Congruences involving $\binom{2k}k^2\binom{3k}km^{-k}$

TL;DR

This paper proves conjectures by Zhi-Wei Sun on congruences involving binomial coefficients and elliptic curves, extending known results to prime moduli and using complex multiplication theory.

Contribution

It provides a proof of Sun's conjectures on binomial coefficient sums modulo prime squares, utilizing elliptic curve theory and complex multiplication.

Findings

Confirmed conjectures on binomial sum congruences modulo p^2.

Connected binomial sums with elliptic curve properties.

Extended previous results to a broader class of primes.

Abstract

Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper, based on the work of Brillhart and Morton, 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 $\sum_{k=0}^{p-1}\binom{2k}k^2\binom{3k}km^{-k}\mod {p^2}$.