Date

Speaker

Title

January 9th

Shrenik Shah (Columbia)

Class number formulae for some Shimura varieties of low dimension
The class number formula connects the residue of the Dedekind zeta function at s=1 to the regulator, which measures the
covolume of the lattice generated by logarithms of units. Beilinson defined a generalized regulator morphism and
conjectural class number formula in the “motivic” setting. His formula provides arithmetic meaning to the orders of
“trivial” zeroes of Lfunctions at integer points as well as the value of the first nonzero derivative at these points.
We study this conjecture for the middle degree cohomology of the Shimura varieties associated to unitary groups of
signature (2,1) and (2,2) over Q. We construct explicit “BeilinsonFlach elements” in the motivic cohomology of
these varieties and compute their regulator. This is joint work with Aaron Pollack. ( Hide
Abstract)

January 16th

No seminar


January 23rd

Sean Howe (Stanford)

Classicality, MayerVietoris, and deRham localglobal
compatibilty Coleman's classicality theorem says that an overconvergent modular
form of weight k is classical if it has slope <k1. We explain how to recover this theorem using the admissible
representation theory of GL_2 and a MayerVietoris sequence for the de Rham cohomology of the tower of modular curves
which separates the contributions of the ordinary and supersingular loci. Time permitting, we will also explain
connections with work of BreuilEmerton and Saito and make a bold conjecture about the de Rham cohomology of the
LubinTate tower.
( Hide
Abstract)

January 30th

Liang Xiao (Connecticut)

Cycles on the special fiber of some Shimura varieties
and Tate conjecture We describe the
irreducible components of the basic locus of Shimura varieties of Hodge type at
a place with good reduction, when the basic locus is of middle dimension. Under
certain genericity condition, we show that they generate the Tate classes of the
special fiber of the Shimura varieties. This is a joint work with Xinwen Zhu.
( Hide
Abstract)

February 6th

Lue Pan (Princeton)

Fontaine–Mazur conjecture in the residually reducible case.
We prove the modularity of some twodimensional residually
reducible padic Galois representations
over Q under certain hypothesis on the residual representation at p.
To do this, we generalize Emerton’s
localglobal compatibility and devise a patching argument for
completed homology in this setting.
( Hide
Abstract)

February 13th

Lucia Mocz (Princeton)

A new Northcott property for Faltings
heights The Faltings height is
a useful invariant for addressing questions in arithmetic geometry. In his
celebrated proof of the Mordell and Shafarevich conjectures, Faltings shows the
Faltings height satisfies a certain Northcott property, which allows him to
deduce his finiteness statements. In this work we prove a new Northcott property
for the Faltings height. Namely we show, assuming the Colmez Conjecture and the
Artin Conjecture, that there are finitely many CM abelian varieties of a fixed
dimension which have bounded Faltings height. The technique developed uses new
tools from integral padic Hodge theory to study the variation of Faltings
height within an isogeny class of CM abelian varieties. In special cases, we are
able to use these techniques to moreover develop new Colmeztype formulas for
the Faltings height. ( Hide
Abstract) 
February 20th

Karl Schaefer (UChicago)

Unramified pExtensions of Kummer Fields via Galois Representations
In their paper "On the Ramification of Hecke Algebras at Eisenstein Primes",
Calegari and Emerton prove a relationship between Merel's number and the rank of the ppart of the class
group of Q(N^1/p) for primes N = 1 mod p. We answer a question of CalegariEmerton on this rank and give
an effective method for computing this rank in the case p = 5. Our results are obtained by relating the
existence of Galois representations satisfying certain local conditions to generalizations of Merel's
number. This is joint work with Eric Stubley. ( Hide
Abstract) 
February 27th

Brandon Levin (Arizona)

Models for Galois deformation rings
An important input into modularity lifting theorems is an understanding of the
geometry of Galois deformation rings, especially local deformation rings with
padic Hodge theory conditions at l = p. Outside of a few cases (ordinary,
FontaineLaffaille,...), the detailed structure of these deformation rings is
very mysterious. I will introduce models which have the same singularities in
generic situations as those of a class of potentially crystalline deformation
rings. I will then discuss applications to the BreuilMézard conjecture and the
weight part of Serre's conjecture. This is joint work in progress with Daniel
Le, Bao V. Le Hung, and Stefano Morra. ( Hide
Abstract)

March 6th

TBC

