =

Y’~I ~4.

REFERENCES

[1]

E. _BOMBIERI, Le _grand Crible dans la Théorie Analytique des _{Nombres, Astérique}

18

_(1973).

[2]

D. _BUMP, S. FRIEDBERG and J. _HOFFSTEIN, Eisenstein series on the

meta-plectic group and non-vanishing theorems for automorphic L-functions and their

derivatives. Ann. Math. 131

_(1990),

53-127.

[3]

J.-M. DESHOUILLERS and H. _IWANIEC, The _{non-vanishing of Rankin-Selberg}

zeta-functions at _{special points,} AMS Contemporary Mathematics Vol. 53

_(1986),

51-95.

[4]

V.A. _KOLYVAGIN, Finateness _of

_E(Q)

and

_III(E,

_Q)

_for a subclass of Weil

curves. Math. USSR Izv.

[5]

K. MURTY and R. _MURTY, Mean values _of derivatives _of modular L-series.

_(to

appear in Ann.

Math.). (See

also K. MURTY, Non-vanishing of L-functions and

their derivatives in _Automorphic Forms and Analytic Number _Theory,

_(edited

_by R.

_Murty),

CRM Publications, Montreal 1990,

89-113).

[6]

R.S. PHILLIPS and P. SARNAK, On _{cusp forms for co-finite subgroups of}

PSL(2,

R).

Invent. Math. 80

_(1985),

339-364.

[7]

D. _ROHRLICH, On _{L-functions of elliptic} curves and _{anticyclotomic} towers.

Invent. Math. 75

_(1984),

383-408.

[8]

D. _ROHRLICH, On _{L-functions of elliptic} curves and _cyclotomic towers. Invent. Math. 75

_(1984),

409-423.

[9]

D. _{ROIIRLICH, L-functions} and division towers. Math. Ann.

_281(1988),

611-632.

[10]

D. _{ROHRLICH, Non-vanishing of L-functions for}

_GL(2).

Invent. Math. 97

(1989),

381-403.

[11]

D. _ROHRLICH, The _{vanishing of} certain _{Rankin-Selberg convolutions,} in

Auto-morphic Forms and _Analytic Number _Theory.

_(edited

_by R.

_Murty)

CRM

Publi-cations, Montreal _1990, 123-133.

[12]

G. _SHIMURA, On modular _{forms of half-integral weight.} Ann. Math. 97

_(1973),

440-481.

Department of Mathematics I

Rutgers University New Brunswick