Romik's Conjecture for the Jacobi Theta Function

TL;DR

This paper investigates the Taylor coefficients of the Jacobi theta function at a special point, proving a refined periodicity property modulo primes for certain coefficients using algebraic relations and p-adic factorials.

Contribution

It refines previous results by establishing an explicit algebraic relation for coefficients modulo primes congruent to 1 mod 4, leading to effective periodicity bounds.

Findings

Proved periodicity of Taylor coefficients modulo primes p ≡ 1 (mod 4).

Derived algebraic relations involving p-adic factorials for these coefficients.

Extended understanding of Romik's conjecture through explicit algebraic formulas.

Abstract

Dan Romik recently considered the Taylor coefficients of the Jacobi theta function around the complex multiplication point $i$. He then conjectured that the Taylor coefficients $d(n)$ either vanish or are periodic modulo any prime ${p}$; this was proved by the combined efforts of Scherer and Guerzhoy-Mertens-Rolen, who considered arbitrary half integral weight modular forms. We refine previous work for $p \equiv 1 \pmod{4}$ by displaying a concise algebraic relation between $d\left( n+ \frac{p-1}{2} \right)$ and $d(n)$ related to the $p$-adic factorial, from which we can deduce periodicity with an effective period.