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In the genus 1 case, mirror symmetry reduces to the statement that a certainfamily of generating functions, relating to an elliptic curve, arequasimodular. In their proof of this fact, Kaneko and Zagier used a relatedfamily 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 $n$.

Kummer congruences arising from the mirror symmetry of an elliptic curve