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Journal Articles Quarterly Journal of Mathematics Year : 2022

Low-lying zeros in families of holomorphic cusp forms: the weight aspect

Abstract

Abstract We study low-lying zeros of L-functions attached to holomorphic cusp forms of level 1 and large even weight. In this family, the Katz–Sarnak heuristic with orthogonal symmetry type was established in the work of Iwaniec, Luo and Sarnak for test functions ϕ satisfying the condition supp$(\widehat \phi) \subset(-2,2)$. We refine their density result by uncovering lower-order terms that exhibit a sharp transition when the support of $\widehat \phi$ reaches the point 1. In particular, the first of these terms involves the quantity $\widehat \phi(1)$ which appeared in the previous work of Fouvry–Iwaniec and Rudnick in symplectic families. Our approach involves a careful analysis of the Petersson formula and circumvents the assumption of the Generalized Riemann Hypothesis (GRH) for higher-degree automorphic L-functions. Finally, when supp$(\widehat \phi)\subset (-1,1)$ we obtain an unconditional estimate which is significantly more precise than the prediction of the L-functions ratios conjecture.

Dates and versions

hal-04008929 , version 1 (28-02-2023)

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Cite

Lucile Devin, Daniel Fiorilli, Anders Södergren. Low-lying zeros in families of holomorphic cusp forms: the weight aspect. Quarterly Journal of Mathematics, 2022, 73 (4), pp.1403-1426. ⟨10.1093/qmath/haac010⟩. ⟨hal-04008929⟩
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