Searching for Kummer congruences in an infinite slope family

The Andrews–Sellers family of partition congruences

In 1994, James Sellers conjectured an infinite family of Ramanujan type congruences for 2-colored Frobenius partitions introduced by George E. Andrews. These congruences arise modulo powers of 5. In 2002 Dennis Eichhorn and Sellers were able to settle the conjecture for powers up to 4. In this article, we prove Sellers' conjecture for all powers of 5. In addition, we discuss why the Andrews-Sel...

The Ramanujan differential operator, a certain CM elliptic curve and Kummer congruences

Let τ be a point in the upper half-plane such that the elliptic curve corresponding to τ can be defined over Q, and let f be a modular form on the full modular group with rational Fourier coefficients. By applying the Ramanujan differential operator D to f , we obtain a family of modular forms Dlf . In this paper we study the behavior of Dl(f)(τ) modulo the powers of a prime p > 3. We show that...

Congruences between Selmer groups ∗

The study of congruences between arithmetically interesting numbers has a long history and plays important roles in several areas of number theory. Examples of such congruences include the Kummer congruences between Bernoulli numbers and congruences between coefficients of modular forms. Many of these congruences could be interpreted as congruences between special values of L-functions of arith...

The Universal Kummer Congruences

Let p be a prime. In this paper, we present detailed p-adic analysis to factorials and double factorials and their congruences. We give good bounds for the p-adic sizes of the coefficients of the divided universal Bernoulli number B̂n n when n is divisible by p−1. Using these we then establish the universal Kummer congruences modulo powers of a prime p for the divided universal Bernoulli numbers...

Universal Kummer Congruences Mod Prime Powers