Dissertations, Theses, and Capstone Projects
Date of Degree
6-2016
Document Type
Dissertation
Degree Name
Ph.D.
Program
Mathematics
Advisor
Krzysztof Klosin
Committee Members
Krzysztof Klosin
Ken Kramer
Brooke Feigon
Subject Categories
Algebra | Number Theory
Keywords
p-adic modular forms
Abstract
A major theme in the theory of $p$-adic deformations of automorphic forms is how $p$-adic $L$-functions over eigenvarieties relate to the geometry of these eigenvarieties. In this talk we explain results in this vein for the ordinary part of the eigencurve (i.e. Hida families). We address how Taylor expansions of one variable $p$-adic $L$-functions varying over families can detect geometric phenomena: crossing components of a certain intersection multiplicity and ramification over the weight space. Our methods involve proving a converse to a result of Vatsal relating congruences between eigenforms to their algebraic special $L$-values and then $p$-adically interpolating congruences using formal models. These methods should extend to the entire eigencurve.
Recommended Citation
Kramer-Miller, Joseph, "p-adic L-functions and the Geometry of Hida Families" (2016). CUNY Academic Works.
https://academicworks.cuny.edu/gc_etds/1325