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Non-isotrivial elliptic surfaces with non-zero average root number

Title:

Non-isotrivial elliptic surfaces with non-zero average root number

Bettin, Sandro, David, Chantal and Delaunay, Christophe (2018) Non-isotrivial elliptic surfaces with non-zero average root number. Journal of Number Theory . ISSN 0022314X (In Press)

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Official URL: http://dx.doi.org/10.1016/j.jnt.2018.03.007

Abstract

We consider the problem of finding non-isotrivial 1-parameter families of elliptic curves whose root number does not average to zero as the parameter varies in Z. We classify all such families when the degree of the coefficients (in the parameter t ) is less than or equal to 2 and we compute the rank over Q(t) of all these families. Also, we compute explicitly the average of the root numbers for some of these families highlighting some special cases. Finally, we prove some results on the possible values average root numbers can take, showing for example that all rational number in [−1,1] are average root numbers for some non-isotrivial 1-parameter family

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Article
Refereed:Yes
Authors:Bettin, Sandro and David, Chantal and Delaunay, Christophe
Journal or Publication:Journal of Number Theory
Date:17 April 2018
Digital Object Identifier (DOI):10.1016/j.jnt.2018.03.007
Keywords:Rational elliptic surface; Rank; Root number; Average root number
ID Code:983801
Deposited By: ALINE SOREL
Deposited On:26 Apr 2018 20:01
Last Modified:17 Apr 2020 00:00

References:

Scott Arms, Álvaro Lozano-Robledo, and Steven J. Miller. Constructing one-parameter families of ellipticcurves with moderate rank. J. Number Theory, 123(2):388–402, 2007

Manjul Bhargava, Daniel M. Kane, Hendrik W. Lenstra, Jr., Bjorn Poonen, and Eric Rains. Modeling the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves. Camb. J. Math., 3(3):275–321, 2015.

Dongho Byeon. Quadratic twists of elliptic curves associated to the simplest cubic fields. Proc. Japan Acad. Ser. A Math. Sci., 73(10):185–186, 1997.

B. Conrad, K. Conrad, and H. Helfgott. Root numbers and ranks in positive characteristic. Adv. Math., 198(2):684–731, 2005.

Ian Connell. Calculating root numbers of elliptic curves over Q. Manuscripta Math., 82(1):93–104, 1994.

Tim Dokchitser and Vladimir Dokchitser. Elliptic curves with all quadratic twists of positive rank. Acta Arith., 137(2):193–197, 2009.

Julie Desjardins. Densité des points rationnels sur les surfaces elliptiques et les surfaces de del pezzo de degré 1. PhD thesis, Université Paris Diderot, Institut de mathematique de Jussieu Rive Gauche, available athttps://webusers.imj-prg.fr/~julie.desjardins/these.pdf, 2016.

Julie Desjardins. On the variation of the root number in families of elliptic curves. Preprint, arXiv:math/1610.07440, 2016.

Christophe Delaunay and Frédéric Jouhet. pℓ-torsion points in finite abelian groups and combinatorial identities. Adv. Math., 258:13–45, 2014.

Sylvain Duquesne. Integral points on elliptic curves defined by simplest cubic fields. Experiment. Math., 10(1):91–102, 2001.
David Farmer. Modeling families of l-functions. Preprint, arXiv:math/0511107v1, 2005.

Emmanuel Halberstadt. Signes locaux des courbes elliptiques en 2 et 3. C. R. Acad. Sci. Paris Sér. I Math., 326(9):1047–1052, 1998.

Harald Helfgott. Root numbers and the parity problem. Preprint, arXiv:math/0305435, 2003.

Harald Helfgott. On the square-free sieve. Acta Arith., 115(4):349–402, 2004.

Harald Helfgott. On the behaviour of root numbers in families of elliptic curves. Preprint, arXiv:math/0408141v3, 2009.

Zev Klagsbrun, Barry Mazur, and Karl Rubin. Disparity in Selmer ranks of quadratic twists of elliptic curves. Ann. of Math. (2), 178(1):287–320, 2013.

Mayumi Kawachi and Shin Nakano. The 2-class groups of cubic fields and 2-descents on elliptic curves. Tohoku Math. J. (2), 44(4):557–565, 1992.

Emmanuel Kowalski. Families of cusp forms. Publ. Math. Besançon Algèbre Théorie Nr., pages 5–40, 2013.

Serge Lang. Algebraic number theory, volume 110 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1994.

Rudolf Lidl and Harald Niederreiter. Finite fields, volume 20 of Encyclopedia of Mathematics and its Applications. Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983. With a foreword by P. M. Cohn.

Rick Miranda. An overview of algebraic surfaces. Notes for Lectures at the Summer School on Algebraic Geometry, Bilkent International Center for Advanced Studies, Bilkent University, Ankara, Turkey, 1995.

Keiji Oguiso and Tetsuji Shioda. The Mordell-Weil lattice of a rational elliptic surface. Comment. Math. Univ. St. Paul., 40(1):83–99, 1991.

Group PARI. PARI/GP, version 2.8.1. Bordeaux, 2016.
Bjorn Poonen and Eric Rains. Random maximal isotropic subspaces and Selmer groups. J. Amer. Math. Soc., 25(1):245–269, 2012.

Ottavio Rizzo. Average root numbers in families of elliptic curves. Proc. Amer. Math. Soc., 127(6):1597–1603, 1999.

Ottavio G. Rizzo. Average root numbers for a nonconstant family of elliptic curves. Compositio Math., 136(1):1–23, 2003.

David E. Rohrlich. Variation of the root number in families of elliptic curves. Compositio Math., 87(2):119–151, 1993.

Fausto Romano. Sulla distribuzione della parità del rango di famiglie di curve ellittiche. Master thesis, supervised by Ottavio Rizzo, Corso di laurea di Matematica, Università degli Studi di Milan, 2005.

Michael Rosen and Joseph H. Silverman. On the rank of an elliptic surface. Invent. Math., 133(1):43–67, 1998.

Peter Sarnak. Definition of families of l-functions. letter to H. Iwaniec, P. Michel, A. Venkatesh and others, 2008.

Charles F. Schwartz. On a family of elliptic surfaces with Mordell-Weil rank 4. Proc. Amer. Math. Soc., 102(1):1–8, 1988.

Matthias Schütt and Tetsuji Shioda. Elliptic surfaces. Preprint, arXiv:math/0907.0298v3, 2010.

Lawrence C. Washington. Class numbers of the simplest cubic fields. Math. Comp., 48(177):371–384, 1987.
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