On the mod p reduction of Fredholm

On the mod p reduction of Fredholm
determinants for modular forms
John Bergdall
Monday, Feb. 2 at 4:15 pm
111 Cummington Mall, MCS B21
Tea and cookies in MCS 144 at 3:45 pm
Fix a prime p. In this talk, we will discuss the p-adic properties of
the coefficients of the characteristic power series of Up acting on spaces of
overconvergent p-adic modular forms. These coefficients are, by a theorem
of Coleman, power series in the weight variable over Zp . Our first goal will
be to show that in tame level one, the simplest case, every coefficient is nonzero mod p and then to give some idea of the (finitely many) roots of each
coefficient. The second goal will be to explain how the previous result fails
in higher levels, along with possible salvages. This will include revisiting the
tame level one case. The progress we’ve made gives examples of recent work
being made elsewhere on the geometric structure of the eigencurve “near its
boundary”. This is joint work with Rob Pollack.