David Spencer

Local origin congruences of modular forms

Modular forms are fundamental objects in modern number theory. They have Fourier expansions which encode interesting arithmetic data (e.g. mod p points on elliptic curves, representation of integers by quadratic forms, etc.). We often find that the Fourier coefficients satisfy congruences with modulus being a prime dividing some zeta value. A famous example is that of the Ramanujan 691 congruence. In this talk I will extend results of Dummigan/Fretwell (on a generalisation of the Ramanujan congruence) and investigate congruences between level N Eisenstein series and level Np cusp forms for various weights. We will see how the moduli in this case now come from Euler factors of Dirichlet L-functions (so give congruences of "local origin"). I will also, time permitting, discuss how this result carries over into the case of weight 1.