Kummer congruences arising from the mirror symmetry of an elliptic curve

TL;DR

This paper explores Kummer congruences related to elliptic curves, demonstrating that certain generating functions are quasimodular and satisfy specific congruences, with implications for p-adic convergence.

Contribution

It establishes Kummer-type congruences for the family of generating functions associated with elliptic curves and analyzes their p-adic properties.

Findings

Generating functions $A_n( au)$ satisfy Kummer-type congruences.

For prime p, $A_n$ coefficients p-adically converge to zero at specific n.

The results connect mirror symmetry, quasimodularity, and p-adic analysis.

Abstract

In the genus 1 case, mirror symmetry reduces to the statement that a certain family of generating functions, relating to an elliptic curve, are quasimodular. In their proof of this fact, Kaneko and Zagier used a related family of generating functions $A_n(\tau)$, which they show to be quasimodular. We show that these $A_n$'s also satisfy Kummer-type congruences. Additionally, we show that for a prime $p$, the $p$th power coefficients of $A_n$ $p$-adically converge to zero, for specific values of $n$.