Distribution of Farey fractions in residue classes and Lang–Trotter conjectures on average

Cojocaru, Alina; Shparlinski, Igor

doi:10.1090/S0002-9939-08-09324-6

Distribution of Farey fractions in residue classes and Lang–Trotter conjectures on average
HTML articles powered by AMS MathViewer

by Alina Carmen Cojocaru and Igor E. Shparlinski PDF

Proc. Amer. Math. Soc. 136 (2008), 1977-1986 Request permission

Abstract:

We prove that the set of Farey fractions of order $T$, that is, the set $\{\alpha /\beta \in \mathbb {Q}\ : \ \operatorname {gcd}(\alpha , \beta ) = 1, \ 1 \le \alpha , \beta \le T\}$, is uniformly distributed in residue classes modulo a prime $p$ provided $T \ge p^{1/2 +\varepsilon }$ for any fixed $\varepsilon >0$. We apply this to obtain upper bounds for the Lang–Trotter conjectures on Frobenius traces and Frobenius fields “on average” over a one-parametric family of elliptic curves.

References

Amir Akbary, Chantal David, and Robert Juricevic, Average distributions and products of special values of $L$-series, Acta Arith. 111 (2004), no. 3, 239–268. MR 2039225, DOI 10.4064/aa111-3-3
S. Baier, ‘The Lang–Trotter conjecture on average’, J. Ramanujan Math. Soc., to appear.
W. D. Banks, and I. E. Shparlinski, ‘Sato–Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height’, Preprint, 2006 (available at http://arxiv.org/abs/math.NT/0609144).
Jonathan Battista, Jonathan Bayless, Dmitriy Ivanov, and Kevin James, Average Frobenius distributions for elliptic curves with nontrivial rational torsion, Acta Arith. 119 (2005), no. 1, 81–91. MR 2163519, DOI 10.4064/aa119-1-6
A. C. Cojocaru and C. David, ‘Frobenius fields for elliptic curves’, Amer. J. Math. (to appear).
Alina Carmen Cojocaru, Etienne Fouvry, and M. Ram Murty, The square sieve and the Lang-Trotter conjecture, Canad. J. Math. 57 (2005), no. 6, 1155–1177. MR 2178556, DOI 10.4153/CJM-2005-045-7
Alina Carmen Cojocaru and Chris Hall, Uniform results for Serre’s theorem for elliptic curves, Int. Math. Res. Not. 50 (2005), 3065–3080. MR 2189500, DOI 10.1155/IMRN.2005.3065
Chantal David and Francesco Pappalardi, Average Frobenius distributions of elliptic curves, Internat. Math. Res. Notices 4 (1999), 165–183. MR 1677267, DOI 10.1155/S1073792899000082
Chantal David and Francesco Pappalardi, Average Frobenius distribution for inerts in $\Bbb Q(i)$, J. Ramanujan Math. Soc. 19 (2004), no. 3, 181–201. MR 2139503
N.D. Elkies, ‘Supersingular primes of a given elliptic curve over a number field’, Ph.D. thesis, Harvard University, Cambridge, MA, 1987.
Noam D. Elkies, The existence of infinitely many supersingular primes for every elliptic curve over $\textbf {Q}$, Invent. Math. 89 (1987), no. 3, 561–567. MR 903384, DOI 10.1007/BF01388985
Noam D. Elkies, Distribution of supersingular primes, Astérisque 198-200 (1991), 127–132 (1992). Journées Arithmétiques, 1989 (Luminy, 1989). MR 1144318
Etienne Fouvry and M. Ram Murty, On the distribution of supersingular primes, Canad. J. Math. 48 (1996), no. 1, 81–104. MR 1382477, DOI 10.4153/CJM-1996-004-7
Ernst-Ulrich Gekeler, Frobenius distributions of elliptic curves over finite prime fields, Int. Math. Res. Not. 37 (2003), 1999–2018. MR 1995144, DOI 10.1155/S1073792803211272
Kevin James, Average Frobenius distributions for elliptic curves with 3-torsion, J. Number Theory 109 (2004), no. 2, 278–298. MR 2106483, DOI 10.1016/j.jnt.2004.06.012
Kevin James and Gang Yu, Average Frobenius distribution of elliptic curves, Acta Arith. 124 (2006), no. 1, 79–100. MR 2262142, DOI 10.4064/aa124-1-7
M. Ram Murty, V. Kumar Murty, and N. Saradha, Modular forms and the Chebotarev density theorem, Amer. J. Math. 110 (1988), no. 2, 253–281. MR 935007, DOI 10.2307/2374502
Vijaya Kumar Murty and John Scherk, Effective versions of the Chebotarev density theorem for function fields, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 6, 523–528 (English, with English and French summaries). MR 1298275
Jean-Pierre Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323–401 (French). MR 644559
I. E. Shparlinski, ‘Distribution of inverses and multiples of small integers and the Sato–Tate conjecture on average’, Michigan Math. J. (to appear).
Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8