Yihang Zhu (Columbia U)
The Hasse-Weil zeta functions of orthogonal Shimura varieties
Initiated by Langlands, the problem of comparing the Hasse-Weil zeta functions of Shimura varieties with automorphic L-functions has received continual study. The strategy proposed by Langlands, later made more precise by Kottwitz, is to compare the Grothendieck-Lefschetz trace formula for Shimura varieties with the trace formula for automorphic forms. Recently the program has been extended to some Shimura varieties not treated before. In the particular case of (non-compact) orthogonal Shimura varieties, we discuss the proof of Kottwitz's conjectural comparison, between the intersection cohomology of their minimal compactifications and the stable trace formulas. Key ingredients include point counting on these Shimura varieties, Morel's theorem on intersection cohomology, and explicit computation in representation theory mostly for real Lie groups.
Hang Xue (U Arizona)
Arithmetic theta lifts and the arithmetic Gan-Gross-Prasad conjecture
I will explain the arithmetic analogue of the Gan-Gross-Prasad conjecture for unitary groups. I will also explain how to use arithmetic theta lift to prove certain endoscopic cases of it.
Ruochuan Liu (BICMR)
Logarithmic p-adic Riemann-Hilbert correspondence and periods on Shimura varieties
In the previous work with Xinwen Zhu we construct a p-adic analogue of the classical Riemann-Hilbert correspondence. As a by-product the de Rham periods of a general Shimura variety are obtained. In a recent joint work with Hansheng Diao, Kai-Wen Lan and Xinwen Zhu, we further establish a logarithmic version of the p-adic Riemann-Hilbert correspondence which enables us to compare the de Rham periods and complex periods for a general Shimura variety.
Chen Wan (IAS)
The local trace formula for the Ginzburg-Rallis model and the generalized Shalika model
We will first discuss a local trace formula for the Ginzburg-Rallis model. This trace formula allows us to prove a multiplicity formula for the Ginzburg-Rallis model, which implies that the summation of the multiplicities on every tempered Vogan L-packet is always equal to 1. Then we will talk about an analogy of this trace formula for the generalized Shalika model, which implies that the multiplicity for the generalized Shalika model is a constant on every discrete Vogan L-packet.
Ellen Eischen (U Oregon)
One approach to studying the p-adic behavior of L-functions relies on understanding p-adic properties of certain modular forms, for example congruences satisfied by their Fourier coefficients. In this talk, I will give an overview of p-adic L-functions and an approach to constructing them, which relies on relating them to certain p-adic families of Eisenstein series. I will start with the earliest examples of p-adic L-functions (due to Serre, Leopoldt, and Kubota) and conclude by mentioning a recent construction of myself, Harris, Li, and Skinner.
Carl Wang-Erickson (Imperial C London)
The rank of the Eisenstein ideal
In his landmark 1976 paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level N. He proved a great deal about these congruences, and also posed some questions: how big is the space of cusp forms that are congruent to the Eisenstein series? How big is the extension generated by their coefficients? In joint work with Preston Wake, we give an answer to these questions in terms of cup products (and Massey products) in Galois cohomology. We will introduce these products and explain what algebraic number-theoretic information they encode. Time permitting, we may be able to indicate some partial generalizations to square-free level.
Koji Shimizu (Harvard U)
Existence of a compatible system of a local system
Fontaine and Mazur conjectured that an l-adic Galois representation of Q comes from algebraic geometry if it is unratified almost everywhere and de Rham at l. The conjecture implies that such a representation should be embedded into a compatible system of l'-adic Galois representations with various l'. My talk is about the relative version of the Fontaine-Mazur conjecture replacing Galois representations by etale local systems on an algebraic variety. I will give supporting evidence of the conjecture by discussing existence of a compatible system of a local system.
Jan Vonk (McGill U)
Singular moduli for real quadratic fields
The theory of complex multiplication, one of the great achievements of number theory in the 19th century, describes finite abelian extensions of imaginary quadratic number fields using singular moduli, which are special values of modular functions at CM points. Hilbert's 12th problem asks for a satisfactory analogue of this theory for arbitrary number fields. I will describe joint work with Henri Darmon in the setting of real quadratic fields, where we construct p-adic analogues of singular moduli through classes of rigid meromorphic cocycles.
Qirui Li (Columbia U)
Intersection number formula on Lubin-Tate spaces
We consider a moduli space classifying deformations of a formal module over \bar F_q. Those spaces are called Lubin Tate deformation spaces. We will construct some CM cycles on this space. By adding Drinfeld level structures, we proved a formula for the intersection number between these CM cycles. As an application, this formula gives a new proof of Keating's results on endomorphism lifting problems for formal modules over \bar F_q.