Skip to main content Accessibility help
×
Home
Hostname: page-component-55b6f6c457-pc5cw Total loading time: 0.408 Render date: 2021-09-25T09:01:26.175Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Large families of elliptic curves ordered by conductor

Published online by Cambridge University Press:  22 June 2021

Ananth N. Shankar
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA ashankar@math.wisc.edu
Arul Shankar
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada ashankar@math.toronto.edu
Xiaoheng Wang
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada x46wang@uwaterloo.ca

Abstract

In this paper we study the family of elliptic curves $E/{{\mathbb {Q}}}$, having good reduction at $2$ and $3$, and whose $j$-invariants are small. Within this set of elliptic curves, we consider the following two subfamilies: first, the set of elliptic curves $E$ such that the quotient $\Delta (E)/C(E)$ of the discriminant divided by the conductor is squarefree; and second, the set of elliptic curves $E$ such that the Szpiro quotient $\beta _E:=\log |\Delta (E)|/\log (C(E))$ is less than $7/4$. Both these families are conjectured to contain a positive proportion of elliptic curves, when ordered by conductor. Our main results determine asymptotics for both these families, when ordered by conductor. Moreover, we prove that the average size of the $2$-Selmer groups of elliptic curves in the first family, again when these curves are ordered by their conductors, is $3$. The key new ingredients necessary for the proofs are ‘uniformity estimates’, namely upper bounds on the number of elliptic curves with bounded height, whose discriminants are divisible by high powers of primes.

Type
Research Article
Copyright
© The Author(s) 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ash, A., Brakenhoff, J. and Zarrabi, T., Equality of polynomial and field discriminants, Exp. Math. 16 (2007), 367374; MR 2367325.CrossRefGoogle Scholar
Barroero, F. and Widmer, M., Counting lattice points and O-minimal structures, Int. Math. Res. Not. IMRN 18 (2014), 49324957; MR 3264671.CrossRefGoogle Scholar
Bhargava, M., Higher composition laws. II. On cubic analogues of Gauss composition, Ann. of Math. (2) 159 (2004), 865886.CrossRefGoogle Scholar
Bhargava, M., The density of discriminants of quartic rings and fields, Ann. of Math. (2) 162 (2005), 10311063.CrossRefGoogle Scholar
Bhargava, M., The geometric squarefree sieve and unramified nonabelian extensions of quadratic fields, Preprint (2014), arXiv:1402.0031.Google Scholar
Bhargava, M. and Harron, P., The equidistribution of lattice shapes of rings of integers in cubic, quartic, and quintic number fields, Compos. Math. 152 (2016), 11111120; MR 3518306.CrossRefGoogle Scholar
Bhargava, M. and Shankar, A., Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves, Ann. of Math. (2) 181 (2015), 191242; MR 3272925.CrossRefGoogle Scholar
Bhargava, M., Shankar, A., Taniguchi, T., Thorne, F., Tsimerman, J. and Zhao, Y., Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves, Preprint (2017), arXiv:1701.02458.Google Scholar
Bhargava, M., Shankar, A. and Tsimerman, J., On the Davenport-Heilbronn theorems and second order terms, Invent. Math. 193 (2013), 439499; MR 3090184.CrossRefGoogle Scholar
Bhargava, M., Shankar, A. and Wang, X., Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces, Preprint (2015), arXiv:1512.03035.Google Scholar
Bhargava, M., Shankar, A. and Wang, X., Squarefree values of polynomial discriminants I, Preprint (2016), arXiv:1611.09806.Google Scholar
Brumer, A., The average rank of elliptic curves. I, Invent. Math. 109 (1992), 445472.CrossRefGoogle Scholar
Brumer, A. and Kramer, K., The rank of elliptic curves, Duke Math. J. 44 (1977), 715743; MR 0457453.CrossRefGoogle Scholar
Brumer, A. and McGuinness, O., The behavior of the Mordell-Weil group of elliptic curves, Bull. Amer. Math. Soc. (N.S.) 23 (1990), 375382.CrossRefGoogle Scholar
Chiche-Lapierre, V., Length of elements in a Minkowski basis for an order in a number field, PhD thesis, University of Toronto (2019).Google Scholar
Davenport, H., On a principle of Lipschitz, J. Lond. Math. Soc. 26 (1951), 179183.CrossRefGoogle Scholar
Delone, B. N. and Faddeev, D. K., Theory of irrationalities of third degree, Acad. Sci. URSS. Trav. Inst. Math. Stekloff 11 (1940), 340; MR 0004269.Google Scholar
Duke, W. and Kowalski, E., A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations, Invent. Math. 139 (2000), 139, with an appendix by Dinakar Ramakrishnan; MR 1728875.CrossRefGoogle Scholar
Eskin, A. and Katznelson, R. Y., Singular symmetric matrices, Duke Math. J. 79 (1995), 515547.CrossRefGoogle Scholar
Fouvry, É., Nair, M. and Tenenbaum, G., L'ensemble exceptionnel dans la conjecture de Szpiro, Bull. Soc. Math. France 120 (1992), 485506; MR 1194273.CrossRefGoogle Scholar
Fujiwara, M., Über die obere Schranke des absoluten Betrages der Wurzeln einer algebraischen Gleichung, Tohoku Math. J., First Series 10 (1916), 167171.Google Scholar
Gan, W. T., Gross, B. and Savin, G., Fourier coefficients of modular forms on $G_2$, Duke Math. J. 115 (2002), 105169; MR 1932327.CrossRefGoogle Scholar
Goldfeld, D., Conjectures on elliptic curves over quadratic fields, in Number theory, Carbondale 1979 Proceedings of the Southern Illinois Number Theory Conference Carbondale, Southern Illinois University, Carbondale, IL, 1979, Lecture Notes in Mathematics, vol. 751 (Springer, Berlin, 1979), 108118.CrossRefGoogle Scholar
Heath-Brown, D. R., The average analytic rank of elliptic curves, Duke Math. J. 122 (2004), 591623.CrossRefGoogle Scholar
Hortsch, R., Counting elliptic curves of bounded Faltings height, Acta Arith. 173 (2016), 239253; MR 3512854.Google Scholar
Katz, N. M. and Sarnak, P., Random matrices, Frobenius eigenvalues, and monodromy, American Mathematical Society Colloquium Publications, vol. 45 (American Mathematical Society, Providence, RI, 1999).Google Scholar
Levi, F., Kubische Zahlkörper und binare kubische Formenklassen, Ber. Sächs. Akad. Wiss. Leipz., Mat.-Nat. Kl. 66 (1914), 2637.Google Scholar
Oesterlé, J., in Nouvelles approches du théorème de fermat, in Séminaire Bourbaki: volume 1987/88, exposés 686-699, Astérisque, vol. 161–162 (Société Mathématique de France, 1988), 165–186; MR 992208.Google Scholar
Silverman, J. H., Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151 (Springer, New York, 1994); MR 1312368.CrossRefGoogle Scholar
Taniguchi, T. and Thorne, F., Levels of distribution for sieve problems in prehomogeneous vector spaces, Math. Ann. 376 (2020), 15371559.CrossRefGoogle Scholar
Watkins, M., Some heuristics about elliptic curves, Exp. Math. 17 (2008), 105125.CrossRefGoogle Scholar
Wilkie, A. J., Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc. 9 (1996), 10511094.CrossRefGoogle Scholar
Young, M. P., Low-lying zeros of families of elliptic curves, J. Amer. Math. Soc. 19 (2006), 205250.CrossRefGoogle Scholar

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Large families of elliptic curves ordered by conductor
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Large families of elliptic curves ordered by conductor
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Large families of elliptic curves ordered by conductor
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *