# Periodicities for Taylor coefficients of half-integral weight modular forms

## TL;DR

This paper proves Romik's conjecture on the periodicity of Taylor coefficients of the Jacobi theta function at  and extends this periodicity phenomenon to a broader class of half-integral weight modular forms.

## Contribution

The paper confirms Romik's conjecture and generalizes the periodicity of Taylor coefficients to a wider class of half-integral weight modular forms.

## Key findings

- Romik's conjecture on the theta function is proven.
- Periodicities in Taylor coefficients are established for a broad class of modular forms.
- The results deepen understanding of the arithmetic properties of modular form expansions.

## Abstract

Congruences of Fourier coefficients of modular forms have long been an object of central study. By comparison, the arithmetic of other expansions of modular forms, in particular Taylor expansions around points in the upper-half plane, has been much less studied. Recently, Romik made a conjecture about the periodicity of coefficients around $\tau=i$ of the classical Jacobi theta function. Here, we prove this conjecture and generalize the phenomenon observed by Romik to a general class of modular forms of half-integral weight.