It is a classical observation of Serre that the Hecke algebra acts locally nilpotently on the graded ring of modular forms modulo 2 for the full modular group. Here we consider the problem of classifying spaces of modular forms for which this phenomenon continues to hold. We give a number of consequences of this investigation as they relate to quadratic forms, partition functions, and central values of twisted modular L-functions.
In a previous article, the authors showed that the L-function attached to a ?-adic .-sheaf is meromorphic in a certain disk depending on the convergence condition of the .-sheaf. That disk is in general best possible. The purpose of the present article is to show that for an affine complete intersection, either the L-function or its reciprocal is actually analytic (i.e., without poles) in the same disk.
1997Academic Press
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