The thesis of Akshay Venkatesh obtains a "Beyond Endoscopy" proof of stable functorial transfer from tori to SL2, by means of the Kuznetsov formula. In this talk, I will show that there is a local statement that underlies this work; namely, there is a local transfer operator taking orbital measures for the Kuznetsov formula to test measures on the torus. The global comparison of trace formulas is then obtained as a Poisson summation formula for this transfer operator.
Affine Deligne-Lusztig varieties (ADLV) naturally arise in the study of Shimura varieties and Rapoport-Zink spaces; their irreducible components give rise to interesting algebraic cycles on the special fiber of Shimura varieties. We prove a conjecture of Miaofen Chen and Xinwen Zhu, which relates the number of irreducible components of ADLV's to a certain weight multiplicity for a representation of the Langlands dual group. Our approach is to use techniques from local harmonic analysis to compute the asymptotics of a certain twisted orbital integral which counts the number of 𝔽q-points on the ADLV as q goes to infinity. This is joint work with Yihang Zhu.
We consider the coherent cohomology of toroidal compactifications of Shimura varieties with coefficients in the canonical extensions of automorphic vector bundles and show that they can be computed as relative Lie algebra cohomology of automorphic representations. Consequently, any Galois representation attached to these coherent cohomology should be automorphic. Our proof is based on Franke’s work on singular cohomology of locally symmteric spaces and via Faltings' BGG spectral sequence we’ve also strengthened Franke’s result in the Shimura variety case.
p-adic period spaces have been introduced by Rapoport and Zink as a generalization of Drinfeld upper half spaces and Lubin-Tate spaces. Those are open subsets of a rigid analytic p-adic flag manifold. An approximation of this open subset is the so called weakly admissible locus obtained by removing a profinite set of closed Schubert varieties. I will explain a recent theorem characterizing when the period space coincides with the weakly admissible locus. The proof consists in a thorough study of modifications of G-bundles on the curve. As an application we can compute the p-adic period space of K3 surfaces with supersingular reduction and other period spaces related to the basic locus in some Shimura varieties. This is joint work with Miaofen Chen and Xu Shen.
Existing unconditional progress on the abc conjecture and Szpiro's conjecture is rather limited and coming from essentially only two approaches: The theory of linear forms in p-adic logarithms, and bounds for the degree of modular parametrizations of elliptic curves by using congruences of modular forms. In this talk I will discuss a new approach as well as some unconditional results that it yields. For a fixed elliptic curve E over the rationals one has several modular parametrizations coming from various Shimura curves X, and our method amounts to using Arakelov theory to bound how these degrees vary as we change the source curve X, keeping E fixed. Unlike linear forms in p-adic logarithms, our method is global and deals with all local contributions at once. Concrete unconditional consequences will be discussed, such as bounding the number of divisors of abc triples polynomially on the radical, bounding the product of the ''fudge factors'' of elliptic curves polynomially on the conductor, and new lower bounds for truncated counting functions in the context of Vojta's arithmetic conjecture.
In mod-p reductions of modular curves, there is a finite set of supersingular points and its open complement corresponding to ordinary elliptic curves. In the study of mod-p reductions of more general Shimura varieties, there is a "Newton stratification" decomposing the reduction into finitely many locally closed subsets, of which exactly one is closed. This closed set is called the basic locus; it recovers the supersingular locus in the classical case of modular curves. In certain cases, the basic locus admits a simple description as a union of classical Deligne-Lusztig varieties. The precise description in these case has proved to be useful for several purposes: to compute intersection numbers of special cycles and to prove the Tate conjecture for certain Shimura varieties. We will describe a group-theoretic approach to understand this phenomenon. We will show that this phenomenon is closely related to the Hodge-Newton decomposition, and many other nice properties on the Shimura varieties. This talk is based on the joint work with Ulrich Gortz and Sian Nie.
Initiated by Langlands, the problem of computing the Hasse-Weil zeta functions of Shimura varieties in terms of automorphic L-functions has received continual study. We will discuss how recent progress in various aspects of the field has allowed the extension of the project to some Shimura varieties not treated before. In the particular case of orthogonal Shimura varieties, we discuss the computation of the Frobenius-Hecke traces on the intersection cohomology of their minimal compactifications, and the comparison to the Arthur-Selberg trace formula via the process of stabilization. Key ingredients include comparing Harish Chandra character formulas to Kostant's theorem on Lie algebra cohomology, and a comparison between different normalizations of the transfer factors for real endoscopy to get all the signs right.
p>A.Venkatesh asked us the question, in the context of torsion automorphic forms: Does the Standard Conjecture (of Grothendieck's) of Künneth type hold with mod p coefficients? We first review the geometric and number-theoretic contexts in which this question becomes interesting, and provide answers: No in general (even for Shimura varieties) but yes in special cases.
In this talk, I will present a formulation of the Gross-Zagier formula over Shimura curves using automorphic representations with algebraic coefficients. It is a joint work with Shou-wu Zhang and Wei Zhang.
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