Date of Degree

6-2016

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor(s)

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.

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