Dan Fretwell

Paramodular Eisenstein congruences of local origin

Let tau(n) be the coefficient of q^n in the power series expansion of q*prod((1-q^n)^24) and let sigma_11(n) be the 11th power divisor sum. A remarkable observation of Ramanujan is that tau(n) = sigma_11(n) mod 691 for all n. It wasn't until many years later that this was proved using the theory of modular forms.

Various generalizations of Ramanujan's congruence have since been found to exist between the Hecke eigenvalues of certain cusp forms and Eisenstein series of the same level, modulo primes dividing numerators of Bernoulli numbers and their variants. One can even go further and prove the existence of Eisenstein congruences between level 1 Eisenstein series and level p cusp forms, where now the moduli have the possibility of coming from divisibility of an Euler factor. Such congruences are said to be of "local origin".

In this talk I will briefly conjecture the existence of local origin congruences for paramodular forms of level p (Siegel modular forms for the paramodular group) and sketch the proof of a necessary condition for a paramodular newform to satisfy such a congruence.