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Akshay Venkatesh

Publications and source records attributed to Akshay Venkatesh.

At least 37 records · Page 2Linked to original sources
arXiv2018

Derived Galois deformation rings

We define a derived version of Mazur's Galois deformation ring. It is a pro-simplicial ring $\mathcal{R}$ classifying deformations of a fixed Galois representation to simplicial coefficient rings; its zeroth homotopy group $π_0 \mathcal{R}$ recovers Mazur's deformation ring. We give evidence that these rings $\mathcal{R}$ occur in the wild: For suitable Galois representations, the Langlands program predicts that $π_0 \mathcal{R}$ should act on the homology of an arithmetic group. We explain how the Taylor--Wiles method can be used to upgrade such an action to a graded action of $π_* \mathcal{R}$ on the homology.

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arXiv2017

Derived Hecke algebra for weight one forms

We study the action of the derived Hecke algebra on the space of weight one forms. By analogy with the topological case, we formulate a conjecture relating this to a certain Stark unit. We verify the truth of the conjecture numerically, for the weight one forms of level $23$ and $31$, and many derived Hecke operators at primes less than $200$. Our computation depends in an essential way on Merel's evaluation of the pairing between the Shimura and cuspidal subgroups of $J_0(q)$.

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arXiv2017

Periods and harmonic analysis on spherical varieties

Given a spherical variety X for a group G over a non-archimedean local field k, the Plancherel decomposition for L^2(X) should be related to "distinguished" Arthur parameters into a dual group closely related to that defined by Gaitsgory and Nadler. Motivated by this, we develop, under some assumptions on the spherical variety, a Plancherel formula for L^2(X) up to discrete (modulo center) spectra of its "boundary degenerations", certain G-varieties with more symmetries which model X at infinity. Along the way, we discuss the asymptotic theory of subrepresentations of C^{infty}(X), and establish conjectures of Ichino-Ikeda and Lapid-Mao. We finally discuss global analogues of our local conjectures, concerning the period integrals of automorphic forms over spherical subgroups.

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arXiv2017

Automorphic cohomology, motivic cohomology, and the adjoint $L$-function

We propose a relationship between the cohomology of arithmetic groups, and the motivic cohomology of certain (Langlands-)attached motives. The motivic cohomology group in question is that related, by Beilinson's conjecture, to the adjoint $L$-function at $s=1$. We present evidence for the conjecture using the theory of periods of automorphic forms, and using analytic torsion.

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arXiv2016

The behavior of random reduced bases

We prove that the number of Siegel-reduced bases for a randomly chosen $n$-dimensional lattice becomes, for $n \rightarrow \infty$, tightly concentrated around its mean. We also show that most reduced bases behave as in the worst-case analysis of lattice reduction. Comparing with experiment, these results suggest that most reduced bases will, in fact, "very rarely" occur as an output of lattice reduction. The concentration result is based on an analysis of the spectral theory of Eisenstein series and uses (probably in a removable way) the Riemann hypothesis.

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arXiv2016

Entropy bounds and quantum unique ergodicity for Hecke eigenfunctions on division algebras

We prove the arithmetic quantum unique ergodicity (AQUE) conjecture for non-degenerate sequences of Hecke eigenfunctions on quotients $Γ\backslash G/K$, where $G\simeq\mathrm{PGL}_{d}(\mathbb{R})$, $K$ is a maximal compact subgroup of $G$ and $Γ<G$ is a lattice associated to a division algebra over $\mathbb{Q}$ of prime degree $d$. More generally, we introduce a new method of proving positive entropy of quantum limits, which applies to higher-rank groups. The result on AQUE is obtained by combining this with a measure-rigidity theorem due to Einsiedler-Katok, following a strategy first pioneered by Lindenstrauss

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arXiv2015

Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields

We prove a homological stabilization theorem for Hurwitz spaces: moduli spaces of branched covers of the complex projective line. This has the following arithmetic consequence: let l>2 be prime and A a finite abelian l-group. Then there exists Q = Q(A) such that, for q greater than Q and not congruent to 1 modulo l, a positive fraction of quadratic extensions of F_q(t) have the l-part of their class group isomorphic to A.

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arXiv2014

Functoriality, Smith theory, and the Brauer homomorphism

If $σ$ is an automorphism of order $p$ of the semisimple group $\mathbf{G}$, there is a natural correspondence between mod $p$ cohomological automorphic forms on $\mathbf{G}$ and $\mathbf{G}^σ$. We describe this correspondence in the global and local settings.

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arXiv2014

Hurwitz Monodromy and Full Number Fields

We give conditions for the monodromy group of a Hurwitz space over the configuration space of branch points to be the full alternating or symmetric group on the degree. Specializing the resulting coverings suggests the existence of many number fields with full Galois group and surprisingly little ramification --- for example, the existence of infinitely many such number fields unramified away from {2,3,5}.

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arXiv2014

Torsion homology growth and cycle complexity of arithmetic manifolds

Let M be an arithmetic hyperbolic 3-manifold, such as a Bianchi manifold. We conjecture that there is a basis for the second homology of M, where each basis element is represented by a surface of `low' genus, and give evidence for this. We explain the relationship between this conjecture and the study of torsion homology growth.

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arXiv2013

Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields, II

We prove a version of the Cohen--Lenstra conjecture over function fields (completing the results of our prior paper). This is deduced from two more general theorems, one topological, one arithmetic: We compute the direct limit of homology, over puncture-stabilization, of spaces of maps from a punctured manifold to a fixed target; and we compute the Galois action on the set of stable components of Hurwitz schemes.

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arXiv2012

A torsion Jacquet--Langlands correspondence

We study torsion in the homology of arithmetic groups and give evidence that it plays a role in the Langlands program. We prove, among other results, a numerical form of a Jacquet--Langlands correspondence in the torsion setting.

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arXiv2011

The distribution of closed geodesics on the modular surface, and Duke's theorem

We give an ergodic theoretic proof of a theorem of Duke about equidistribution of closed geodesics on the modular surface. The proof is closely related to the work of Yu. Linnik and B. Skubenko, who in particular proved this equidistribution under an additional congruence assumption on the discriminant. We give a more conceptual treatment using entropy theory, and show how to use positivity of the discriminant as a substitute for Linnik's congruence condition.

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arXiv2010

The subconvexity problem for $\GL_{2}$

Generalizing and unifying prior results, we solve the subconvexity problem for the $L$-functions of $\GL_{1}$ and $\GL_{2}$ automorphic representations over a fixed number field, uniformly in all aspects. A novel feature of the present method is the softness of our arguments; this is largely due to a consistent use of canonically normalized period relations, such as those supplied by the work of Waldspurger and Ichino--Ikeda.

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arXiv2010

Distribution of periodic torus orbits and Duke's theorem for cubic fields

We study periodic torus orbits on spaces of lattices. Using the action of the group of adelic points of the underlying tori, we define a natural equivalence relation on these orbits, and show that the equivalence classes become uniformly distributed. This is a cubic analogue of Duke's theorem about the distribution of closed geodesics on the modular surface: suitably interpreted, the ideal classes of a cubic totally real field are equidistributed in the modular 5-fold SL_3(Z)\SL_3(R)/SO_3(R). In particular, this proves (a stronger form of) the folklore conjecture that the collection of maximal compact flats in SL_3(Z)\SL_3(R)/SO_3(R) of volume less than V becomes equidistributed as V goes to infinity. The proof combines subconvexity estimates, measure classification, and local harmonic analysis.

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arXiv2010

Linnik's ergodic method and the distribution of integer points on spheres

We discuss Linnik's work on the distribution of integral solutions to $x^2+y^2+z^2 =d$, as $d$ goes to infinity. We give an exposition of Linnik's ergodic method; indeed, by using large-deviation results for random walks on expander graphs, we establish a refinement of his equidistribution theorem. We discuss the connection of these ideas with modern developments (ergodic theory on homogeneous spaces, $L$-functions).

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arXiv2006

Sparse equidistribution problems, period bounds, and subconvexity

We introduce a ``geometric'' method to bound periods of automorphic forms. The key features of this method are the use of equidistribution results in place of mean value theorems, and the systematic use of mixing and the spectral gap. Applications are given to equidistribution of sparse subsets of horocycles and to equidistribution of CM points; to subconvexity of the triple product period in the level aspect over number fields, which implies subconvexity for certain standard and Rankin-Selberg $L$-functions; and to bounding Fourier coefficients of automorphic forms.

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