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SUMMARY:Raphael Steiner (ETH Zürich)
DTSTART:20210120T160000Z
DTEND:20210120T170000Z
DTSTAMP:20260418T065436Z
UID:LNTS/25
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/LNTS/25/">Fo
 urth moments and sup-norms with the aid of theta functions</a>\nby Raphael
  Steiner (ETH Zürich) as part of London number theory seminar\n\n\nAbstra
 ct\nIt is a classical problem in harmonic analysis to bound $L^p$-norms of
  eigenfunctions of the Laplacian on (compact) Riemannian manifolds in term
 s of the eigenvalue. A general sharp result in that direction was given by
  Hörmander and Sogge. However\, in an arithmetic setting\, one ought to d
 o better. Indeed\, it is a classical result of Iwaniec and Sarnak that exa
 ctly that is true for Hecke-Maass forms on arithmetic hyperbolic surfaces.
  They achieved their result by considering an amplified second moment of H
 ecke eigenforms. Their technique has since been adapted to numerous other 
 settings. In my talk\, I shall explain how to use Shimizu's theta function
  to express a fourth moment of Hecke eigenforms in geometric terms (second
  moment of matrix counts). In joint work with Ilya Khayutin and Paul Nelso
 n\, we give sharp bounds for said matrix counts and thus a sharp bound on 
 the fourth moment in the weight and level aspect. As a consequence\, we im
 prove upon the best known bounds for the sup-norm in these aspects. In par
 ticular\, we prove a stronger than Weyl-type sub-convexity result.\n
LOCATION:https://researchseminars.org/talk/LNTS/25/
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