Quasimodular forms arising from Jacobi's theta function and special symmetric polynomials

TL;DR

This paper constructs new quasimodular forms from Jacobi's theta function, connects them to symmetric polynomials related to numerical semigroups, and proves conjectures about these polynomials using geometric invariants.

Contribution

It introduces a novel sequence of quasimodular forms derived from theta functions and links them to symmetric polynomials in the context of algebraic and geometric structures.

Findings

Constructed a sequence of quasimodular forms of all weights from minimal input.

Resolved two conjectures regarding symmetric polynomials associated with numerical semigroups.

Connected symmetric polynomials to the $ ext{A}$-genus of spin manifolds.

Abstract

Ramanujan derived a sequence of even weight $2n$ quasimodular forms $U_{2n}(q)$ from derivatives of Jacobi's weight $3/2$ theta function. Using the generating function for this sequence, one can construct sequences of quasimodular forms of all nonnegative integer weights with minimal input: a weight 1 modular form and a power series $F(X)$. Using the weight 1 form $\theta(q)^2$ and $F(X)=\exp(X/2)$, we obtain a sequence $\{Y_n(q)\}$ of weight $n$ quasimodular forms on $\Gamma_0(4)$ whose symmetric function avatars $\widetilde{Y}_n(\pmb{x}^k)$ are the symmetric polynomials $T_n(\pmb{x}^k)$ that arise naturally in the study of syzygies of numerical semigroups. With this information, we settle two conjectures about the $T_n(\pmb{x}^k).$ Finally, we note that these polynomials are systematically given in terms of the Borel-Hirzebruch $\widehat{A}$-genus for spin manifolds, where one identifies power sum symmetric functions $p_i$ with Pontryagin classes.