Explicit Forms and Proofs of Zagier's Rank Three Examples for Nahm's Problem

主讲人 Speaker:Wang Liuquan (Wuhan University)
时间 Time:Tues.,15:00-16:00,Dec.6,2022
地点 Venue:Tencent Meeting ID: 245869045


Let $r\geq 1$ be a positive integer, $A$ a real positive definite symmetric $r\times r$ rational matrix, $B$ a rational vector of length $r$, and $C$ a rational scalar. Nahm's problem is to find all triples $(A,B,C)$ such that the $r$-fold $q$-hypergeometric series

$$f_{A,B,C}(q):=\sum_{n=(n_1,\dots,n_r)^\mathrm{T}\in (\mathbb{Z}_{\geq 0})^r} \frac{q^{\frac{1}{2}n^\mathrm{T} An+n^\mathrm{T} B+C}}{(q;q)_{n_1}\cdots (q;q)_{n_r}}$$

becomes a modular form, and we call such $(A,B,C)$ a modular triple. When the rank $r=3$, after extensive computer searches, Zagier provided twelve sets of conjectural modular triples and proved three of them. We prove a number of Rogers-Ramanujan type identities involving triple sums. These identities give modular form representations for and thereby verify all of Zagier's rank three examples. In particular, we prove a conjectural identity of Zagier. 



王六权博士,2014年本科毕业于浙江大学,2017年博士毕业于新加坡国立大学,现为武汉大学教授。他主要从事数论与组合数学领域的研究,研究课题多集中在q-级数、整数分拆、特殊函数、模形式等方面。迄今在《Advances in Mathematics》, 《Transactions of the American Mathematical Society》、《Advances in Applied Mathematics》、《Journal of Number Theory》、《Ramanujan Journal》等期刊上发表论文40多篇,先后主持国家自然科学基金青年基金和面上项目各一项,2021年入选国家级青年人才计划。