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고려대학교 수학과 세미나 



Distribution of Hecke eigenvalues: large discrepancy 

Vertical SatoTatetheorem 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 $k+Nto infty$. 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 probabilitymeasure $mu_{p,k}$ 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+epsilon}). ] 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 NaserTalebizadehSardariand Simon Marshall. Texas A&M 







