Date

Speaker

Title

January 7

Andrew Sutherland (MIT)

The SatoTate conjecture for abelian varieties
The original SatoTate conjecture addresses the statistical distribution
of the number of points on the reductions modulo primes of a fixed elliptic
curve defined over the rational numbers. It predicts that this
distribution can be explained in terms of a random matrix model, using the
Haar measure on the special unitary group SU(2)....
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January 14

Ronen Mukamel (UChicago)

Billiards, Hilbert modular forms and explicit algebraic models for
Teichmuller curves
For each real quadratic ring O, the Hilbert modular surface parametrizing
principally polarized abelian varieties with real multiplication by O
contains a Weierstrass curve W(O). The curve W(O) emerges from the study
of billiards in polygons and is important in Teichmuller theory because the
natural immersion into the moduli space of genus two curves is isometric....
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January 21

Preston Wake (UChicago)

The padic EichlerShimura isomorphism
A theorem of Eichler and Shimura says that the space of cusp forms with complex coefficients appears as a direct summand of the cohomology of the compactified modular curve. Ohta has proven an analog of this theorem for the space of ordinary padic cusp forms with integral coefficients. Ohta's result has important applications in the Iwasawa theory....
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January 28

Keerthi Madapusi Pera (Harvard)

The irreducibility of the moduli of polarized K3 surfaces
We show that the moduli of polarized K3 surfaces of fixed degree
is irreducible in characteristic p>2. The key idea is to exhibit pHecke
correspondences between the ordinary loci of moduli spaces of different
degrees and so reduce to the case where the degree is not too divisible by
p. For the construction of these correspondences, we make crucial use of....
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February 4

Robert Pollack (Boston University)

On μinvariants and congruences with Eisenstein series
For any irregular prime p, one has a Hida family of cuspidal
eigenforms of level 1 whose residual Galois representations are all
reducible. This family has already played a starring role in Wiles’ proof
of Iwasawa’s main conjecture for totally real fields. In this talk, we
instead focus on the Iwasawa theory of these modular forms in their own
right. We will discuss new phenomena.....
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February 11

Liang Xiao (UC Irvine)

Slopes of the eigencurve over boundary disks
Despite the importance of eigencurve in the padic number theory,
the geometry of the eigencurve is still poorly understood. The amazing
calculation of BuzzardKilford in the case of p=2 suggests that the slopes
of the eigencurve should behave reasonably well near the boundary of the
weight space. In this talk, I will report on some evidence regarding this
expectation.....
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February 26 (Wed) Note special day!
(2:304PM in E206) 
Frank Calegari
(Northwestern Univ.)

The stable homology of congruence subgroups (Joint seminar with Alg.Top. and Geom./Top.).
Let F be a number field. A
stability result due to Charney and Maazen says that the
homology groups
H_{d}(SL_{N}(O_{F}),Z)
(for d fixed) are independent of N for
N sufficiently large. The resulting stable
cohomology groups are intimately related to the algebraic
Ktheory of
O_{F}. In these talks, we
shall explore the homology of the ppower
congruence subgroups of
SL_{N}(O_{F}) in
fixed degree d as N becomes large. We
show that the resulting homology groups consist of two
parts: an “unstable” part which depends only on
local behavior concerning how the prime p splits
in F, and a “stable” part which
contains global information concerning padic
regulator maps. Our argument consists of two parts. The
first part (which is joint work with Matthew Emerton)
explains how to modify the homology of congruence subgroups
in a suitable way (using completed homology) to obtain
groups which are literally stable for large N.....
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February 28 (Fri) Note special day!
(2:304PM in E206) 
March 4

Anders Södergren
(Univ. of Copenhagen)

Poisson statistics and the value distribution of the Epstein zeta
function
In this talk I will discuss certain questions concerning the asymptotic behavior of the Epstein zeta function E_{n}(L,s) in the limit of large dimension n. In particular, I will describe the value distribution of E_{n}(L,s) for a random lattice L of large dimension n, giving partial answers to questions raised by Sarnak and Strömbergsson in their study of minima of E_{n}(L,s).....
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March 11

Yongqiang Zhao
(Univ. of Waterloo)

On sieve methods for varieties over finite fields
Although sieve methods in classical analytic number theory have a
long and very fruitful history, its appearance in algebraic geometry is
relatively new, and was introduced by Bjorn Poonen about ten years ago. In
this talk, we will first discuss Poonen's sieve through a concrete example,
then we will introduce a new interpolation technique to sieve methods.....
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