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Large Selmer groups over number fields

Published online by Cambridge University Press:  15 July 2009

ALEX BARTEL
ALEX BARTEL*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Wilber force Road, Cambridge, CB3 0WB. e-mail: [email protected]
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Abstract

Let p be a prime number and M a quadratic number field, M ≠ ℚ() if p ≡ 1 mod 4. We will prove that for any positive integer d there exists a Galois extension F/ℚ with Galois group D2p and an elliptic curve E/ℚ such that F contains M and the p-Selmer group of E/F has size at least pd.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 148 , Issue 1 , January 2010 , pp. 73 - 86
Copyright
Copyright © Cambridge Philosophical Society 2009

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References

REFERENCES

[1]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 (1964), 65–70.CrossRefGoogle Scholar
[2]Clark, P.The period-index problem in WC-groups i: elliptic curves. J. Number Theory. 114 (1) (2005), 193–208.CrossRefGoogle Scholar
[3]Clark, P. L. and Sharif, S. Period, index and potential sha. arXiv:0811.3019v1 (2008).Google Scholar
[4]Cohen, H.Advanced Topics in Computational Number Theory. GTM 193 (Springer-Verlag, 2000).CrossRefGoogle Scholar
[5]Curtis, C. and Reiner, I.Methods of Representation Theory with Applications to Finite Groups and Orders, volume II (John Wiley and Sons, 1987).Google Scholar
[6]Dokchitser, T. and Dokchitser, V. On the Birch–Swinnerton–Dyer quotients modulo squares. arXiv:math/0610290v3. To appear in Annals of Math. (2007).Google Scholar
[7]Dokchitser, T. and Dokchitser, V. Regulator constants and the parity conjecture. arXiv:0709.2852v3. To appear in Invent. Math. (2007).Google Scholar
[8]Fisher, T.Some examples of 5 and 7 decent for elliptic curves over ℚ. J. Eur. Math. Soc. 3 (2001), 169–201.CrossRefGoogle Scholar
[9]Kloosterman, R.The p-part of the Tate–Shafarevich groups of elliptic curves can be arbitrarily large. Journal de Theorie des Nombres de Bordeaux 17 (2005), 787–800.Google Scholar
[10]Kloosterman, R. and Schaefer, E.Selmer groups of elliptic curves that can be arbitrarily large. J. Number Theory 99 (2003), 148–163.CrossRefGoogle Scholar
[11]Kramer, K.A family of semistable elliptic curves with large Tate–Shafarevitch groups. Proc. Amer. Math. Soc. 89 (1983), 379–386.CrossRefGoogle Scholar
[12]Matsuno, K. Elliptic curves with large Tate–Shafarevich groups over a number field. To appear in Math. Res. Lett.Google Scholar
[13]Matsuno, K.Construction of elliptic curves with large Iwasawa λ-invariants and large Tate–Shafarevich groups. Manuscripta Math. 122 (3) (2007), 289–304.CrossRefGoogle Scholar
[14]Mazur, B.Modular curves and the Eisenstein ideal. Inst. Hautes Etudes Sci. Publ. Math. 47 (1977), 33–186.CrossRefGoogle Scholar
[15]Mazur, B.Rational isogenies of prime degree. Invent. Math. 44 (1978), 129–162.CrossRefGoogle Scholar
[16]Milne, J. S.On the arithmetic of abelian varieties. Invent. Math. 17 (1972), 177–190.CrossRefGoogle Scholar
[17]Milne, J. S.Arithmetic Duality Theorems. Number 1 in Perspectives in Mathematics. (Academic Press, 1986).Google Scholar
[18]Serre, J.-P.Propriétés galoisiennes des points d'ordre fini des courbes elliptiques. Invent. Math. 15 (1972), 259–331.CrossRefGoogle Scholar
[19]Silverman, J. H.The Arithmetic of Elliptic Curves. GTM 106 (Springer Verlag, 1985).Google Scholar
[20]Silverman, J. H.Advanced Topics in the Arithmetic of Elliptic Curves. GTM 151 (Springer-Verlag, 1994).CrossRefGoogle Scholar
[21]Tate, J. On the conjecture of Birch and Swinnerton–Dyer and a geometric analog. Séminaire Bourbaki 18e année (no. 306) (1965/66).Google Scholar
[22]Zassenhaus, H.Neuer Beweis der Endlichkeit der Klassenzahl bei unimodularer Äquivalenz endlicher ganzzahliger Substitutionsgruppen. Hamb. Abh. 12 (1938), 276–288.CrossRefGoogle Scholar