# Schneider-Siegel theorem for a family of values of a harmonic weak Maass form at Hecke orbits

## TL;DR

This paper extends classical results on the transcendence of special values of the modular j-invariant to a family of values of harmonic weak Maass forms evaluated at Hecke orbits, showing they are transcendental for infinitely many primes under certain conditions.

## Contribution

It generalizes Schneider and Siegel's theorem to harmonic weak Maass forms on Hecke orbits, establishing transcendence of their values at algebraic points for infinitely many primes.

## Key findings

- Values of harmonic weak Maass forms at Hecke orbits are transcendental for infinitely many primes.
- The result applies to a broad class of harmonic weak Maass forms with algebraic principal parts.
- Transcendence holds under mild assumptions on the form and for primes outside a specific finite set.

## Abstract

Let $j(z)$ be the modular $j$-invariant function. Let $\tau$ be an algebraic number in the complex upper half plane $\mathbb{H}$. It was proved by Schneider and Siegel that if $\tau$ is not a CM point, i.e., $[\mathbb{Q}(\tau):\mathbb{Q}]\neq2$, then $j(\tau)$ is transcendental. Let $f$ be a harmonic weak Maass form of weight $0$ on $\Gamma_0(N)$. In this paper, we consider an extension of the results of Schneider and Siegel to a family of values of $f$ on Hecke orbits of $\tau$. For a positive integer $m$, let $T_m$ denote the $m$-th Hecke operator. Suppose that the coefficients of the principal part of $f$ at the cusp $i \infty$ are algebraic, and that $f$ has its poles only at cusps equivalent to $i \infty$. We prove, under a mild assumption on $f$, that for any fixed $\tau$, if $N$ is a prime such that $ N\geq 23 \text{ and } N \not \in \{23, 29, 31, 41, 47, 59, 71\},$ then $f(T_m.\tau)$ are transcendental for infinitely many positive integers $m$ prime to $N$.