Number Theory Seminar @ University of Melbourne

It is a folklore conjecture that the sup norm of a Laplace eigenfunction on a compact hyperbolic surface grows more slowly than any positive power of the eigenvalue. In dimensions three and higher, this was shown to be false by Iwaniec-Sarnak and Donnelly. I will present joint work with Farrell Brumley that strengthens these results, and extends them to locally symmetric spaces associated to SO(p,q).

I will give an introduction to common work with Marcelo De Martino and Eric Opdam, where we determine (for the moment in the number field case for a split group and supported in the trivial representation of the maximal torus) the automorphic spherical spectrum following the method initiated by Langlands using contour shift. It is based on a residue distribution approach developed previously by Eric Opdam and Gerrit Heckman. For exceptional groups, computer assistance was needed.

Recent years have seen a confluence between ideas coming from each of the arithmetic and geometric sides of the Langlands program. One manifestation of this is the rising importance of the consideration of moduli stacks of local and global Galois representations and related objects, going under the general heading of Langlands parameters.

In this talk I will describe these moduli stacks --- their definitions and basic structure, illustrated through examples --- in various local and global contexts. I will also briefly indicate the role they play in contemporary formulations of the Langlands program.

The talk will explain the work of many different researchers, but will include discussions of recent and ongoing work of the speaker in collaboration with Toby Gee and Xinwen Zhu.

In this talk, by using the trace formula method, I will prove a multiplicity formula of K-types for all representations of real reductive groups in terms of the Harish-Chandra character.

This talk is based on my thesis work and joint work (in progress) with Congling Qiu. I will introduce a Gross-Zagier formula for CM cycles over Shimura curves. The formula connects the global height pairing of CM cycles in Kuga varieties over Shimura curves with the derivatives of the L-functions associated to weight-2k modular forms. As a key original ingredient of the proof, I will introduce some harmonic analysis on local systems over graphs, including an explicit construction of Green's function, which we apply to compute suitable integral extensions of the CM cycles, and local intersections at primes where the quaternion algebra associated with the Shimura curve ramifies.

Iwasawa theory is the study of the growth of arithmetic invariants in Galois extensions of global fields with Galois group a p-adic Lie group. Beginning with Iwasawa's seminal work in which he proved that the p-primary part of the class group in ℤ_p-extensions of number fields grows with striking and unexpected regularity, Iwasawa theory has become a central strand of modern number theory and arithmetic geometry. While the theory has traditionally focused on towers of number fields, the function field setting has been studied extensively, and has important applications to the theory of p-adic modular forms. This talk will introduce an exciting new kind of p-adic Iwasawa theory for towers of function fields over finite fields of characteristic p, and discuss some applications to problems and open conjectures in the field.