This lecture will partly survey branching laws for real and p-adic groups which often is related to period integrals of automorphic representations, discuss some of the more recent developments, focusing attention on homological aspects and the Bernstein decomposition.
Half-integral weight modular forms are classical objects with many important arithmetic applications. In terms of automorphic representations, these correspond to objects on the metaplectic double cover of SL2. In this talk, I will outline a theory of modular forms of half-integral weight on double covers of exceptional groups, generalizing the integral weight theory developed by Gross-Wallach, Gan-Gross-Savin, and Pollack. Furthermore, I discuss a particular example of a weight 1/2 modular form on G2 whose Fourier coefficients encode the 2-torsion in the narrow class groups of totally real cubic fields. This is built by studying a remarkable automorphic representation of the double cover of the exceptional group F4.
Recently, Fargues and Scholze attached a semi-simple L-parameter to any smooth irreducible representation of a p-adic reductive group, realizing the local Langlands correspondence as a geometric Langlands correspondence over the Fargues-Fontaine curve. They conjectured that there should exist an analogue of the geometric Langlands conjecture in this setting, known as the categorical local Langlands correspondence. Concretely, this conjecture translates to the belief that certain Shtuka spaces, generalizing the Lubin-Tate and Drinfeld towers appearing in the work of Harris-Taylor, should have cohomology dictated by the semi-simple L-parameter that they construct. In this talk, we will explain how one can make some progress on this conjecture by showing the Fargues-Scholze correspondence is compatible with known instances of the local Langlands correspondence through global methods, and then using this compatibility together with techniques from geometric Langlands to fully describe the cohomology of these Shtuka spaces in certain cases.
In 1986, Hooley applied (what practically amounts to) the general Langlands reciprocity (modularity) conjecture and GRH in a fresh new way, over certain families of cubic 3-folds. This eventually led to conditional near-optimal bounds for the number of integral solutions to x13+...+x63 in expanding boxes.
Building on Hooley's work, I will sketch new applications of large-sieve hypotheses, the Square-free Sieve Conjecture, and predictions of Random Matrix Theory type, over the same geometric families - e.g. conditional statistical results on sums of three integer cubes (a project suggested by Amit Ghosh and Peter Sarnak). These form the bulk of my thesis work (advised by Sarnak), and involve phenomena both random and structured, average- and worst-case, and multiplicative and additive.
How does the dimension of the first cohomology grow in a tower of covering spaces? After a tour of examples of behaviours for low-dimensional spaces, I will focus on arithmetic manifolds. Specifically, for towers of complex hyperbolic manifolds, I will describe how to bound the rates of growth using results from Langlands functoriality.
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(â„š). 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.
We consider the standard L-function attached to a cuspidal automorphic representation of a general linear group. We present a proof of a subconvex bound in the t-aspect. More generally, we address the spectral aspect in the case of uniform parameter growth.
I will explain some recent work on special cases of the Bloch-Kato conjecture for the symmetric cube of certain modular Galois representations. Under certain standard conjectures, this work constructs non-trivial elements in the Selmer groups of these symmetric cube Galois representations; this works by p-adically deforming critical Eisenstein series in a generically cuspidal family of automorphic representations, and then constructing a lattice in the associated family of Galois representations, all for the exceptional group G2. While I will touch on all of these aspects of the construction, I will mainly focus on the Galois side in this talk.
Consider the function field F of a smooth curve over đť”˝q, with q>2. L-functions of automorphic representations of GL2 over F are important objects for studying the arithmetic properties of the field F. Unfortunately, they can be defined in two different ways: one by Godement-Jacquet, and one by Jacquet-Langlands. Classically, one shows that the resulting L-functions coincide using a complicated computation. Each of these L-functions is the GCD of a family of zeta integrals associated to test data. I will categorify the question, by showing that there is a correspondence between the two families of zeta integrals, instead of just their L-functions. The resulting comparison of test data will induce an exotic symmetric monoidal structure on the category of representations of GL2. It turns out that an appropriate space of automorphic functions is a commutative algebra with respect to this symmetric monoidal structure. I will outline this construction, and show how it can be used to construct a category of automorphic representations.
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.
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