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Weighted Distribution of Low-lying Zeros of GL(2)  $L$ -functions

  • Andrew Knightly (a1) and Caroline Reno (a1)
Abstract

We show that if the zeros of an automorphic $L$ -function are weighted by the central value of the $L$ -function or a quadratic imaginary base change, then for certain families of holomorphic GL(2) newforms, it has the effect of changing the distribution type of low-lying zeros from orthogonal to symplectic, for test functions whose Fourier transforms have sufficiently restricted support. However, if the $L$ -value is twisted by a nontrivial quadratic character, the distribution type remains orthogonal. The proofs involve two vertical equidistribution results for Hecke eigenvalues weighted by central twisted $L$ -values. One of these is due to Feigon and Whitehouse, and the other is new and involves an asymmetric probability measure that has not appeared in previous equidistribution results for GL(2).

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This work was partially supported by grant #317659 from the Simons Foundation to author A. K. Section 3 is based in part on the University of Maine MA thesis of author C. R.

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