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Elliptic Curves with Large Analytic Order of Ш(E)

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Algebra, Arithmetic, and Geometry

Part of the book series: Progress in Mathematics ((PM,volume 269))

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Summary

We present the results of our search for elliptic curves over \(\mathbb{Q}\) with exceptionally large analytic orders of the Tate-Shafarevich group. We exibit \(134\) examples of rank zero curves with |

E| > 18322 which was the largest known value for any explicit curve. Our record is a curve with

.

We also present examples of curves of rank zero with the value of \(L(E,1)\) much smaller, or much bigger, than in any previously known example. Finally, we present an example of a pair of non-isogeneous curves whose values of \(L(E,1)\) coincide in the first 11 digits after the point!

Partially supported by NSF Grants DMS-9707965 and DMS-0503401

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Notes

  1. 1.

    In this article we adhere to the following notational convention. Let \(A(E)\) and \(B(E)\) be some quantities dependent on a curve E belonging to a specified class \(\mathcal{C}\) of elliptic curves defined over \({\mathbb Q}\). We say that \(A(E)\ll B(E)\) if for any \(K> 0\), there exists \(N_0\) such that \(A(E)< KB(E)\) for all curves in \(\mathcal C\) with conductor \(N(E)> N_0\). This is meaningful only if \(\mathcal C\) contains infinitely many nonisomorphic curves. If either \(A(E)\) or \(B(E)\) depends on some parameter \(\epsilon\), then the choice of \(N_0\) is allowed to depend on \(\epsilon\).

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Authors and Affiliations

  1. Instytut Matematyki, Uniwersytet Szczeciński, ul. Wielkopolska 15, 70-451, Szczecin, Poland

    Andrzej Dąbrowski

  2. Department of Mathematics, University of California, Berkeley, CA, 94720-3840, USA

    Mariusz Wodzicki

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  1. Andrzej Dąbrowski
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Correspondence to Andrzej Dąbrowski .

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Editors and Affiliations

  1. Department of Mathematics, Courant Institute of Math. Sciences, New York University, Mercer St. 251, New York, 10012, New York, USA

    Yuri Tschinkel

  2. Eberly College of Science, Dept. Mathematics, Pennsylvania State University, University Park, 16802, Pennsylvania, USA

    Yuri Zarhin

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Dąbrowski, A., Wodzicki, M. (2009). Elliptic Curves with Large Analytic Order of Ш(E). In: Tschinkel, Y., Zarhin, Y. (eds) Algebra, Arithmetic, and Geometry. Progress in Mathematics, vol 269. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4745-2_9

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