Subject: Re: L(f,1), Mestre-Oesterle
From: Loic Merel <[email-masked]>
To: [email-masked]
Cc: [email-masked]
Date: Tue, 22 Sep 1998 16:19:49 +0100

Dear William,
Many thanks for your tables. I already found the previous versions
you sent to me useful. I used them to compute the points of X_0(43)
over the cyclotomic field Q(\mu_43). I mentioned this during a
conference this summer in Crete.

About my paper "Universal...", there are a few mistakes there
(one reason other than my own fault, is the fact that that I never
saw the proofs). Some of them are insignificant. One you have to
be careful about is the fact that  the parameterization of the cusps
of X_1(N) is wrong. It is easy to fix it anyway. One should replace
the set I called P_N by the disjoint union of the (Z/gcd(d,N)Z)^* where
d runs through Z/NZ. Then everything works mutatis mutandis.

You accurately pointed out another mistake. In fact my definition
of \epsilon'_t is not compatible with the property of the matrix T
given just above. Of course the definition you suggested is the
correct one.

> I noticed while doing these computations that for prime level p>=73,
> there always seems to be an eigenform f in S_2(Gamma_0(N)) such that
> L(f,1)=0.  Do you know of any theorems of the shape:
>   "for all N>=N_0 the Jacobian J_0(N) has positive (analytic) rank"

Here is a theorem (or at any rate a consequence of the conjecture of BSD):
For N prime, J_0(N) has finitely many rational points if and
only if N divides the order of the Monster (a.k.a. the Fischer-Griesz
group, a.k.a. the largest sporadic group).
You can ask Andy Ogg about this. In fact this is simply the fact that
these values of N (which includes all N>73) coincide with those for
which the genus of the curve X_0(N)/W_N is zero. When this genus
is nonzero there exists a newform with minus sign for the functional
equation, in that case the L-function of that modular form vanishes,
hence infinitely many points.

> Nonetheless, in about half the cases I looked at, the guess was correct.
> For example, for all prime conductors between 79 and 199.

Could you be more precise? Are there values of N modulo 12
for which the guess is correct? My guess was of course very rough;
I thought that maybe by weighting the sum slightly one might get
the right answer.

> If we could find an analogue of the winding element in terms of the
> supersingular basis, would that be a good step towards answering this
> question.  What is known along these lines?

This is exactly the problem: finding the winding element in terms
of the supersingular basis. I asked the question to Barry Mazur,
who directed me, not to a paper of Gross, but to a less known
article of Klingenberg. I'll try to study this text as soon as
I'll have time to do so.

Loic