Curve counting and modular forms: elliptic curve case
Time: Thu, 4 Apr 2019, 13:30-15:05
Place: First conference room at Jinchunyuan West Building (近春园西楼)
Speaker: Jie Zhou (YMSC, Tsinghua University)
Abstract:
In this talk, I will start by a gentle introduction of Gromov-Witten theory which roughly is a theory of the enumeration of holomorphic maps from complex curves to a fixed target space, focusing on the elliptic curve (as the target space) example. Then I will explain some ingredients from mirror symmetry, as well as a Hodge-theoretic description of quasi-modular and modular forms and their relations to periods of elliptic curves. After that I will show how to prove the enumeration of holomorphic maps are related to modular and quasi-modular forms, following the approach developed by Yefeng Shen and myself. Finally I will discuss the Taylor expansions near elliptic points of the resulting quasi-modular forms and their enumerative meanings. If time permits, I will also talk about some interesting works by Candelas-de la Ossa-Rodriguez-Villegas regarding the counting of points on and the counting of holomorphic maps to elliptic curves over finite fields.