报告时间：2021年3月17日（星期三）10:30-11:30

报告地点：腾讯会议 ID：743 305 326 密码：1234  腾讯会议链接https://meeting.tencent.com/s/7h8Ews1aVgBV

报告人: 王六权(武汉大学)

报告摘要: Atkin and Garvan introduced the functions $N_k(n)$ and $M_k(n)$, which denote the $k$-th moments of ranks and cranks in the theory of partitions.

Let $e_{2r}(n)$ be the $n$-th Fourier coefficient of $E_{2r}(\tau)/\eta(\tau)$, where $E_{2r}(\tau)$ is the classical Eisenstein series of weight $2r$ and $\eta(\tau)$ is the Dedekind eta function. Via the theory of quasi-modular forms, we find that for $k \leq 5$, $N_k(n)$ and $M_k(n)$ can be expressed using $e_{2r}(n)$ ($0\leq r \leq k$), $p(n)$ and $N_2(n)$. For $k>5$, additional functions are required for such expressions. For $r\in \{2, 3, 4, 5, 7\}$, by studying the action of Hecke operators on $E_{2r}(\tau)/\eta(\tau)$, we provide explicit congruences modulo arbitrary powers of primes for $e_{2r}(n)$. Moreover, for $\ell \in \{5, 7, 11, 13\}$ and any $k\geq 1$, we present uniform methods for finding nice representations for $\sum_{n=0}^\infty e_{2r}\left(\frac{\ell^{k}n+1}{24}\right)q^n$, which work for every $r\geq 2$.  These representations allow us to prove congruences modulo powers of $\ell$, and we have done so for $e_4(n)$ and $e_6(n)$ as examples. Based on the congruences satisfied by $e_{2r}(n)$, we establish  congruences modulo arbitrary powers of $\ell$ for the  moments and symmetrized moments of ranks and cranks as well as higher order $\spt$-functions.

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邀请人：毛仁荣