Petridis, YN;
Risager, MS;
(2018)
Arithmetic statistics of modular symbols.
Inventiones Mathematicae
, 212
(3)
pp. 997-1053.
10.1007/s00222-017-0784-7.
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Abstract
Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve L-functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by modular symbols. Another of their conjectures predicts the Gaussian distribution of normalized modular symbols ordered according to the size of the denominator of the cusps. We prove this conjecture in a refined version that also allows restrictions on the location of the cusps.
| Type: | Article |
|---|---|
| Title: | Arithmetic statistics of modular symbols |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1007/s00222-017-0784-7 |
| Publisher version: | http://doi.org/10.1007/s00222-017-0784-7 |
| Language: | English |
| Additional information: | © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. |
| Keywords: | math.NT, math.NT, Primary 11F67, Secondary 11E45, 11M36 |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/1551347 |
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