Abstract
Using the Ratios Conjecture as introduced by Conrey, Farmer and Zirnbauer, we obtain closed formulas for the one-level density for two families of L-functions attached to elliptic curves, and we can then determine the underlying symmetry types of the families. The one-level scaling density for the first family corresponds to the orthogonal distribution as predicted by the conjectures of Katz and Sarnak, and the one-level scaling density for the second family is the sum of the Dirac distribution and the even orthogonal distribution. This is a new phenomenon for a family of curves with odd rank: the trivial zero at the central point accounts for the Dirac distribution, and also affects the remaining part of the scaling density which is then (maybe surprisingly) the even orthogonal distribution. The one-level density for this family was studied in the past for test functions with Fourier transforms of limited support, but since the Fourier transforms of the even orthogonal and odd orthogonal distributions are undistinguishable for small support, it was not possible to identify the distribution with those techniques. This can be done with the Ratios Conjecture, and it sheds more light on “independent” and “non-independent” zeroes, and the repulsion phenomenon.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Breuil, C, Conrad, B, Diamond, F, Taylor, R: On the modularity of elliptic curves over
: wild 3-adic exercises. J. Amer. Math. Soc. 14(4), 843–939 (2001).
Birch, BJ, Swinnerton-Dyer, HPF: Notes on elliptic curves. II. J. Reine und Ang. Math. 218, 79–108 (1965).
Conrey, B, Farmer, DW, Zirnbauer, MR: Autocorrelation of ratios of L-functions. Commun. Number Theory Phys. 2(3), 593–636 (2008).
Conrey, B, Snaith, NC: Applications of the L-functions ratios conjectures. Proc. Lond. Math. Soc. 94(3), 594–646 (2007).
Farmer, D: Mean values of ζ ′/ζ and the Gaussian unitary ensemble hypothesis. Internat. Math. Res. Notices. 2, 71–82 (1995).
Fiorilli, D, Miller, SJ: Surpassing the ratios conjecture in the 1-level density of Dirichlet L-functions. Algebra and Number Theory. to appear, arXiv:1111.3896v3.
Goes, J, Jackson, S, Miller, SJ, Montague, D, Ninsuwan, K, Peckner, R, Pham, T: A unitary test of the L-functions Ratios Conjecture. J. Number Theory. 130(10), 2238–2258 (2010).
Helfgott, H: Root numbers and the parity problem, Ph. D. thesis, Princeton University, arXiv:math.NT/0305435 (2003).
Helfgott, H: On the behaviour of root numbers in families of elliptic curves, arXiv:math/0408141 (2009).
Huynh, DK, JP Keating, Snaith, NC: Lower order terms for the one-level density of elliptic curve L-functions. J. Number Theory. 129, 2883–2902 (2009).
Huynh, DK, Miller, SJ, Morrison, R: An elliptic curve family test of the Ratios Conjecture. J. Number Theory. 131, 1117–1147 (2011).
Iwaniec, H, Luo, W, Sarnak, P: Lowlying zeros of families of L-functions. Inst. Hautes Études Sci. Publ. Math. 91(2000), 55–131 (2001).
Iwaniec, H, Kowalski, E: Analytic number theory. AAmerican Mathematical Society Colloquium Publications, 53. American Mathematical Society, Providence, RI, xii+615 pp (2004). ISBN: 0-8218-3633-1.
Katz, NM, Sarnak, P: Zeroes of zeta functions and symmetry. Bull. Amer. Math. Soc. 36, 1–26 (1999).
Katz, NM, Sarnak, P: Random matrices, Frobenius eigenvalues, and monodromy. American Mathematical Society Colloquium Publications, 45. American Mathematical Society, Providence, RI, xii+419 pp (1999). ISBN: 0-8218-1017-0.
Michel, P: Rang moyen de courbes elliptiques et lois de Sato-Tate. Monat. Math. 120, 127–136 (1995).
Miller, SJ: 1- and 2-Level Densities for Families of Elliptic Curves: Evidence for the Underlying Group Symmetries, Ph. D. thesis, Princeton University (2002).
Miller, SJ: One- and two-level densities for rational families of elliptic curves: evidence for the underlying group symmetries. Compos. Math. 140(4), 952–992 (2004).
Miller, SJ: Investigations of zeros near the central point of elliptic curve L-functions. Experimental Math. 15, 257–279 (2006).
Miller, SJ: An orthogonal test of the L-functions ratios conjecture. Proc. Lond. Math. Soc. (3). 99(2), 484–520 (2009).
Miller, SJ, Montague, D: An Orthogonal Test of the L-functions Ratios Conjecture, II. Acta Arith. 146, 53–90 (2011).
Montgomery, H: The pair correlation of zeros of the zeta function. In: Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 181–193. Amer. Math. Soc., Providence, R.I (1973).
Nonnenmacher, S, Zirnbauer, M: Det-Det correlations for quantum maps: dual pair and saddle-point analyses. J. Math. Phys. 43(5), 2214–2240 (2002).
Rizzo, O: Average root numbers for a non-constant family of elliptic curves. Compositio Math. 136(1), 1–23 (2003).
Silverman, J: The average rank of an algebraic family of elliptic curves. J. Reine Angew. Math. 504, 227–236 (1998).
Stepanov, SA: Arithmetic of Algebraic Curves Translated from the Russian by Irene Aleksanova. Monographs in Contemporary Mathematics. Consultants Bureau, New York (1994).
Taylor, R, Wiles, A: Ring-theoretic properties of certain Hecke algebras. Ann. Math. 141(3), 553–572 (1995).
Wiles, A: Modular elliptic curves and Fermat’s last theorem. Ann. Math. 141(3), 443–551 (1995).
Young, MP: Low-lying zeros of families of elliptic curves. J. Amer. Math. Soc. 19(1), 205–250 (2006).
Young, MP: Moments of the critical values of families of elliptic curves, with applications. Canad. J. Math. 62(5), 1155–1181 (2010).
Washington, L: Class numbers of the simplest cubic fields. Math. Comp. 48(177), 371–384 (1987).
Acknowledgements
The authors are very grateful to Sandro Bettin, Barry Mazur, Steve J. Miller, Mike Rubinstein, Nina Snaith, Matt Young and the anonymous referee for helpful discussions and comments related to this paper. A large part of this work was done when the second author was visiting the CRM and Concordia University, and he thanks both institutions for their hospitality.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as 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.
About this article
Cite this article
David, C., Huynh, D.K. & Parks, J. One-level density of families of elliptic curves and the Ratios Conjecture. Res. number theory 1, 6 (2015). https://doi.org/10.1007/s40993-015-0005-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40993-015-0005-7