综合报告会：The distribution of zeros of modular forms

中国科学院数学与系统科学研究院

数学研究所

数学科学全国重点实验室

综合报告会

Speaker: Zeev Rudnick (Tel Aviv University and Shandong University）

Language: English

Title: The distribution of zeros of modular forms

Time & Venue: 2026年9月4日（星期五） 15:30-17:00 南楼N204
Abstract: I will discuss old and new results about the distribution of zeros of modular forms, and relation to Quantum Unique Ergodicity. It is known that a modular form of weight k has about k/12 zeros in the fundamental domain. A classical question in the analytic theory of modular forms is “can we locate the zeros of a distinguished family of modular forms?”. In 1970, F. Rankin and Swinnerton-Dyer proved that the zeros of the Eisenstein series all lie on the circular part of the boundary of the fundamental domain. In the beginning of this century, I discovered that for cuspidal Hecke eigenforms, the picture is very different — the zeros are not localized, and in fact become uniformly distributed in the fundamental domain. Very recently, we have investigated other families of modular forms, such as the Miller basis (ZR 2024, Roei Raveh 2025, Adi Zilka 2026), Poincaré series (RA Rankin 1982, Noam Kimmel 2025) and theta functions (Roei Raveh 2026), finding a variety of possible distributions of the zeroes.

报告人简介: Prof. Zeev Rudnick 是以色列特拉维夫大学Cissie and Aaron Beare数论讲席教授。他于1990在Ilya Piatetski-Shapiro和 Roger Evans Howe教授指导下获得博士学位。他曾获得Sloan奖学金，于2001年获得Erdős奖，于2012年当选为美国数学学会会士，于2014年应邀在韩国首尔举行的国际数学家大会上作报告。Rudnick教授的研究领域非常广泛，主要关心量子混沌中的问题，以及函数域中的解析数论问题。他长期担任Geometric and Functional Analysis、International Mathematics Research Notices等国际顶尖期刊的主编。