Skip to main content Accessibility help


  • SUMAN AHMED (a1) and MENG FAI LIM (a2)


Let $p$ be an odd prime number and $E$ an elliptic curve defined over a number field $F$ with good reduction at every prime of $F$ above $p$ . We compute the Euler characteristics of the signed Selmer groups of $E$ over the cyclotomic $\mathbb{Z}_{p}$ -extension. The novelty of our result is that we allow the elliptic curve to have mixed reduction types for primes above $p$ and mixed signs in the definition of the signed Selmer groups.


Corresponding author


Hide All

M. F. Lim is supported by the National Natural Science Foundation of China under Grant Nos. 11550110172 and 11771164.



Hide All
[1] Büyükboduk, K. and Lei, A., ‘Integral Iwasawa theory of Galois representations for non-ordinary primes’, Math. Z. 286 (2017), 361398.
[2] Coates, J. and Sujatha, R., Galois Cohomology of Elliptic Curves, 2nd edn, Tata Institute of Fundamental Research Lectures on Mathematics, 88 (Narosa, New Delhi–Mumbai, 2010).
[3] Greenberg, R., ‘Iwasawa theory for p-adic representations’, in: Algebraic Number Theory—in Honor of K. Iwasawa, Advanced Studies in Pure Mathematics, 17 (eds. Coates, J., Greenberg, R., Mazur, B. and Satake, I.) (Kinokuniya–Mathematical Society of Japan, Tokyo, 1989), 97137.
[4] Greenberg, R., ‘Trivial zeros of p-adic L-functions’, in: p-adic Monodromy and the Birch and Swinnerton-Dyer Conjecture (Boston, MA, 1991), Contemporary Mathematics, 165 (American Mathematical Society, Providence, RI, 1994), 149174.
[5] Greenberg, R., ‘Iwasawa theory for elliptic curves’, in: Arithmetic Theory of Elliptic Curves (Cetraro, 1997), Lecture Notes in Mathematics, 1716 (ed. Viola, C.) (Springer, Berlin, 1999), 51144.
[6] Kato, K., ‘ p-adic Hodge theory and values of zeta functions of modular forms’, in: Cohomologies p-adiques et applications arithmétiques. III, Astérisque, 295 (Société Mathématique de France, Paris, 2004), 117290.
[7] Kim, B. D., ‘The parity conjecture for elliptic curves at supersingular reduction primes’, Compos. Math. 143 (2007), 4772.
[8] Kim, B. D., ‘The plus/minus Selmer groups for supersingular primes’, J. Aust. Math. Soc. 95(2) (2013), 189200.
[9] Kitajima, T. and Otsuki, R., ‘On the plus and the minus Selmer groups for elliptic curves at supersingular primes’, Tokyo J. Math. 41(1) (2018), 273303.
[10] Kobayashi, S., ‘Iwasawa theory for elliptic curves at supersingular primes’, Invent. Math. 152(1) (2003), 136.
[11] Lei, A. and Sujatha, R., ‘On Selmer groups in the supersingular reduction case’, Preprint.
[12] Mattuck, A., ‘Abelian varieties over p-adic ground fields’, Ann. of Math. (2) 62 (1955), 92119.
[13] Mazur, B., ‘Rational points of abelian varieties with values in towers of number fields’, Invent. Math. 18 (1972), 183266.
[14] Mazur, B., Tate, J. and Teitelbaum, J., ‘On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer’, Invent. Math. 84 (1986), 148.
[15] Neukirch, J., Schmidt, A. and Wingberg, K., Cohomology of Number Fields, 2nd edn, Grundlehren der Mathematischen Wissenschaften, 323 (Springer, Berlin, 2008).
[16] Perrin-Riou, B., ‘Arithmétique des courbes elliptiques et theórie d’Iwasawa’, Mém. Soc. Math. Fr. 17 (1984), 1129.
[17] Schneider, P., ‘Iwasawa L-functions of varieties over algebraic number fields. A first approach’, Invent. Math. 71 (1983), 251293.
MathJax is a JavaScript display engine for mathematics. For more information see


MSC classification


  • SUMAN AHMED (a1) and MENG FAI LIM (a2)


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed