Tag - Dirichlet L-functions
Hilbert's 12th problem is to provide explicit analytic formulae for elements generating the maximal abelian extension of a given number field. In this talk I will describe an approach to Hilbert's 12th that involves proving exact p-adic formulae for Gross-Stark units. This builds on prior joint work with Kakde and Ventullo in which we proved Gross’s conjectural leading term formula for Deligne-Ribet p-adic L-functions at s=0. This is joint work with Mahesh Kakde.
The analytic theory of Poincaré series and Maass cusp forms and their L-functions for SL3(ℤ) has, so far, been limited to the spherical Maass forms, i.e. elements of a spectral basis for L2(SL3(ℤ)\PSL3(ℝ)/SO3(ℝ)). I will describe the Maass cusp forms of L2(SL3(ℤ)\PSL3(ℝ)) which are minimal with respect to the action of the Lie algebra and give a (relatively) simple method for constructing Kuznetsov-type trace formulas by considering Fourier coefficients of certain Poincaré series. In recent work with Valentin Blomer, we have extended our proof of spectral-aspect subconvexity for L-functions of SL3(ℤ) Maass forms to the non-spherical case, and I will discuss the structure of that proof, as well.
Braverman and Kazhdan have conjectured the existence of summation formulae that are essentially equivalent to the analytic continuation and functional equation of Langlands L-functions in great generality. Motivated by their conjectures and related conjectures of L. Lafforgue, Ngo, and Sakellaridis, Baiying Liu and I have proven a summation formula analogous to the Poisson summation formula for the subscheme cut out of three quadratic spaces (Vi,Qi) of even dimension by the equation Q1(v1)=Q2(v2)=Q3(v3). I will sketch the proof of this formula in the first portion of the talk. In the second portion, time permitting, I will discuss how these summation formulae lead to functional equations for period integrals for automorphic representations of GLn1 × GLn2 × GLn3 where the ni are arbitrary, and speculate on the relationship between these period integrals and Langlands L-functions.
If two motives are congruent, is it the case that the special values of their respective L-functions are congruent? More precisely, can the formula predicting special values of motivic L-functions be interpolated in p-adic families of motives? I will explain how the formalism of the Weight-Monodromy filtration for p-adic families of Galois representations sheds light on this question (and suggests a perhaps surprising answer).
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.
In joint work with Emmanuel Kowalski and Philippe Michel, we prove two different estimates on sums of coefficients of modular forms - one related to L-functions and another to the level of distribution. A key step in the argument is a careful analysis of vanishing cycles, a tool originally developed by Lefschetz to study the topology of algebraic varieties. We will explain why this is helpful for these problems.
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