Congruence properties of Borcherds product exponents

TL;DR

This paper investigates the congruence properties of Borcherds product exponents, linking them to modular forms and representations, extending understanding of their arithmetic behavior.

Contribution

It establishes a connection between the exponents of Borcherds products and modular representations of the logarithmic derivative of Hilbert class polynomials, advancing the theory of modular form coefficients.

Findings

Identifies congruence relations of $A(n,d)$ modulo primes.

Relates exponents to modular representations.

Provides new insights into the arithmetic of Borcherds product exponents.

Abstract

In his striking 1995 paper, Borcherds found an infinite product expansion for certain modular forms with CM divisors. In particular, this applies to the Hilbert class polynomial of discriminant $-d$ evaluated at the modular $j$-function. Among a number of powerful generalizations of Borcherds' work, Zagier made an analogous statement for twisted versions of this polynomial. He proves that the exponents of these product expansions, $A(n,d)$, are the coefficients of certain special half-integral weight modular forms. We study the congruence properties of $A(n,d)$ modulo a prime $\ell$ by relating it to a modular representation of the logarithmic derivative of the Hilbert class polynomial.