LOW-LYING ZEROS IN FAMILIES OF HOLOMORPHIC CUSP FORMS: THE WEIGHT ASPECT

Journal article, 2022

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 phi satisfying the condition supp((phi) over cap) subset of (-2, 2). We refine their density result by uncovering lower-order terms that exhibit a sharp transition when the support of (phi) over cap reaches the point 1. In particular, the first of these terms involves the quantity (phi) over cap (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((phi) over cap) subset of (-1, 1) we obtain an unconditional estimate which is significantly more precise than the prediction of the L-functions ratios conjecture.