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Akshay Venkatesh

Publications and source records attributed to Akshay Venkatesh.

41 records · Page 3Linked to original sources
arXiv2006

The distribution of periodic torus orbits on homogeneous spaces

We prove results towards the equidistribution of certain families of periodic torus orbits on homogeneous spaces, with particular focus on the case of the diagonal torus acting on quotients of $\PGL_n(\R)$. After attaching to each periodic orbit an integral invariant (the discriminant) our results have the following flavour: certain standard conjectures about the distribution of such orbits hold up to exceptional sets of at most $O(Δ^ε)$ orbits of discriminant $\leq Δ$. The proof relies on the well-separatedness of periodic orbits together with measure rigidity for torus actions. We also give examples of sequences of periodic orbits of this action that fail to become equidistributed, even in higher rank. We give an application of our results to sharpen a theorem of Minkowski on ideal classes in totally real number fields of cubic and higher degrees.

math.DS↗
arXiv2006

Local-global principles for representations of quadratic forms

We prove the local-global principle holds for the problem of representations of quadratic forms by quadratic forms, in codimension $\geq 7$. The proof uses the ergodic theory of $p$-adic groups, together with a fairly general observation on the structure of orbits of an arithmetic group acting on integral points of a variety.

math.NT↗
arXiv2005

Existence and Weyl's law for spherical cusp forms

Let G be a split adjoint semisimple group over Q and K a maximal compact subgroup of the real points G(R). We shall give a uniform, short and essentially elementary proof of the Weyl law for cusp forms on congruence quotients of G(R)/K. This proves a conjecture of Sarnak for Q-split groups, previously known only for the case of PGL(n). The key idea amounts to a new type of simple trace formula.

math.NT↗
arXiv2004

On Quantum unique ergodicity for locally symmetric spaces I

We construct an equivariant microlocal lift for locally symmetric spaces. In other words, we demonstrate how to lift, in a ``semi-canonical'' fashion, limits of eigenfunction measures on locally symmetric spaces to Cartan-invariant measures on an appropriate bundle. The construction uses elementary features of the representation theory of semisimple real Lie groups, and can be considered a generalization of Zelditch's results from the upper half-plane to all locally symmetric spaces of noncompact type. This will be applied in a sequel to settle a version of the quantum unique ergodicity problem on certain locally symmetric spaces.

math.RT↗