PhD Defense Dino Destefano

Title: Investigating slopes of overconvergent modular forms

We study the slopes of the Atkin’s U operator acting on overconvergent p-adic modular forms.  In the case of tame level 1 and for p =5,7,13; we compute a quadratic lower bound for the Newton polygon of U.  The methods of proof are explicit and rely on a certain deformation of the U operator and its characteristic power series.

This gives us the possibility to compute the smallest possible slope for p=5,7 and to prove necessary and sufficient conditions on the weight such that the dimension of the cuspidal space is one.  This result allows us to exhibit some p-adic analytic families of modular forms in the framework of Coleman’s theory.

We then formulate a conjecture that would allow us to extend our analysis to all the congruence classes modulo p−1.

Supervisor: Prof. Ian Kiming, Math, University of Copenhagen

Assessment committee:

Ass. Prof. Fabien Mehdi Pazuki (Chairman), MATH, University of Copenhagen

Prof. Matthias Schütt, Universität Hannover  

Prof. Gabor Wiese, University Luxembourg