BIMSA >
BIMSA Lecture
A canonical splitting of the Hodge filtration and applications to p-adic L-functions (2)
A canonical splitting of the Hodge filtration and applications to p-adic L-functions (2)
Organizer
Speaker
Daniel Kriz
Time
Monday, January 26, 2026 1:30 PM - 2:30 PM
Venue
A6-101
Online
Zoom 815 762 8413
(BIMSA)
Abstract
This is a two-part talk. In the first part, I will go over a construction of a functorial splitting of the p-adic Hodge filtration on first universal de Rham cohomology on Shimura curves which is defined over a large open neighborhood of the infinite-level Shimura curve and specializes to the unit root splitting on the ordinary locus. The construction of this splitting involves defining new period sheaves via completing OB_{dR} along ideals generated by explicit p-adic periods coming from the geometry of the tower of Shimura curves. Using these sheaves and this splitting, one can formulate new theories of p-adic modular forms and define differential operators acting on them which specialize to Katz's theory on the ordinary locus. In the second part of the talk, I will go over applications of these objects to constructions of Katz-type, Bertolini-Darmon-Prasanna-type and Liu-Zhang-Zhang-type p-adic L-functions for p inert or ramified in the CM field K, generalizing and unifying my and Andreatta-Iovita's previous work. I will also discuss special value formulas for these p-adic L-functions and their arithmetic applications, including Sylvester's conjecture on primes expressible as sums of rational cubes, showing 100% of positive squarefree integers congruent to 5,6,7 mod 8 are congruent numbers, and results toward Goldfeld's conjecture.