Romik's Conjecture for the Jacobi Theta Function

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 $p\equiv 1(\mathrm{mod}4)$ 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.

15 pages