Congruences involving $\binom{2k}k^2\binom{4k}{2k}m^{-k}$

TL;DR

This paper proves conjectures by Zhi-Wei Sun related to congruences involving binomial coefficient sums modulo prime squares, using elliptic curve theory and complex multiplication techniques.

Contribution

It introduces new proofs of Sun's conjectures on binomial sum congruences using elliptic curve methods and complex multiplication, extending previous combinatorial results.

Findings

Confirmed several of Sun's conjectures on binomial sum congruences.

Established new congruence relations involving binomial coefficients and elliptic curves.

Extended the understanding of binomial sums modulo prime squares.

Abstract

Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper, by using the work of Ishii and Deuring's theorem for elliptic curves with complex multiplication we solve some conjectures of Zhi-Wei Sun concerning $\sum_{k=0}^{p-1}\binom{2k}k^2\binom{4k}{2k}m^{-k}\mod {p^2}$.