Cotorsion of anti-cyclotomic Selmer groups on average

Kundu, Debanjana; Sprung, Florian

doi:10.1090/proc/16543

Cotorsion of anti-cyclotomic Selmer groups on average
HTML articles powered by AMS MathViewer

by Debanjana Kundu and Florian Sprung

Proc. Amer. Math. Soc. 152 (2024), 521-535

DOI: https://doi.org/10.1090/proc/16543

Published electronically: November 7, 2023

HTML | PDF | Request permission

Abstract:

For an elliptic curve, we study how many Selmer groups are cotorsion over the anti-cyclotomic $\mathbb {Z}_p$-extension as one varies the prime $p$ or the quadratic imaginary field in question.

References

M. Bertolini and H. Darmon, Iwasawa’s main conjecture for elliptic curves over anticyclotomic $\Bbb Z_p$-extensions, Ann. of Math. (2) 162 (2005), no. 1, 1–64. MR 2178960, DOI 10.4007/annals.2005.162.1
Massimo Bertolini, Selmer groups and Heegner points in anticyclotomic $\mathbf Z_p$-extensions, Compositio Math. 99 (1995), no. 2, 153–182. MR 1351834
Manjul Bhargava, Mass formulae for extensions of local fields, and conjectures on the density of number field discriminants, Int. Math. Res. Not. IMRN 17 (2007), Art. ID rnm052, 20. MR 2354798, DOI 10.1093/imrn/rnm052
Ajit Bhand and M. Ram Murty, Class numbers of quadratic fields, Hardy-Ramanujan J. 42 (2019), 17–25. MR 4221215
David Brink, Prime decomposition in the anti-cyclotomic extension, Math. Comp. 76 (2007), no. 260, 2127–2138. MR 2336287, DOI 10.1090/S0025-5718-07-01964-3
Henri Cohen, Francisco Diaz y Diaz, and Michel Olivier, On the density of discriminants of cyclic extensions of prime degree, J. Reine Angew. Math. 550 (2002), 169–209. MR 1925912, DOI 10.1515/crll.2002.071
H. Cohen and H. W. Lenstra Jr., Heuristics on class groups of number fields, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983) Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 33–62. MR 756082, DOI 10.1007/BFb0099440
Alina Carmen Cojocaru, On the surjectivity of the Galois representations associated to non-CM elliptic curves, Canad. Math. Bull. 48 (2005), no. 1, 16–31. With an appendix by Ernst Kani. MR 2118760, DOI 10.4153/CMB-2005-002-x
David A. Cox, Primes of the form $x^2 + ny^2$, 2nd ed., Pure and Applied Mathematics (Hoboken), John Wiley & Sons, Inc., Hoboken, NJ, 2013. Fermat, class field theory, and complex multiplication. MR 3236783, DOI 10.1002/9781118400722
J. Coates and R. Sujatha, Galois cohomology of elliptic curves, Tata Institute of Fundamental Research Lectures on Mathematics, vol. 88, Published by Narosa Publishing House, New Delhi; for the Tata Institute of Fundamental Research, Mumbai, 2000. MR 1759312
Noam D. Elkies, The existence of infinitely many supersingular primes for every elliptic curve over $\textbf {Q}$, Invent. Math. 89 (1987), no. 3, 561–567. MR 903384, DOI 10.1007/BF01388985
Ralph Greenberg, Iwasawa theory for $p$-adic representations, Algebraic number theory, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1989, pp. 97–137. MR 1097613, DOI 10.2969/aspm/01710097
Ralph Greenberg, Introduction to Iwasawa theory for elliptic curves. Arithmetic algebraic geometry (Park City, UT, 1999), IAS/Park City Math. Ser. 9 (1999), 407–464.
Jeffrey Hatley, Debanjana Kundu, and Anwesh Ray, Statistics for anticyclotomic Iwasawa invariants of elliptic curves, Preprint, arXiv:2106.01517, 2021.
Kuniaki Horie and Yoshihiro Ônishi, The existence of certain infinite families of imaginary quadratic fields, J. Reine Angew. Math. 390 (1988), 97–113. MR 953679
Peter Humphries, The density of square-free integers satisfying a congruence relation, Mathematics Stack Exchange, 2017, URL:https://math.stackexchange.com/q/2093697 (version: 2017-01-12).
Chan-Ho Kim, personal communication.
Winfried Kohnen and Ken Ono, Indivisibility of class numbers of imaginary quadratic fields and orders of Tate-Shafarevich groups of elliptic curves with complex multiplication, Invent. Math. 135 (1999), no. 2, 387–398. MR 1666783, DOI 10.1007/s002220050290
Shin-ichi Kobayashi, Iwasawa theory for elliptic curves at supersingular primes, Invent. Math. 152 (2003), no. 1, 1–36. MR 1965358, DOI 10.1007/s00222-002-0265-4
Chan-Ho Kim, Robert Pollack, and Tom Weston, On the freeness of anticyclotomic Selmer groups of modular forms, Int. J. Number Theory 13 (2017), no. 6, 1443–1455. MR 3656202, DOI 10.1142/S1793042117500804
Alain Kraus, Une remarque sur les points de torsion des courbes elliptiques, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 9, 1143–1146 (French, with English and French summaries). MR 1360773
Matteo Longo and Stefano Vigni, Plus/minus Heegner points and Iwasawa theory of elliptic curves at supersingular primes, Boll. Unione Mat. Ital. 12 (2019), no. 3, 315–347. MR 3989744, DOI 10.1007/s40574-018-0162-4
J. S. Milne, Arithmetic duality theorems, 2nd ed., BookSurge, LLC, Charleston, SC, 2006. MR 2261462
Karl Prachar, Über die kleinste quadratfreie Zahl einer arithmetischen Reihe, Monatsh. Math. 62 (1958), 173–176 (German). MR 92806, DOI 10.1007/BF01301288
Robert Pollack and Tom Weston, On anticyclotomic $\mu$-invariants of modular forms, Compos. Math. 147 (2011), no. 5, 1353–1381. MR 2834724, DOI 10.1112/S0010437X11005318
Jean-Pierre Serre, Galois properties of finite order points of elliptic curves, Invent. Math. 15 (1971), no. 4, 259–331.
Jean-Pierre Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323–401 (French). MR 644559
Pablo Soberón, Problem-solving methods in combinatorics, Birkhäuser/Springer Basel AG, Basel, 2013. An approach to olympiad problems. MR 3057774, DOI 10.1007/978-3-0348-0597-1
Nahid Walji, Supersingular distribution on average for congruence classes of primes, Acta Arith. 142 (2010), no. 4, 387–400. MR 2640067, DOI 10.4064/aa142-4-7
Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575, DOI 10.1007/978-1-4612-1934-7