math.NT — Number Theory

Sun's supercongruence for an Apéry-like sequence

Contributed by Oleksiy Babanskyy

Let $a_n=\sum_{j=0}^n\binom{n}{j}^2\binom{2j}{j}$. We prove that, for every prime $p>3$, the sum $(-1)^{(p-1)/2}\sum_{n=0}^{p-1}\binom{2n}{n}a_n/54^n$ is congruent modulo $p^2$ to $4x^2-2p$ when $p=x^2+3y^2\equiv1\pmod3$, and to zero when $p\equiv2\pmod3$. For the latter primes a quadratic differential invariant of the truncated generating polynomial lifts Sun's congruence modulo $p$. For the former primes we certify two modular periods directly by identities of weight eight, construct a Hecke polynomial with an explicit degree bound, and specialize a finite polynomial identity at a complex multiplication point. The coefficient at degree $p$ and both parity cases of the complex multiplication matrices are treated explicitly. A corresponding evaluation of the Franel sum with denominator $50$ follows by the same arguments.