Tag - Representation theory
Establishing the conjectured analytic properties of triple product L-functions is a crucial case of Langlands functoriality. However, little is known. I will present work in progress on the case of triples of automorphic representations on GL3; in some sense this is the smallest case that appears out of reach via standard techniques. The approach is based on a the beautiful fibration method of Braverman and Kazhdan for constructing Schwartz spaces and proving analogues of the Poisson summation formula.
The factorization of Fourier coefficients of automorphic forms plays an important role in a wide range of topics, from the study of L-functions to the interpretation of scattering amplitudes in string theory.
In this talk I will present a transfer theorem which derives the Eulerianity of certain Fourier coefficients from that of another coefficient. I will also discuss some applications of this theorem to Fourier coefficients of automorphic forms in minimal and next-to-minimal representations.
Based on recent work with Dmitry Gourevitch, Axel Kleinschmidt, Daniel Persson and Siddhartha Sahi.
Let F be a CM field. Scholze constructed Galois representations associated to classes in the cohomology of locally symmetric spaces for GLn over F with p-torsion coefficients. These Galois representations are expected to satisfy local-global compatibility at primes above p. Even the precise formulation of this property is subtle in general, and uses Kisin’s potentially semistable deformation rings. However, this property is crucial for proving modularity lifting theorems.
I will discuss joint work with J. Newton, where we establish local-global compatibility in the crystalline case under mild technical assumptions. This relies on a new idea of using P-ordinary parts, and improves on earlier results obtained in joint work with P. Allen, F. Calegari, T. Gee, D. Helm, B. Le Hung, J. Newton, P. Scholze, R. Taylor, and J. Thorne in certain Fontaine-Laffaille cases.
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.
Let G be a real semisimple Lie group with an irreducible unitary representation π. The non-temperedness of π is measured by the parameter p(π), which is defined as the infimum of p≥2 such that π has matrix coefficients in Lp(G). Sarnak and Xue conjectured that for any arithmetic lattice Γ⊂G and principal congruence subgroup Γ(q)⊂Γ, the multiplicity of π in L2(G/Γ(q)) is at most O(V(q)2/p(π)+ε) where V(q) is the covolume of Γ(q). In some contexts such estimate is a decent substitute for the Ramanujan conjecture. For G of real rank 1 Sarnak and Xue translate the estimate into a Diophantine counting problem which they managed to solve for SL2(ℝ) and SL2(ℂ).
In this talk I will explain how one can get the same multiplicity bounds for families of pairwise non-commensurable lattices in G=SL2(ℝ), SL2(ℂ) given as unit groups of maximal orders of quaternion algebras over number fields (“horizontal families”). Namely: m(π,Γ)≪V2/p(π)+ε, where V is the covolume of Γ. I will also discuss similar bounds on multiplicities of representations π1×π2 of G=SL2(ℝ)2 where π1 is fixed non-tempered but π2 is allowed to vary together with the lattice.
The point of this talk is to give three examples of derived structures influencing representations that have connections with number theory. These structures arise from the differential graded algebra of group cochains valued in the endomorphism ring of a representation.
Two examples have to do with representations of a Galois group. One of these realizes a number theoretic criterion for the modulo p multiplicity one condition for Jacobians of modular curves at an Eisenstein maximal ideal of a Hecke algebra; this is joint work with Preston Wake. Another furnishes a realization as a derived Galois deformation ring of an exterior algebra considered in works of Galatius-Venkatesh, Hansen-Thorne, and Venkatesh. The third example features smooth modulo p representations of a p-adic Lie group, answering some questions of Sorensen about the relationship between its Iwasawa algebra and the associated derived Hecke algebra.
The study of eigenvarieties began with Coleman and Mazur, who constructed the first eigencurve, a rigid analytic space parametrizing p-adic modular Hecke eigenforms. Since then various authors have constructed eigenvarieties for automorphic forms on many other groups. We will give bounds on the eigenvalues of the Up Hecke operator appearing in Chenevier's eigenvarieties for definite unitary groups. These bounds generalize ones of Liu-Wan-Xiao for dimension 2, which they used to prove a conjecture of Coleman-Mazur-Buzzard-Kilford in that setting, to all dimensions. We will then discuss the ideas of the proof, which goes through the classification of automorphic representations that are principal series at p, and a geometric consequence.
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