On the modulo $p$ zeros of modular forms congruent to theta series

TL;DR

This paper investigates the zeros of certain modular forms related to theta series, revealing their algebraic properties modulo primes and their connection to hypergeometric functions.

Contribution

It introduces theta modular forms for the full modular group and analyzes the algebraic and congruence properties of their zeros modulo primes.

Findings

Zeros are in the ground field modulo p for Jacobi theta series

Zeros are at most quadratic over the ground field for hexagonal lattice theta series

Zeros are congruent to zeros of truncated hypergeometric functions

Abstract

For a prime $p$ larger than $7$, the Eisenstein series of weight $p-1$ has some remarkable congruence properties modulo $p$. Those imply, for example, that the $j$-invariants of its zeros (which are known to be real algebraic numbers in the interval $[0,1728]$), are at most quadratic over the field with $p$ elements and are congruent modulo $p$ to the zeros of a certain truncated hypergeometric series. In this paper we introduce "theta modular forms" of weight $k \geq 4$ for the full modular group as the modular forms for which the first dim$(M_k)$ Fourier coefficients are identical to certain theta series. We consider these theta modular forms for both the Jacobi theta series and the theta series of the hexagonal lattice. We show that the $j$-invariant of the zeros of the theta modular forms for the Jacobi theta series are modulo $p$ all in the ground field with $p$ elements. For the theta modular form of the hexagonal lattice we show that its zeros are at most quadratic over the ground field with $p$ elements. Furthermore, we show that these zeros in both cases are congruent to the zeros of certain truncated hypergeometric functions.