A Kaneko-Zagier equation for Jacobi forms

Abstract: The Kaneko-Zagier equation is a second order differential equation depending on a parameter k which gives rise to an infinite family of modular forms as solutions. These solutions are closely related to Weierstrass p function, which becomes clear by considering the inverse (under composition) of a suitably normalized generating series of the solutions for integer values of k. In this talk, we study an analogue of the Kaneko-Zagier differential equation for Jacobi forms. We point to three features of the infinite family of solutions. First of all, the solutions are quasi-Jacobi forms, and we determine their transformation under the Jacobi group. Secondly, the inverse of a suitable normalized generating series of these solutions is again a well-known function, namely a ratio of theta functions. Finally, a special feature of the solutions is the polynomial dependence of the index parameter. (Joint with Georg Oberdieck and Aaron Pixton)

number theory

Audience: researchers in the topic

Japan Europe Number Theory Exchange Seminar

Series comments: The purpose of the Japan Europe Number Theory Exchange Seminar is to give a opportunity for researchers in Japan and Europe to exchange their research projects by giving short talks (30 min). The target audience are researchers of any level in the area of number theory.

Start for Fall 2021: 26th October