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Discrete and local averages in the hyperbolic circle problem

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Yiannis Petridis
University College London
Wed, 2017-02-22 11:15 - 12:15
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

For the hyperbolic plane, a point $z$ and $\Gamma$ a discrete subgroup of $\hbox{SL}(2, \mathbb R)  $ we want to estimate how many points in the orbit $\Gamma z$ lie within distance $s$ from $z$. This is the analogue of the Gauss circle problem.
It exhibits various differences, necessitating the use of automorphic forms and, in particular, Maa{\ss} cusp forms. In this case the main term grows exponentially ${\pi e^s }/{\hbox{vol}(\Gamma\setminus \mathbb H)}.$ Here the best error term is $O(e^{2s/3 })$, due to Selberg and has not been improved for approximately 50 years.
Combining arithmetic methods, e.g. averages of cubic moments of quadratic twists of $L$-functions,  sup-norms of Maa{\ss} cusp forms and Eisenstein series,  the rate of quantum ergodicity of Maa{\ss} cusp forms, and estimates on spectral exponential sums, we improve the exponent $2/3$, when we average locally over the center of the hyperbolic circle $z$ for the group $\Gamma=\hbox{SL}(2, \mathbb Z)$ or do a   discrete average over Heegner points for  $\hbox{SL}(2, \mathbb Z). $

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