Arithmetic properties of meromorphic modular forms

Abstract: While investigating the Doi-Naganuma lift, Zagier studied certain cusp forms f_{k,d} of weight 2k associated to positive discriminants d. These cusp forms also appear prominently in the kernel function for the Shimura-Shintani correspondence. Moreover, Kohnen and Zagier showed that they have rational periods and geodesic cycle integrals. The natural generalization of the function f_{k,d} to negative discriminants d yields a meromorphic modular form with poles at the CM points of discriminant d. Together with C. Alfes-Neumann, K. Bringmann, S. Löbrich, and J. Males, we showed that these meromorphic modular forms have interesting arithmetic properties, too. Indeed, they have rational periods and cycle integrals, and integral Fourier coefficients which satisfy strong divisibility conditions. Moreover, their Fourier coefficients are non-vanishing and have very regular sign changes. If time permits, we will also discuss a surprising relation with the coefficients of the modular j-invariant and the partition function.

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