Kummer congruences arising from the mirror symmetry of an elliptic curve

2014/02/05 by Adele Lopez, Lopez, Adele
Mathematics · #11F30 #11F33 (Primary) 14J33 (Secondary) #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11F30 #msc:11F33 #msc:14J33

paper · pdf · doi:10.48550/arxiv.1402.1116

arxiv created 2014/02/05 · arxiv updated 2014/02/06

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(τ), 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 pth power coefficients of A_n p-adically converge to zero, for specific values of n.

Related

• p-adic congruences in iterated derivatives of the Weierstrass elliptic function
• Classification of p-adic functions satisfying Kummer type congruences
• On Power Sums and the Kummer Congruences
• Explicit Kummer Theory for Elliptic Curves
• Congruences between Heegner points and quadratic twists of elliptic curves