|
September 27 |
ORGANIZATIONAL MEETING |
|
October 4 |
NO MEETING |
|
October 11 |
Nolan Wallach (UC San Diego) Today the main emphasis in local number theory (i.e the Local Langlands correspondence) is on the finite places. In characteristic 0 the infinite place is the "elephant in the room". This is especially true in the Whittaker Theory in which serious difficulties separate the infinite from the finite places. Whittaker models were developed to help the study of Fourier coefficients at cusps of non-holomophic cusp forms (i.e Maass cusp forms) through representation theory. The first of these lectures will start with an explanation of the role of Whittaker models in the theory of automorphic forms. It will continue with a description of the main results. The second lecture will explain the proof of the Whittaker Plancherel Theorem. |
|
October 18 |
Nolan Wallach (UC San Diego) Today the main emphasis in local number theory (i.e the Local Langlands correspondence) is on the finite places. In characteristic 0 the infinite place is the "elephant in the room". This is especially true in the Whittaker Theory in which serious difficulties separate the infinite from the finite places. Whittaker models were developed to help the study of Fourier coefficients at cusps of non-holomophic cusp forms (i.e Maass cusp forms) through representation theory. The first of these lectures will start with an explanation of the role of Whittaker models in the theory of automorphic forms. It will continue with a description of the main results. The second lecture will explain the proof of the Whittaker Plancherel Theorem. |
|
October 25 |
Rachel Newton (Reading) In 1970, Manin observed that the Brauer group Br(X) of a variety X over a number field K can obstruct the Hasse principle on X. In other words, the lack of a K-point on X despite the existence of points over every completion of K is sometimes explained by non-trivial elements in Br(X). This so-called Brauer-Manin obstruction may not always suffice to explain the failure of the Hasse principle but it is known to be sufficient for some classes of varieties (e.g. torsors under connected algebraic groups) and conjectured to be sufficient for rationally connected varieties and K3 surfaces. A zero-cycle on X is a formal sum of closed points of X. A rational point of X over K is a zero-cycle of degree 1. It is interesting to study the zero-cycles of degree 1 on X, as a generalisation of the rational points. Yongqi Liang has shown that for rationally connected varieties, sufficiency of the Brauer-Manin obstruction to the Hasse principle for rational points over all finite extensions of K implies sufficiency of the Brauer-Manin obstruction to the Hasse principle for zero-cycles of degree 1 over K. In this talk, I will discuss joint work with Francesca Balestrieri where we extend Liang's result to Kummer varieties. |
|
November 1 |
NO MEETING |
|
November 8 |
Daniel Le (Toronto); 1-2pm in APM 7421 -- no pre-talk A conjecture of Serre (now a theorem of Gross, Edixhoven, and Coleman-Voloch) classifies pairs of weights where one finds modular forms congruent modulo a prime p in terms of local behavior at p. We discuss a generalization of this conjecture in higher rank. A key step in our work is the study of a certain subscheme of Gaitsgory's A^1 affine Grassmannian which shares properties with some affine Springer fibers. This is joint work with B. Le Hung, B. Levin, and S. Morra. |
|
|
Joe Ferrara (UC San Diego); 2-3pm in APM 7421 -- no pre-talk In the 1970's Stark made precise conjectures about the leading term of the Taylor series at s=0 for Artin L-functions. In the rank one setting when the order vanishing is exactly one, these conjectures relate the derivative of the L-function at s=0 to the logarithm of a unit in an abelian extension of the base field. In this talk, we will define a p-adic L-function and state a p-adic Stark conjecture in the rank one setting when the base field is a quadratic field. We prove our conjecture in the case when the base field is imaginary quadratic and the prime p is split, and discuss numerical evidence in the other cases. |
|
November 15 |
Nathan Green (UC San Diego) For each function field multiple zeta value (defined by Thakur), we construct a t-module with an attached logarithmic vector such that a specific coordinate of the logarithmic vector is a rational multiple of that multiple zeta value. We then show that the other coordinates of this logarithmic vector contain hyperderivatives of a deformation of these multiple zeta values, which we call t-motivic multiple zeta values. This allows us to give a logarithmic expression for monomials of multiple zeta values. Joint work with Chieh-Yu Chang and Yoshinori Mishiba. |
|
November 22 |
NO MEETING (Thanksgiving) |
|
November 29 |
Jacob Tsimerman (Toronto and UCSD) We discuss a new method to bound 5-torsion in class groups using elliptic curves. The most natural "trivial" bound on the n-torsion is to bound it by the size of the entire class group, for which one has a global class number formula. We explain how to make sense of the n-torsion of a class group intrinsically as a "dimension 0 selmer group", and by embedding it into an appropriate Elliptic curve we can bound its size by the Tate-Shafarevich group which we can bound using the BSD conjecture. This fits into a general paradigm where one bounds "dimension 0 selmer groups" by embedding into global objects, and using class number formulas. |
|
December 6 |
Jacob Lurie (Harvard and UCSD) Let k be a perfect field of characteristic p, and let Gal(k) denote the absolute Galois group of k. By a classical result of Katz, the category of finite-dimensional F_p-vector spaces with an action of Gal(k) is equivalent to the category of finite-dimensional vector spaces over k with a Frobenius-semilinear automorphism. In this talk, I'll discuss some joint work with Bhargav Bhatt which generalizes Katz's result, replacing the field k by an arbitrary F_p-scheme X. In this case, there is a correspondence relating p-torsion etale sheaves on X to quasi-coherent sheaves on X equipped with a Frobenius-semilinear automorphism, which can be viewed as a "mod p" version of the Riemann-Hilbert correspondence for complex algebraic varieties. |