Date

Speaker

Title

March 27th

Ellen Eischen (Oregon)

padic Lfunctions for unitary groups
I will discuss a construction of padic Lfunctions, especially in the setting of unitary groups. I will highlight how this construction relates to those of Serre, Katz, and Hida, and I will emphasize the role of properties of certain automorphic forms (analogous to the role played by modular forms in their work). This includes joint work with Michael Harris, JianShu Li, and Christopher Skinner.
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April 3rd

Ari Shnidman (Boston College)

A higher order GrossKohnenZagier formula
I'll present a formula relating the intersection of two different HeegnerDrinfeld cycles to the rth
derivative of an automorphic Lfunction. This is a "higherorder" generalization of the GrossKohnenZagier formula,
in the function field setting, and is inspired by recent work of Yun and W. Zhang. Our formula gives strong evidence
that all HeegnerDrinfeld cycles are colinear in cohomology. This is joint work with Ben Howard.
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April 10th

Jack Shotton (Chicago)

Ihara's lemma for Shimura curves via
patching Ihara's lemma is a statement about the
structure of the mod l cohomology of modular curves that was a key ingredient in
Ribet's results on level raising. It was generalised to Shimura curves over Q by
Diamond and Taylor, by a totally different method. Under an assumption on the image
of the residual Galois representation, we give another proof of Diamond and Taylor's
result that extends to some new cases and that we hope will generalise to Shimura
curves over totally real fields. This proof uses the TaylorWilesKisin `patching'
method and the geometry of local deformation rings to reduce to the case of definite
quaternion algebras. This is joint work in progress with Jeffrey Manning.
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April 17th

Anthony VárillyAlvarado (Rice)

Vojta's conjecture and uniform boundedness of
fulllevel structures on abelian varieties over number
fields In 1977, Mazur proved that the
torsion subgroup of an elliptic curve over Q is, up to isomorphism, one of only
15 groups. Before Merel gave a qualitative generalization of this result to
arbitrary number fields, it was known that variants of the abc conjecture would
imply uniform boundedness of torsion on elliptic curves over number fields of
bounded degree. In this talk, I will explain how, using Vojta's conjecture as a
higherdimensional generalization of the abc conjecture, one can deduce similar
uniform boundedness statements for fulllevel structures on abelian varieties of
fixed dimension over number fields. This is joint work with Dan Abramovich and
Keerthi MadapusiPera.
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April 24th

Ilya Khayutin (Princeton)

Joint equidistribution of CM
points A celebrated theorem of Duke states that Picard/Galois orbits of CM points
on a complex modular curve equidistribute in the limit when the absolute value of the discriminant goes to infinity. The
equidistribution of Picard and Galois orbits of special points in products of modular curves was conjectured by Michel
and Venkatesh and as part of the equidistribution strengthening of the AndréOort conjecture. I will discuss the proof
of a recent theorem making progress towards this conjecture.
Currently, this problem does not seem to be amenable to methods of automorphic forms even assuming GRH. Nevertheless,
assuming a splitting condition at two primes the joining rigidity theorem of Einsiedler and Lindenstrauss applies. As a
result the obstacle to proving equidistribution is the potential concentration of mass on graphs of Hecke
correspondences and translates thereof. I will present a method to discard this possibility using a geometric expansion
of a relative trace, description of the relative orbital integrals in terms of integral ideals and a sieve
argument. ( Hide Abstract)

May 1st

Kannan Soundararajan (Stanford)

Weak subconvexity for automorphic
Lfunctions I will discuss recent work with
Jesse Thorner which establishes a weak subconvexity for central values of
automorphic Lfunctions. I will explain what the subconvexity problem is, why
we care (in some situations), and give an overview of the ideas behind our
recent work.
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May 8th

KaiWen Lan (Minnesota)

Nearby cycles of automorphic étale
sheaves I will explain that,
in most cases where integral models are available in the literature, the
automorphic étale cohomology of a (possibly noncompact) Shimura variety in
characteristic zero is canonically isomorphic to the cohomology of the
associated (ladic) nearby cycles in positive characteristics. If time permits,
I will also talk about some applications or related results. (This is joint work
with Stroh.)
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May 15th

Robin Bartlett (King's College London)

The reduction modulo p of crystalline representations of
small weight I will discuss an extension to
GLn of a result of GeeLiuSavitt on the reduction modulo p of crystalline
representations with HodgeTate weights in the interval [0,p]. The methods are
purely local, using tools from padic Hodge theory. I will explain how these
results are related to the weight part of Serre's conjecture beyond dimension 2.
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May 22nd

Hiroki Kato (Tokyo)

Wild ramification and restrictions to
curves I will discuss the
characteristic cycle of an ladic sheaf defined by Saito, which knows the Euler
characteristic, Milnor formula for isolated singularities, and many other
invariants attached to the ladic sheaf. Saito and Yatagawa proved that it is
determined by wild ramification. I will explain it is determined by the
conductors of the restrictions to curves.
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May 29th

Martin Weissman (UC Santa Cruz)

Lgroups and covering
groups To extend the Langlands program to
covering groups, such as the metaplectic group, one needs to construct a dual
group G^\vee (a connected reductive group) and an Lgroup (an extension of the
absolute Galois group, or Weil group, by G^\vee). In this talk I will
introduce covering groups in a framework suggested by Brylinski and Deligne, and
describe the construction of the dual group and Lgroup for such covering
groups. To finish the talk, I will describe the current state of the art  the
evidence that this Lgroup is the "correct" choice, and some open questions.
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