Congruences involving ${\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^2\left(\genfrac{}{}{0ex}{}{4k}{2k}\right)m^{-k}$

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 ${\sum }_{k=0}^{p-1}{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^2\left(\genfrac{}{}{0ex}{}{4k}{2k}\right)m^{-k}\mathrm{mod}{p}^2$.

14 pages

References in corpus (3)

• Open Conjectures on Congruences
• Point counting on reductions of CM elliptic curves
• On sums involving products of three binomial coefficients

Cited by in corpus (1)

• Jacobsthal sums, Legendre polynomials and binary quadratic forms