Arithmetic statistics of modular symbols

• 成果类型: Article
• 署名作者: Petridis, Yiannis N.; Risager, Morten S.
• 署名单位: University of London; University College London; University of Copenhagen
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-017-0784-7
• 发表日期: 2018
• 页码: 997-1053
• 关键词: fourier coefficients; forms; sums

摘要

Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve L-functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by modular symbols. Another of their conjectures predicts the Gaussian distribution of normalized modular symbols ordered according to the size of the denominator of the cusps. We prove this conjecture in a refined version that also allows restrictions on the location of the cusps.