PhD Defense Bryan W Advocaat
Title: Explicit Overconvergence Rates Related to Eisenstein Series
Abstract:
In this thesis, we study explicit overconvergence rates related to Eisenstein series. We start by providing the necessary theoretical background on the theory of overconvergent modular forms. This includes the theory of Katz expansions, which is the main tool in the rest of the thesis to deduce overconvergence rates. Our main object will then be the family of modular functions E∗κ/V (E∗κ), which is derived from the (p-stabilized) Eisenstein series. These functions appear naturally when one moves between different weights of overconvergent modular forms and hence a good understanding of them is crucial to the
entire theory. In particular, their overconvergence rates show up when studying the slopes (i.e. the valuations of eigenvalues of a p-adic Hecke-operator) which have been the object of a lot of research.
The first result will then be, assuming p ≥ 5, an explicit bound on its overconvergence rate, involving a on p-depending constant. To prove this, we introduce the notion of a “formal Katz expansion”, which is an interpolation of the regular Katz expansions. A technical argument involving valuations of Vandermonde matrices will then allow us to deduce bounds on the overconvergence rates. We show that these rates are not optimal and give some improved bounds in certain specific cases. We comment on its relation to a conjecture made by Coleman, and on its connections to statements, such as the Halo conjecture. We furthermore demonstrate why it would be desirable to know the exact overconvergence rate, which prompts the remaining part of the thesis.
In the final chapter, we give a computation counterpart to these statements. We provide (again under the assumption that p ≥ 5) two algorithms. The first one allows us to compute the Katz expansion of an overconvergent modular form (given its q-expansion as input). The second algorithm, the more important one, uses the first algorithm to compute valuations of terms appearing in the formal Katz expansion. As these valuations are key in bounding the overconvergence rates of ∗ κ /V (E ∗ κ), this algorithm gives us insight in these rates. Based on data obtained through this algorithm, we provide a conjecture that would directly imply an improved bound on the overconvergence rate as proved in the previous chapter. The correctness of both algorithms is proved as well.
Advisors: Ian Kiming, Math, University of Copenhagen and
Gabor Wiese, Math, University of Luxembourg
Assessment Committee -
Professor Chair Fabien Pazuki, Math, University of Copenhagen
Professor, Sergei Merkulov, University of Luxembourg
Professor Mladen Dimitrov, University of Lille, France