Hecke equivariance of generalized Borcherds products of type $O(2,1)$

TL;DR

This paper proves that generalized Borcherds' lifting operators of type O(2,1) are Hecke equivariant, leading to new relations and congruences for twisted traces of singular moduli and class numbers.

Contribution

It establishes Hecke equivariance of the generalized Borcherds' lifting operator and the logarithmic differential operator for modular forms of type O(2,1).

Findings

Hecke equivariance of the Borcherds' lifting operator under extended Hecke operators.

Hecke equivariance of the logarithmic differential operator.

Derivation of relations for twisted traces and class numbers modulo primes.

Abstract

Recently, a weak converse theorem for Borcherds' lifting operator of type $O(2,1)$ for $\G_0(N)$ is proved and the logarithmic derivative of a modular form for $\G_0(N)$ is explicitly described in terms of the values of Niebur-Poincar\'e series at its divisors in the complex upper half-plane. In this paper, we prove that the generalized Borcherds' lifting operator of type $O(2,1)$ is Hecke equivariant under the extension of Guerzhoy's multiplicative Hecke operator on the integral weight meromorphic modular forms and the Hecke operator on half-integral weight vector-valued harmonic weak Maass forms. Additionally, we show that the logarithmic differential operator is also Hecke equivariant under the multiplicative Hecke operator and the Hecke operator on integral weight meromorphic modular forms. As applications of Hecke equivariance of the two operators, we obtain relations for twisted traces of singular moduli modulo prime powers and congruences for twisted class numbers modulo primes, including those associated to genus $1$ modular curves.