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Information Center for Mathematical Science

세미나

Information Center for Mathematical Science

세미나

고려대학교 수학과 세미나
Title Distribution of Hecke eigenvalues: large discrepancy
Date 2019-01-07
Speaker 정준혁(Texas A&M)
Sponsors 2019-01-07
Host KOREA University
Place 아산이학관 526호
Abstract Vertical Sato-Tatetheore​m for holomorphic modular forms concerns the distribution of eigenvaluesof a fixed Heckeoperator $T_p$acting on the space of weight $k$ and level $N$ modular forms, as $k+N\to \infty$. Itwas proven by Serre (and independently by Sarnak)that there exists a limiting measure $\mu_p$, which depends only on $p$, such thatthe eigenvalues become equidistributed​ relatively to $\mu_p$. Fix $N$ for simplicity. Then this can berestated in terms of the discrepancy between two measures: a probabilitymeas​ure $\mu_{p,k}$ supported on the eigenvalues of the Heckeoperator, and $\mu_p$, i.e., it is equivalent to $D(\mu_{p,k}, \mu_p) \to0$. Regarding the rate of convergence, in the context of arithmetic quantumchaos, it was suggested both by speculation and numerical test that \[ D(\mu_{p​,k}, \mu_p) =O(k^{-1/2+\eps​ilon}). \] In this talk, I'm going to disprove thisby showing that \[ D(\mu_{p,​k}, \mu_p) =\Omega(k^{-1/3​}\log^2 k). \] This is a joint work with NaserTalebizade​hSardariand Simon Marshall. Texas A&M