Dick Gross (Harvard)
Restriction problems for classical groups
I will discuss several restriction problems in the representation theory of classical groups over local fields, including restriction of irreducible representations from U(n) to U(n-1). This work, which is joint with W-T Gan and D Prasad, attempts to predict the multiplicities in the restriction from number-theoretic data of the Langlands parameters.
Brian Conrad (Stanford)
One of the most beautiful topics in pure mathematics is the structure theory
of connected reductive algebraic groups over general fields. This is especially nice over
separably closed fields, where it is given in terms of root systems. Over
general perfect fields (such as Q) one can often reduce questions involving general smooth connected affine
algebraic groups to the reductive case, but this is not at all possible over imperfect
fields, such as function fields of curves over finite fields. This leads one to seek a weakening
of reductivity (equivalent to it over perfect fields) for which one can nonetheless develop a useful structure theory.
The right weakening, called pseudo-reductivity, was introduced and studied by Borel and Tits,
but they were unable to develop a classification suitable for arithmetic applications.
In joint work with O. Gabber and G. Prasad we have established such a classification
(away from characteristic 2 for now). I will explain some concrete problems that motivate
the desire to study pseudo-reductive groups, and show a number of examples to illustrate various
aspects of the classification. The precise form of the classification theorem will also be given.
Barry Mazur (Harvard)
Elliptic Curves and Hilbert's Tenth problem
This lecture is about joint work with Karl Rubin regarding the Mordell-Weil group of elliptic curves over arbitrary number fields. As a consequence of this work, under appropriate hypotheses we can find elliptic curves that have many quadratic twists with trivial Mordell-Weil group, and (assuming the Shafarevich-Tate conjecture) many others with infinite cyclic Mordell-Weil group over an arbitrary number field K. Moreover we can find such elliptic curves E that have the following stability property: for a given cyclic extension-field L of K of prime degree the Mordell-Weil rank of E over L remains equal to 1. Using work of Poonen and Shlapentokh, it follows from our results that if the Shafarevich-Tate conjecture holds, then Hilbert’s Tenth Problem has a negative answer over the ring of integers of every number field.
Matthew Emerton (Northwestern)
p-adic L-functions and Iwasawa theory --- an introduction
I will explain the construction of Kubota--Leopoldt
p-adic L-function, which is a p-adic analytic function obtained
by interpolating special values of the Riemann zeta function.
I will then go on to explain the role that this p-adic L-function
plays in describing the arithmetic of the cyclotomic extensions of
the rational numbers. The key result in this direction is the so-called Main Conjecture of Iwasawa theory, which was proved
by Mazur and Wiles
in the 1980s. It represents the culmination of a long tradition of
number theoretic investigation, reaching all the way back to the work of Kummer from the mid 1800s, who made the first general
study of the
arithmetic of cyclotomic fields.
Matthew Emerton (Northwestern)
p-adic L-functions and Iwasawa theory --- an introduction, part II
This is a continuation of last week's talk.
Mark Kisin (Chicago)
The Grothendieck-Katz conjecture for certain locally symmetric varieties
We show how results from Margulis rigidity can be combined
with results of Katz and Andre to prove the Grothendieck-Katz conjecture
for a large class of locally symmetric varieties including the moduli
space of polarized abelian varieties.
This is joint work with B. Farb.
Tom Barnet-Lamb (Harvard)
Potential automorphy for certain Galois representations to GL(n)
I will describe recent generalizations of mine to a theorem of Harris,
Shepherd-Barron, and Taylor, showing that certain Galois
representations become automorphic after one makes a suitably large
totally-real extension to the base field. The main innovation is that the
result applies to Galois representations to GL_n, where the previous work
dealt with representations to Sp_n; I can also relax certain
congruence conditions which existed in the earlier work, and work over
a CM, rather than a totally-real, field. The main technique is the
consideration of the cohomology the Dwork hypersurface, and in
particular, of pieces of this cohomology other than the invariants under the
natural group action.
Jordan Ellenberg (Madison)
Stable topology of Hurwitz spaces and arithmetic counting problems
We will discuss some arithmetic counting problems, ranging from the antique (how many squarefree integers are there in [0..N]?) to the au courant (conjectures of Bhargava and Cohen-Lenstra about the distributions of discriminants and of class groups.) When considered over function fields, these conjectures reveal themselves as having to do with stabilization of cohomology of moduli spaces of covers of curves, or Hurwitz spaces. We will report on progress on the topological study of Hurwitz spaces, which leads to information about arithmetic counting problems over function fields over finite fields; for instance, a version of Cohen-Lenstra "correct up to the constant" for F_q(t). If time permits I will try to give a picture of the rather general ensemble of arithmetic counting conjectures suggested by the method (e.g. -- for how many squarefree integers in [0..N] is there a totally real quintic extension of Q with discriminant N?) and explain how to prove versions of these conjectures in the much easier regime where "q goes to infinity first." The theorems discussed will include the ones I gestured at somewhat tentatively in my talk at the Langlands conference last year in Evanston, and which are now on firmer footing.
(joint work with Akshay Venkatesh and Craig Westerland)