Kummer Congruences and Formal Groups

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...

Kummer Type Congruences and Stickelberger Subideals

Universal Kummer Congruences Mod Prime Powers

Bounds of Divided Universal Bernoulli Numbers and Universal Kummer Congruences

Let p be a prime. We obtain 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. As an application, we give a simple proof of Clarke’s 1989 universal von Staudt theorem. We also establish the universal Kummer congruences modulo p for the divided universal Bernoulli numbers for the case (p − 1)|n, which is a new result.

Searching for Kummer congruences in an infinite slope family

We consider powers of a grossencharacter, the corresponding Lfunctions twisted with quadratic Dirichlet characters, and their central critical values. We state several conjectures concerning Kummer-type congruences between these numbers for a ramified prime and describe specific numerical data in support of these conjectures.