On a Linnik problem for elliptic curves
HTML articles powered by AMS MathViewer
- by Andrzej Dąbrowski and Jacek Pomykała PDF
- Proc. Amer. Math. Soc. 147 (2019), 3759-3763 Request permission
Abstract:
Let $S(Q,B)$ denote the number of moduli $q\leq Q$ for which a primitive character $\chi$ mod $q$ exists such that $n_{\chi }>B$, where $n_{\chi }$ denotes the smallest natural number such that $\chi (n) \not =1$. Baier showed that for any $\beta >2$ we have $S(Q,(\log Q)^{\beta }) \ll Q^{\frac {1}{\beta -1}+\varepsilon }$ and asked for an analogue of this result for elliptic curves. It is the aim of this note to establish such an analogue.References
- Stephan Baier, A remark on the least $n$ with $\chi (n)\neq 1$, Arch. Math. (Basel) 86 (2006), no. 1, 67–72. MR 2201299, DOI 10.1007/s00013-005-1382-2
- W. Duke and E. Kowalski, A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations, Invent. Math. 139 (2000), no. 1, 1–39. With an appendix by Dinakar Ramakrishnan. MR 1728875, DOI 10.1007/s002229900017
- Stephen Gelbart and Hervé Jacquet, A relation between automorphic representations of $\textrm {GL}(2)$ and $\textrm {GL}(3)$, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 471–542. MR 533066, DOI 10.24033/asens.1355
- D. R. Heath-Brown, The number of primes in a short interval, J. Reine Angew. Math. 389 (1988), 22–63. MR 953665, DOI 10.1515/crll.1988.389.22
- Hugh L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, vol. 84, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. MR 1297543, DOI 10.1090/cbms/084
- Jean-Pierre Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323–401 (French). MR 644559
Additional Information
- Andrzej Dąbrowski
- Affiliation: Institute of Mathematics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland
- MR Author ID: 357378
- Email: andrzej.dabrowski@usz.edu.pl, dabrowskiandrzej7@gmail.com
- Jacek Pomykała
- Affiliation: Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland
- Email: pomykala@mimuw.edu.pl
- Received by editor(s): September 27, 2018
- Received by editor(s) in revised form: January 9, 2019
- Published electronically: May 9, 2019
- Communicated by: Amanda Folsom
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 3759-3763
- MSC (2010): Primary 11F30, 11G05, 11N36
- DOI: https://doi.org/10.1090/proc/14589
- MathSciNet review: 3993768