Tag - Shimura varieties

Jack Thorne: Symmetric Power Functoriality For Hilbert Modular Forms

18th May, 2023

Symmetric power functoriality is one of the basic cases of Langlands's functoriality conjectures and is the route to the proof of the Sato-Tate conjecture (concerning the distribution of the modulo p point counts of an elliptic curve over ℚ, as the prime p varies). I will discuss the proof of the existence of the symmetric power liftings of Hilbert modular forms of regular weight. The proof uses automorphy lifting theorems, automorphic forms on unitary groups, and the geometry of Shimura varieties, as well as the fact that Spec(ℤ) is simply connected.

Congling Qiu: Modularity and Heights of CM cycles on Kuga-Sato varieties

14th October, 2021

We study CM cycles on Kuga-Sato varieties over X(N) via theta lifting and relative trace formula. Our first result is the modularity of CM cycles, in the sense that the Hecke modules they generate are semisimple whose irreducible components are associated to higher-weight holomorphic cuspidal automorphic representations of GL2(ℚ). This is proved via theta lifting. Our second result is a higher weight analogue of the general Gross-Zagier formula of Yuan, S. Zhang and W. Zhang.

This is proved via relative trace formula, provided the modularity of CM cycles.

Salim Tayou: Exceptional Jumps of Picard Rank of K3 Surfaces over Number Fields

18th February, 2021

Given a K3 surface X over a number field K, we prove that the set of primes of K where the geometric Picard rank jumps is infinite, assuming that X has everywhere potentially good reduction. This result is formulated in the general framework of GSpin Shimura varieties and I will explain other applications to abelian surfaces. I will also discuss applications to the existence of rational curves on K3 surfaces.

The results in this talk are joint work with Ananth Shankar, Arul Shankar and Yunqing Tang.

Joel Nagloo: Ax-Lindemann-Weierstrass Theorem for Fuchsian automorphic functions

21st January, 2021

Over the last decades, following works around the Pila-Wilkie counting theorem in the context of o-minimality, there has been a surge in interest around functional transcendence results, in part due to their connection with special points conjectures. A prime example is Pila's modular Ax-Lindemann-Weierstrass (ALW) Theorem and its role in his proof of the André-Oort conjecture.

In this talk we will discuss how an entirely new approach, using the model theory of differential fields, can be used to prove the ALW Theorem with derivatives for Fuchsian automorphic functions - a direct generalization of Pila’s ALW theorem. We will also explain how new cases of the André-Pink conjecture can be obtained using this new approach.

This is joint work with G. Casale and J. Freitag.

Jingwei Xiao: A Unitary Analogue of Friedberg-Jacquet Periods and Central Values of Standard L Functions on GL(2n)

3rd December, 2020

Let G be a reductive group over a number field F and H a subgroup. Automorphic periods study the integrals of cuspidal automorphic forms on G over H(F)\H(AF). They are often related to special values of certain L-functions. One of the most notable cases is when (G,H)=(U(n+1)☓U(n), U(n)), and these periods are related to central values of Rankin-Selberg L-functions on GL(n+1)☓GL(n). In this talk, I will explain my work in progress with Wei Zhang that studies central values of standard L-functions on GL(2n) using (G,H)=(U(2n), U(n)☓U(n)) and some variants. I shall explain the conjecture and a relative trace formula approach to study it. We prove the required fundamental lemma using a limit of the Jacquet-Rallis fundamental lemma and Hironaka’s characterization of spherical functions on the space of non-degenerate Hermitian matrices. Also, the question admits an arithmetic analogue.

Levent Alpoge: Effective height bounds for odd-degree totally real points on some curves

12th November, 2020

Let o be an order in a totally real field, say F. Let K be an odd-degree totally real field. Let S be a finite set of places of K. We study S-integral K-points on integral models Ho of Hilbert modular varieties because not only do said varieties admit complete curves (thus reducing questions about such curves' K-rational points to questions about S-integral K-points on these integral models), they also have their S-integral K-points controlled by known cases of modularity, in the following way. First assume for clarity modularity of all GL2-type abelian varieties over K: then all S-integral K-points on Ho arise from K-isogeny factors of the [F:ℚ]-th power of the Jacobian of a single Shimura curve with level structure (by Jacquet-Langlands transfer). By a generalization of an argument of von Känel, isogeny estimates of Raynaud/Masser-Wüstholz and Bost's lower bound on the Faltings height suffice to then bound the heights of all points in Ho(oK,S). As for the assumption, though modularity is of course not known in this generality, by following Taylor's (sufficiently explicit for us) proof of his potential modularity theorem we are able to make the above unconditional.

Finally we use the hypergeometric abelian varieties associated to the arithmetic triangle group Δ(3,6,6) to give explicit examples of curves to which the above height bounds apply. Specifically, we prove that, for a∈ ℚ̅ x totally real of odd degree (e.g. a = 1) and L/ℚ(a) totally real of odd degree, there is an effectively computable c = ca,L∈ ℤ+ such that all x,y∈L satisfying x6 + 4y3 = a2 satisfy h(x) < c. Note that this gives infinitely many curves for each of which Faltings' theorem is now effective over infinitely many number fields.

Chao Li: On the Kudla-Rapoport conjecture

9th April, 2020

The Kudla-Rapoport conjecture predicts a precise identity between the arithmetic intersection number of special cycles on unitary Rapoport-Zink spaces and the derivative of local representation densities of hermitian forms. It is a key local ingredient to establish the arithmetic Siegel-Weil formula and the arithmetic Rallis inner product formula, relating the height of special cycles on Shimura varieties to the derivative of Siegel Eisenstein series and L-functions. We will motivate this conjecture, explain a proof and discuss global applications.

Jared Weinstein: Moduli spaces of shtukas over function fields

13th February, 2020

We present some work in progress, on moduli spaces of Drinfeld shtukas. These spaces are the function field analogous to Shimura varieties. In fact they are more versatile; there are r-legged versions for any r. Tate's conjecture predicts some interesting relations between shtuka spaces and function field arithmetic. For instance, there should be a notion of modularity for the r-fold product of an elliptic curve. We verify these predictions in a few cases.

This is partly joint work with Noam Elkies.

Thomas Haines: Singularities in reductions of Shimura varieties

2nd May, 2019

The singularities in the reduction modulo p of the modular curve Y0(p) are visualized by the famous picture of two curves meeting transversally at the supersingular points. It is a fundamental question to understand the singularities which arise in the reductions modulo p of integral models of Shimura varieties. For PEL type Shimura varieties with parahoric level structure at p, this question has been studied since the 1990s. Due to the recent  construction of Kisin and Pappas, it now makes sense to pursue this question for abelian type Shimura varieties with parahoric level structure. Recently He-Pappas-Rapoport gave a classification of the Shimura varieties in this class which have either good or semistable reduction. But what is the strongest statement we can make about the nature of the singularities in general? For some time it has been expected that the integral models are Cohen-Macaulay.

This talk will discuss recent work with Timo Richarz, in which we prove that, with mild restrictions on p, all Pappas-Zhu parahoric local models, and therefore all Kisin-Pappas Shimura varieties, are Cohen-Macaulay.