p-adic L-functions and the Geometry of Hida Families

<p><br />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.</p>

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Mathematics

Grantor

The Graduate School and University Center of The City University of New York

Year dc:date.available

2016

Author dc:creator

• Kramer-Miller, Joseph

Advisor dc:contributor.advisor

• Krzysztof Klosin

Committee members dc:contributor.committeemember

• Krzysztof Klosin
• Ken Kramer
• Brooke Feigon

• Algebra
• Number Theory
• p-adic modular forms

Repository record dc:identifier

https://academicworks.cuny.edu/gc_etds/1325

OAI identifier oai:identifier

oai:academicworks.cuny.edu:gc_etds-2341