## Statistics for Iwasawa invariants of elliptic curves

HTML articles powered by AMS MathViewer

- by Debanjana Kundu and Anwesh Ray PDF
- Trans. Amer. Math. Soc.
**374**(2021), 7945-7965 Request permission

Corrigendum: Trans. Amer. Math. Soc. (to appear).

## Abstract:

We study the average behaviour of the Iwasawa invariants for the Selmer groups of elliptic curves, setting out new directions in arithmetic statistics and Iwasawa theory.## References

- L. Babinkostova, J. C. Bahr, Y. H. Kim, E. Neyman, and G. K. Taylor,
*Anomalous primes and the elliptic Korselt criterion*, J. Number Theory**201**(2019), 108–123. MR**3958043**, DOI 10.1016/j.jnt.2019.02.013 - Dominique Bernardi and Bernadette Perrin-Riou,
*Variante $p$-adique de la conjecture de Birch et Swinnerton-Dyer (le cas supersingulier)*, C. R. Acad. Sci. Paris Sér. I Math.**317**(1993), no. 3, 227–232 (French, with English and French summaries). MR**1233417** - Armand Brumer,
*The average rank of elliptic curves. I*, Invent. Math.**109**(1992), no. 3, 445–472. MR**1176198**, DOI 10.1007/BF01232033 - John Coates and Gary McConnell,
*Iwasawa theory of modular elliptic curves of analytic rank at most $1$*, J. London Math. Soc. (2)**50**(1994), no. 2, 243–264. MR**1291735**, DOI 10.1112/jlms/50.2.243 - 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** - J. Coates and R. Sujatha,
*Fine Selmer groups of elliptic curves over $p$-adic Lie extensions*, Math. Ann.**331**(2005), no. 4, 809–839. MR**2148798**, DOI 10.1007/s00208-004-0609-z - J. Cremona and M. Sadek,
*Local and global densities for Weierstrass models of elliptic curves.*preprint, arXiv:2003.08454, 2020. - Christophe Delaunay,
*Heuristics on Tate-Shafarevitch groups of elliptic curves defined over $\Bbb Q$*, Experiment. Math.**10**(2001), no. 2, 191–196. MR**1837670** - Byoung Du Kim,
*The plus/minus Selmer groups for supersingular primes*, J. Aust. Math. Soc.**95**(2013), no. 2, 189–200. MR**3142355**, DOI 10.1017/S1446788713000165 - Dorian Goldfeld,
*Conjectures on elliptic curves over quadratic fields*, Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979) Lecture Notes in Math., vol. 751, Springer, Berlin, 1979, pp. 108–118. MR**564926** - R. Greenberg.
*Arithmetic Theory of Elliptic Curves: Lectures given at the 3rd Session of the Centro Internazionale Matematico Estivo (CIME) held in Cetaro, Italy, July 12-19, 1997*. Springer, 1999. - Susan Howson,
*Euler characteristics as invariants of Iwasawa modules*, Proc. London Math. Soc. (3)**85**(2002), no. 3, 634–658. MR**1936815**, DOI 10.1112/S0024611502013680 - Kazuya Kato,
*$p$-adic Hodge theory and values of zeta functions of modular forms*, Astérisque**295**(2004), ix, 117–290 (English, with English and French summaries). Cohomologies $p$-adiques et applications arithmétiques. III. MR**2104361** - Nicholas M. Katz and Peter Sarnak,
*Random matrices, Frobenius eigenvalues, and monodromy*, American Mathematical Society Colloquium Publications, vol. 45, American Mathematical Society, Providence, RI, 1999. MR**1659828**, DOI 10.1090/coll/045 - 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 - V. A. Kolyvagin,
*Finiteness of $E(\textbf {Q})$ and SH$(E,\textbf {Q})$ for a subclass of Weil curves*, Izv. Akad. Nauk SSSR Ser. Mat.**52**(1988), no. 3, 522–540, 670–671 (Russian); English transl., Math. USSR-Izv.**32**(1989), no. 3, 523–541. MR**954295**, DOI 10.1070/IM1989v032n03ABEH000779 - Antonio Lei and Ramdorai Sujatha,
*On Selmer groups in the supersingular reduction case*, Tokyo J. Math.**43**(2020), no. 2, 455–479. MR**4185844**, DOI 10.3836/tjm/1502179319 - Barry Mazur,
*Rational points of abelian varieties with values in towers of number fields*, Invent. Math.**18**(1972), 183–266. MR**444670**, DOI 10.1007/BF01389815 - Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg,
*Cohomology of number fields*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2000. MR**1737196** - Loren D. Olson,
*Hasse invariants and anomalous primes for elliptic curves with complex multiplication*, J. Number Theory**8**(1976), no. 4, 397–414. MR**450191**, DOI 10.1016/0022-314X(76)90087-1 - Bernadette Perrin-Riou,
*Fonctions $L$ $p$-adiques d’une courbe elliptique et points rationnels*, Ann. Inst. Fourier (Grenoble)**43**(1993), no. 4, 945–995 (French, with English and French summaries). MR**1252935** - Bernadette Perrin-Riou,
*Théorie d’Iwasawa des représentations $p$-adiques sur un corps local*, Invent. Math.**115**(1994), no. 1, 81–161 (French). With an appendix by Jean-Marc Fontaine. MR**1248080**, DOI 10.1007/BF01231755 - Hourong Qin,
*Anomalous primes of the elliptic curve $E_D\colon y^2=x^3+D$*, Proc. Lond. Math. Soc. (3)**112**(2016), no. 2, 415–453. MR**3471254**, DOI 10.1112/plms/pdv072 - A. Ray and R. Sujatha,
*Euler characteristics and their congruences for multi-signed Selmer groups*, preprint, arXiv:2011.05387, 2020. - Penny C. Ridgdill,
*On the frequency of finitely anomalous elliptic curves*, ProQuest LLC, Ann Arbor, MI, 2010. Thesis (Ph.D.)–University of Massachusetts Amherst. MR**2941460** - M. Sadek,
*Counting elliptic curves with bad reduction over a prescribed set of primes*, preprint, arXiv:1704.02056, 2017. - Peter Schneider,
*$p$-adic height pairings. I*, Invent. Math.**69**(1982), no. 3, 401–409. MR**679765**, DOI 10.1007/BF01389362 - Peter Schneider,
*$p$-adic height pairings. II*, Invent. Math.**79**(1985), no. 2, 329–374. MR**778132**, DOI 10.1007/BF01388978 - René Schoof,
*Nonsingular plane cubic curves over finite fields*, J. Combin. Theory Ser. A**46**(1987), no. 2, 183–211. MR**914657**, DOI 10.1016/0097-3165(87)90003-3 - A. N. Shankar, A. Shankar, and X. Wang,
*Families of elliptic curves ordered by conductor.**Compos. Math.*, to appear. - Joseph H. Silverman,
*The arithmetic of elliptic curves*, 2nd ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR**2514094**, DOI 10.1007/978-0-387-09494-6 - Florian Sprung,
*A formulation of $p$-adic versions of the Birch and Swinnerton-Dyer conjectures in the supersingular case*, Res. Number Theory**1**(2015), Paper No. 17, 13. MR**3501001**, DOI 10.1007/s40993-015-0018-2 - X. Wan,
*Iwasawa main conjecture for supersingular elliptic curves and BSD conjecture*, preprint, arXiv:1411.6352, 2014. - Mark Watkins,
*Some heuristics about elliptic curves*, Experiment. Math.**17**(2008), no. 1, 105–125. MR**2410120** - C. Wuthrich,
*The fine Selmer group and height pairings*. PhD thesis, University of Cambridge, 2004. - Christian Wuthrich,
*On $p$-adic heights in families of elliptic curves*, J. London Math. Soc. (2)**70**(2004), no. 1, 23–40. MR**2064750**, DOI 10.1112/S0024610704005277 - Sarah Livia Zerbes,
*Generalised Euler characteristics of Selmer groups*, Proc. Lond. Math. Soc. (3)**98**(2009), no. 3, 775–796. MR**2500872**, DOI 10.1112/plms/pdn049

## Additional Information

**Debanjana Kundu**- Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
- MR Author ID: 1409674
- ORCID: 0000-0002-1545-3841
- Email: dkundu@math.ubc.ca
**Anwesh Ray**- Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
- MR Author ID: 1376642
- Email: anweshray@math.ubc.ca
- Received by editor(s): February 10, 2021
- Published electronically: July 27, 2021
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**374**(2021), 7945-7965 - MSC (2020): Primary 11R23, 11R45, 11G05
- DOI: https://doi.org/10.1090/tran/8478
- MathSciNet review: 4328687