Date: Sun, 22 Aug 1999 16:11:21 -0400 (EDT)
From: Dick Gross <gross@abel.MATH.HARVARD.EDU>
To: was@math.berkeley.edu
Subject: Re: Mestre-Oesterle module

You can find this in a paper I published with Kudla, in Compositio, on the
special
value of triple product L-functions. I don't have the precise reference at
home,
but I think it is around 1990.

The criterion works for all eigenforms of weight 2, not just for those coming
from elliptic curves, although we assume the level N is squarefree. However, it
is
NOT an if and only if statement for L(f,1) = 0. What is true is that the sum of
the
cubes vanishes if and only if the triple product L-function L(fxfxf,s) vanishes
at
s=2. Since L(fxfxf,2) = L(Sym^3(f),2)*L(f,1)^2, the vanishing of L(f,1) implies
the vanishing of the sum of the cubes. But not necessarily the other way... In
particular, Merel's question is (as far as I know) still open.

I hope this helps with your example (which sounds pretty spectacular).

Dick Gross