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Thursday, February 17, 2022
4:00 PM - 5:00 PM
Linde Hall 387

Number Theory Seminar

$L^p$-norm bounds for automorphic forms via spectral reciprocity
Peter Humphries, Department of Mathematics, University of Virginia,

A major area of study in analysis involves the distribution of mass of Laplacian eigenfunctions on a Riemannian manifold. A key result towards this is explicit $L^p$-norm bounds for Laplacian eigenfunctions in terms of their Laplacian eigenvalue, due to Sogge in 1988. Sogge's bounds are sharp on the sphere, but need not be sharp on other manifolds. I will discuss some aspects of this problem for the modular surface; in this setting, the Laplacian eigenfunctions are automorphic forms, and certain $L^p$-norms can be shown to be closely related to certain mixed moments of $L$-functions. In turn, I will discuss how these mixed moments of $L$-functions can be studied via spectral reciprocity: explicit identities relating two different moments of $L$-functions. This is joint with with Rizwanur Khan.

For more information, please contact Math Department by phone at 626-395-4335 or by email at [email protected].