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Ranks of elliptic curves over \(\mathbb{Z}_p^2\)-extensions

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Abstract

Let E be an elliptic curve with good reduction at a fixed odd prime p and K an imaginary quadratic field where p splits. We give a growth estimate for the Mordell-Weil rank of E over finite extensions inside the \(\mathbb{Z}_p^2\)-extension of K.

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References

  1. M. Bertolini, Selmer groups and Heegner points in anticyclotomic Zp-extensions, Compositio Mathematica 99 1995, 153–182.

    MathSciNet  MATH  Google Scholar 

  2. M. Bertolini, Growth of Mordell-Weil groups in anticyclotomic towers, in Arithmetic geometry (Cortona, 1994), Symposia Mathematica, Vol. 37, Cambridge University Press, Cambridge, 1997, pp. 23–44.

    Google Scholar 

  3. S. Bloch and K. Kato, L-functions and Tamagawa numbers of motives, in The Grothendieck Festschrift, Vol. I, Progress in Mathematics, Vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 333–400.

    Google Scholar 

  4. K. Büyükboduk and A. Lei, Iwasawa theory of elliptic modular forms over imaginary quadratic fields at non-ordinary primes, International Mathematics Research Notices, to appear, https://doi.org/10.1093/imrn/rnz117.

  5. F. Castella, M. Çiperiani, C. Skinner and F. Sprung, On the Iwasawa main conjectures for modular forms at non-ordinary primes, https://arxiv.org/abs/1804.10993.

  6. C. Cornut, Mazur’s conjecture on higher Heegner points, Inventiones mathematicae 148 2002, 495–523.

    Article  MathSciNet  Google Scholar 

  7. A. Cuoco and P. Monsky, Class numbers in \(\mathbb{Z}_p^d\)-extensions, Mathematische Annalen 255 1981, 235–258.

    Article  MathSciNet  Google Scholar 

  8. A. Cuoco, The growth of Iwasawa invariants in a family, Compositio Mathematica 41 1980, 415–437.

    MathSciNet  MATH  Google Scholar 

  9. R. Greenberg, Introduction to the Iwasawa Theory for elliptic curves, in Arithmetic algebraic geometry (Park City, UT, 1999), IAS/Park City Mathematics Series, Vol. 9, American Mathematical Society, Providence, RI, 2001, pp. 407–464.

    Google Scholar 

  10. R. Greenberg, Galois theory for the Selmer group of an abelian variety, Compositio Mathematica 136 (2003), 255–297.

    Article  MathSciNet  Google Scholar 

  11. M. Harris, Correction to: “p-adic representations arising from descent on abelian varieties”, Compositio Mathematica 121 2000, 105–108.

    Article  MathSciNet  Google Scholar 

  12. K. Kato, p-adic Hodge theory and values of zeta functions of modular forms, Astérisque 295 2004, 117–290.

    MathSciNet  MATH  Google Scholar 

  13. S.-I. Kobayashi, Iwasawa theory for elliptic curves at supersingular primes, Inventiones Mathematicae 152 2003, 1–36.

    Article  MathSciNet  Google Scholar 

  14. B. D. Kim, Signed-Selmer groups over the \(\mathbb{Z}_p^2\)-extension of an imaginary quadratic field, Canadian Journal of Mathematics 66 2014, 826–843.

    Article  MathSciNet  Google Scholar 

  15. T. Kitajima and R. Otsuki, On the plus and minus Selmer groups for elliptic curves at supersingular primes, Tokyo Journal of Mathematics 41 2018, 273–303.

    Article  MathSciNet  Google Scholar 

  16. A. Lei, Factorisation of two-variable p-adic L-functions, Canadian Mathematical Bulletin 57 2014, 845–852.

    Article  MathSciNet  Google Scholar 

  17. A. Lei, D. Loeffler and S. L. Zerbes, On the asymptotic growth of Bloch-Kato-Shafarevich-Tate groups of modular forms over cyclotomic extensions, Canadian Journal of Mathematics 69 2017, 826–850.

    Article  MathSciNet  Google Scholar 

  18. A. Lei, D. Loeffler and S. L. Zerbes, Wach modules and Iwasawa theory for modular forms, Asian Journal of Mathematics 14 2010, 475–528.

    Article  MathSciNet  Google Scholar 

  19. D. Loeffler and S. L. Zerbes, Iwasawa theory and p-adic L-functions over \(\mathbb{Z}_p^2\)-extensions, International Journal of Number Theory 10 2014, 2045–2095.

    Article  MathSciNet  Google Scholar 

  20. M. Longo and S. Vigni, Plus/minus Heegner points and Iwasawa theory of elliptic curves at supersingular primes, Bollettino dell’Unione Matematica Italiana 12 2019, 315–347.

    Article  MathSciNet  Google Scholar 

  21. B. Mazur, Modular curves and arithmetic, in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Warsaw, 1983), PWN, Warsaw, 1984, pp. 185–211.

    Google Scholar 

  22. B. Mazur and K. Rubin, Elliptic curves and class field theory, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 185–195.

    Google Scholar 

  23. B. Mazur and K. Rubin, Studying the growth of Mordell-Weil, Documenta Mathematics Extra Volume (2003), 585–607.

  24. D. E. Rohrlich, On L-functions of elliptic curves and cyclotomic towers, Inventiones Mathematicae 75 1984, 409–423.

    Article  MathSciNet  Google Scholar 

  25. F. Sprung, Iwasawa theory for elliptic curves at supersingular primes: a pair of main conjectures, Journal of Number Theory 132 2012, 1483–1506.

    Article  MathSciNet  Google Scholar 

  26. F. Sprung, The Šafarevič-Tate group in cyclotomic ℤp-extensions at supersingular primes, Journal für die Reine und Angewandte Mathematik 681 2013, 199–218.

    MathSciNet  MATH  Google Scholar 

  27. F. Sprung, The Iwasawa main conjecture for elliptic curves at odd supersingular primes, https://arxiv.org/abs/1610.10017.

  28. V. Vatsal, Special values of anticyclotomic L-functions, Duke Mathematical Journal 116 2003, 219–261.

    Article  MathSciNet  Google Scholar 

  29. J. Van Order, Some remarks on the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication, Journal of Algebra 350 2012, 273–299.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Département de mathématiques et de statistique, Université Laval Pavillon Alexandre-Vachon, 1045 avenue de la médecine, Québec, QC, G1V 0A6, Canada

    Antonio Lei

  2. School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287-1804, USA

    Florian Sprung

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  1. Antonio Lei
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  2. Florian Sprung
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Correspondence to Florian Sprung.

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Lei, A., Sprung, F. Ranks of elliptic curves over \(\mathbb{Z}_p^2\)-extensions. Isr. J. Math. 236, 183–206 (2020). https://doi.org/10.1007/s11856-020-1969-0

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  • DOI: https://doi.org/10.1007/s11856-020-1969-0

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