Date

Speaker

Title

January 5

Jack Shotton (UChicago)

Local deformation rings when l is not equal to p
Given a mod p representation of the absolute Galois group of Q_l , consider the universal framed deformation ring R parametrising its lifts. When l and p are distinct I will explain a relation between the mod p geometry of R and the mod p representation theory of GL_n(Z_l), that is parallel to
the BreuilMézard conjecture in the l = p case. I will give examples and say something about the proof, which uses automorphy lifting techniques.
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January 12

Jeff Achter (Colorado State)

On descending cohomology geometrically
Mazur has drawn attention to the question of determining when the
cohomology of a smooth, projective variety over a number field can be modeled by an abelian variety. I will discuss recent work with
CasalainaMartin and Vial which constructs such a "phantom" abelian variety for varieties with maximal geometric coniveau. In the special case of cohomology in degree three, we show that the image of the (complex) AbelJacobi map admits a distinguished model over the base field, and that an algebraic correspondence realizes this descended intermediate Jacobian as a phantom.
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January 19

Melanie Wood (UWMadison)

The CohenLenstra Heuristics and and Random Groups
We will introduce the CohenLenstra Heuristics that conjecturally give the distribution of class groups of imaginary quadratic fields, and discuss features of this probability distribution on finite abelian groups motivating this conjecture. In particular, we will explain a new theorem that this distribution is ``universal'' (in the sense that the Central Limit Theorem shows that the normal distribution is universal). Further we will explain nonabelian analogs of these conjectures and the related random nonabelian groups, based on conjectures of Boston, Bush, and Hajir and joint work with Boston, including evidence for these nonabelian analogs in the function field case.
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January 26

Yunqing Tang (Harvard)

Algebraic solutions of differential equations over the projective line minus three points
The Grothendieck–Katz pcurvature conjecture predicts that an arithmetic differential equation whose reduction modulo p has vanishing pcurvatures for almost all p, has finite monodromy. It is known that it suffices to prove the conjecture for differential equations on the projective line minus three points. In this talk, I will first focus on this case and introduce a padic convergence condition, which would hold if the pcurvature is defined and vanishes. Using the algebraicity criteria established by Andr\'e, Bost, and ChambertLoir, I will prove a variant of this conjecture for the projective line minus three points, which asserts that if the equation satisfies the above convergence condition for all p, then its monodromy is trivial. I will also prove a similar variant of the pcurvature conjecture for a certain elliptic curve with jinvariant 1728 minus its identity point.
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February 2

Bao V. Le Hung (UChicago)

Some computations with potentially crystalline deformation rings
We explain how to explicitly compute some potentially crystalline deformation ring for three dimensional local Galois representation, and explain some applications. This is joint work in progress with D.Le, B.Levin and S.Morra.
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February 9

Keerthi Madapusi Pera (UChicago)

On the average height of abelian varieties with CM
In the 90s, generalizing the classical ChowlaSelberg formula, P. Colmez formulated a conjectural formula for the Faltings heights of abelian varieties with multiplication by the ring of integers in a CM field, which expresses them in terms of logarithmic derivatives at 1 of certain Artin Lfunctions. Using ideas of Gross, he also proved his conjecture for abelian CM extensions. In this talk, I will explain a proof of Colmez's conjecture in the average for an arbitrary CM field. This is joint work with F. Andreatta, E. Goren and B. Howard.
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February 16

Yifeng Liu (Northwestern)

Bad reduction of Hilbert modular varieties and application to
BlochKato conjecture
In this talk, we will study the reduction of Shimura varieties
attached to certain quaternion algebras at some ramified prime. We explain
how the global structure of the bad reduction is related to the level
raising phenomenon for modular forms. As an application, we will use this
to bound the Selmer groups of certain motives of high rank, providing new
cases of the BlochKato conjecture.
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February 23

Stefan Patrikis
(Univ. of Utah)

Deformations of Galois representations and exceptional monodromy
I will explain how to realize the exceptional algebraic groups
as algebraic monodromy groups of geometric (in the sense of
FontaineMazur) Galois representations, in particular obtaining the first
such examples in types F4 and E6. The argument relies on lifting
wellchosen mod p representations to characteristic zero, using a
generalization (to essentially any reductive group) of a technique
developed by Ravi Ramakrishna for type A1.
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March 1

Simon Marshall
(UWMadison)

The asymptotic behaviour of periods of automorphic forms
If f is an automorphic form on a group G, the integral of f over the adelic points of a subgroup of G is known as a period of f. I will describe how period integrals can give us useful information about the cohomology or harmonic analysis of arithmetic manifolds, and present results on the asymptotics of certain periods that can be interpreted as upper and lower bounds for the sup norms of Maass forms. I will discuss the links between these results and topics such as theta lifting, special Lvalues, trace formulae, and the work of Sakellaridis and Venkatesh on the spectra of spherical varieties. Part of this is joint work with Farrell Brumley.
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March 8

Matthias Strauch
(Indiana)

Arithmetic differential operators on the padic upper half plane
and padic representations of GL(2)
The padic upper half plane comes equipped with a remarkable tower
of GL(2)equivariant etale covering spaces, as was shown by Drinfeld.
It has been an open question for some time whether the spaces of global sections
of the structure sheaf on such coverings provide admissible locally analytic representations.
Using global methods and the padic Langlands correspondence for GL(2,Qp), this is now known
to be the case by the work of Dospinescu and Le Bras. For the first layer of this tower Teitelbaum
exhibited a nice formal model which we use to provide a local proof for the admissibility
of the representation (when the base field is any finite extension of Qp). The other key ingredients
are suitably defined sheaves of arithmetic differential operators and Daffinity results
for formal models of the rigid analytic projective line, generalizing those of Christine Huyghe.
This is joint work with Deepam Patel and Tobias Schmidt.
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