The size of newforms

Abstract: Given an $L^2$-normalised newform for $\mathrm{SL}(n)$, how large are its values in terms of its level? The theory of quantum chaos suggests such a newform should be small. I will give an overview of how to show interesting upper bounds using spectral analysis, Hecke operators, and geometry of numbers. I will present the first "non-trivial" bounds in higher rank and talk about intermediate results of perhaps independent interest, such as Atkin--Lehner operators for $\mathrm{SL}(n)$ and a reduction theory with level structure.

number theory

Audience: researchers in the topic

London number theory seminar

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For a record of talks predating this website see: wwwf.imperial.ac.uk/~buzzard/LNTS/numbtheo_past.html