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The Density Hypothesis for Horizontal Families of Lattices

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Djordje Milicevic
Bryn Mawr College/MPIM
Wed, 2020-11-04 14:30 - 15:30
Parent event: 
Number theory lunch seminar

Zoom Online Meeting ID: 919 6497 4060
For password see the email or contact Pieter Moree (moree@mpim...).


Selberg’s celebrated Eigenvalue Conjecture states that all nonzero Laplacian eigenvalues on congruence quotients of the upper half-plane are at least 1/4. This particularly strong form of the “spectral gap” property can be thought of as the archimedean counterpart of the Ramanujan-Petersson conjecture for Hecke eigenvalues of cusp forms, is expected to suitably hold for more general Lie groups and their arithmetic quotients, and remains far from resolution.

For analytic applications in a family of automorphic forms, in the absence of Selberg’s conjecture, the non-tempered spectrum can often be satisfactorily handled if the exceptions in the family are known to be “sparse” and “not too bad”, in a sense made precise by the so-called density hypothesis evoking the classical density estimates of prime number theory. In this talk, we will present our recent result establishing the density hypothesis for a broad natural “horizontal” family of not necessarily commensurable arithmetic orbifolds, with uniform power-saving estimates in the volume and spectral aspects. This work is joint with Mikolaj Fraczyk, Gergely Harcos, and Peter Maga.

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