Simple zeros of $\operatorname{GL}(2)$ $L$-functions

Alexandre de Faveri

doi:10.4171/jems/1559

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Simple zeros of $GL (2)$ $L$ -functions

Alexandre de Faveri
Stanford University, Stanford, USA

$Simple zeros of $\operatorname{GL}(2)$ $L$-functions cover$

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Abstract

Let $f \in S_{k} (Γ_{1} (N))$ be a primitive holomorphic form of arbitrary weight $k$ and level $N$ . We show that the completed $L$ -function of $f$ has $Ω (T^{δ})$ simple zeros with imaginary part in $[- T, T]$ , for any $δ < 2/27$ . This is the first power bound in this problem for $f$ of non-trivial level, where previously the best results were $Ω (lo g lo g lo g T)$ for $N$ odd, due to Booker, Milinovich, and Ng, and infinitely many simple zeros for $N$ even, due to Booker. In addition, for $f$ of trivial level ( $N = 1$ ), we also improve an old result of Conrey and Ghosh on the number of simple zeros.

Cite this article

Alexandre de Faveri, Simple zeros of $GL (2)$ $L$ -functions. J. Eur. Math. Soc. (2024), published online first

DOI 10.4171/JEMS/1559