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The non-vanishing of central values of automorphic \(L\)-functions and Landau-Siegel zeros. (English) Zbl 0992.11037

This paper is a survey and announcement of results, the complete proofs of which are being prepared for publication elsewhere. The authors consider the positivity (even with a given positive lower bound) for the central values of certain families of automorphic \(L\)-functions and the highly interesting relation of this problem with the existence of exceptional zeros for quadratic \(L\)-functions. The families of forms in question are:
(i) the holomorphic cusp forms of even weight \(k\leq K\) for the full modular group that are Hecke eigenfunctions,
(ii) the holomorphic cusp forms of fixed even weight that are newforms for the congruence group \(\Gamma _0(N)\) as \(N\) varies over squarefree positive integers and tends to infinity.
In both cases, for at least one half of the related \(L\)-functions, a certain positive lower bound can be shown, but unfortunately a slightly bigger frequency would be needed for either family in order to eliminate the exceptional zeros. True, in the \(N\)-aspect, the percentage exceeds fifty in mean, but this fact does not entail the same striking consequence as the same hypothetical property for each large \(N\).

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11M20 Real zeros of \(L(s, \chi)\); results on \(L(1, \chi)\)
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