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Selmer groups of elliptic curves that can be arbitrarily large. (English) Zbl 1074.11032

Summary: In this article, it is shown that certain kinds of Selmer groups of elliptic curves can be arbitrarily large. The main result is that if \(p\) is a prime at least 5, then \(p\)-Selmer groups of elliptic curves can be arbitrarily large if one ranges over number fields of degree at most \(g+1\) over the rationals, where g is the genus of \(X_0(p)\). As a corollary, one sees that \(p\)-Selmer groups of elliptic curves over the rationals can be arbitrarily large for \(p=5,7\) and 13 (the cases \(p\leq7\) were already known). It is also shown that the number of elements of order \(N\) in the \(N\)-Selmer group of an elliptic curve over the rationals can be arbitrarily large for \(N=9,10,12,16\) and \(25\).

MSC:

11G05 Elliptic curves over global fields
14G25 Global ground fields in algebraic geometry
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[1] Bölling, R., Die Ordnung der Schafarewitsch-Tate Gruppe kann beliebig groß werden, Math. Nachr., 67, 157-179 (1975) · Zbl 0314.14008
[2] Cassels, J. W.S., Arithmetic on curves of genus 1 (VI). The Tate-Šafarevič group can be arbitrarily large, J. Reine Angew. Math., 214/215, 65-70 (1964) · Zbl 0236.14012
[3] Cassels, J. W.S., Arithmetic on curves of genus 1 (VIII). On the conjectures of Birch and Swinnerton-Dyer, J. Reine Angew. Math, 217, 180-189 (1965) · Zbl 0241.14017
[4] Fisher, T., Some examples of 5 and 7 descent for elliptic curves over \(Q\), J. Eur. Math. Soc., 3, 169-201 (2001) · Zbl 1007.11031
[5] Fricke, R., Lehrbuch der Algebra, III (1928), F. Vieweg & Sohn: F. Vieweg & Sohn Braunschweig · JFM 54.0187.20
[6] Halberstam, H.; Richert, H.-E., Sieve Methods (1974), Academic Press: Academic Press London · Zbl 0298.10026
[7] Knapp, A. W., Elliptic curves (1992), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0804.14013
[8] Kramer, K., A family of semistable elliptic curves with large Tate-Shafarevitch groups, Proc. Amer. Math. Soc., 89, 379-386 (1983) · Zbl 0567.14018
[9] S. Lang, Elliptic functions, Addison-Wesley Publishing Co., Inc., Reading, MA-London-Amsterdam, 1973.; S. Lang, Elliptic functions, Addison-Wesley Publishing Co., Inc., Reading, MA-London-Amsterdam, 1973. · Zbl 0316.14001
[10] Lang, S., Introduction to Modular Forms (1976), Springer-Verlag: Springer-Verlag Berlin
[11] Schaefer, E. F., Class groups and Selmer groups, J. Number Theory, 56, 79-114 (1996) · Zbl 0859.11034
[12] E.F. Schaefer, M. Stoll, How to do a \(p\); E.F. Schaefer, M. Stoll, How to do a \(p\) · Zbl 1119.11029
[13] Shimura, G., Introduction to the Arithmetic Theory of Automorphic Functions (1971), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0221.10029
[14] Silverman, J., The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, Vol. 106 (1986), Springer-Verlag: Springer-Verlag New York · Zbl 0585.14026
[15] J. Tate, Algorithm for determining the type of a singular fibre in an elliptic pencil, in: Modular Functions of One Variable IV, Lecture Notes in Mathematics, Vol. 476, Springer-Verlag, Berlin, 1975.; J. Tate, Algorithm for determining the type of a singular fibre in an elliptic pencil, in: Modular Functions of One Variable IV, Lecture Notes in Mathematics, Vol. 476, Springer-Verlag, Berlin, 1975.
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