## Arithmetic of elliptic curves with supersingular reduction in $$p$$. (Arithmétique des courbes elliptiques à réduction supersingulière en $$p$$.)(French. English summary)Zbl 1061.11031

One of the main conjectures on elliptic curves gives information on the behavior of the rank of an elliptic curve in the cyclotomic $$\mathbb Z_p$$-extension of $$\mathbb Q$$. For primes with good ordinary reduction, parts of these conjectures were proved by B. Mazur [Invent. Math. 18, 183–266 (1972; Zbl 0245.14015)]; primes with supersingular reduction have resisted such a treatment until now, and are the subject of this article.
Part of the author’s main result (which is too long to be quoted in full) reads as follows: let $$E$$ be an elliptic curve defined over $$\mathbb Q$$ and with supersingular reduction at the prime $$p$$, and let $$K_n$$ be the field of $$p^n$$-th roots of unity.
1. The rank of $$E(K_\infty)$$ is finite, and the corank of $$\text Ш(E/K_n)$$ is bounded.
2. If $$E(K_n)$$ and $$\text Ш(E/K_n)$$ are finite for all $$n$$, then there exist integers $$\mu_+$$, $$\mu_-$$, $$\lambda_+$$, $$\lambda_-$$ and a rational number $$\nu$$ such that $$\text{ord}_p\text Ш(E/K_n)$$ is given by a simple expression involving only $$p$$, $$n$$, and the Iwasawa constants above.
3. If $$L(E,1)/\Omega_E$$ is a $$p$$-unit, and if $$E(K_n)$$ and $$\text Ш(E/K_n)(p)$$ are finite for all $$n$$, then
$$\text{ord}_p \text Ш(E/K_n)(p) = \lfloor p^{n+1}/(p^2-1) - n/2 \rfloor$$ .
The author also provides many numerical examples obtained with the help of PARI.

### MSC:

 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11G05 Elliptic curves over global fields 11R23 Iwasawa theory

Zbl 0245.14015

ecdata; PARI/GP
Full Text:

### References:

  Benois D., ”La conjecture de Bloch et Kato pour les motifs Q(m) sur un corps abélien.” (2000)  Bernardi D., C. R. Acad. Sci. Paris Sér. I Math. 317 (3) pp 227– (1993)  DOI: 10.1007/BF01390344 · Zbl 0358.10016  Bloch S., The Grothendieck Festschrift pp 333– (1990)  Colmez P., Astérisque pp 21– (2000)  Cremona J., ”Algorithms for Modular Elliptic Curves.” (1997) · Zbl 0872.14041  [GDR Medicis] GDR Medicis. Available from World Wide Web (http://www.medicis.polytechnique.fr  DOI: 10.5802/aif.763 · Zbl 0403.12006  DOI: 10.1007/BFb0093453  Kato K., ”p-adic Hodge Theory and Values of Zeta Functions of Modular Forms.” (2001)  Kurihara M., ”On the Tate-Shafarevich Groups Over Cyclotomic Fields of an Elliptic Curve with Supersingular Reduction.” (2000) · Zbl 1033.11028  Manin Yu. I., Izv. Akad. Nauk SSSR, Ser. Mat. 36 (1972)  DOI: 10.1007/BF01389815 · Zbl 0245.14015  DOI: 10.1215/S0012-7094-91-06229-0 · Zbl 0735.14020  DOI: 10.1007/BF01388731 · Zbl 0699.14028  PARI/GP. 2003. [PARI/GP 03], logiciel, version 2.1.5, Bordeaux. Available from World Wide Web (http ://www.parigphome.de)  [Perrin-Riou] Available from World Wide Web (http://www.math.u-psud.fr/ bpr  DOI: 10.1007/BF01234420 · Zbl 0715.11030  DOI: 10.1007/BF01232022 · Zbl 0781.14013  DOI: 10.5802/aif.1362 · Zbl 0840.11024  DOI: 10.1007/BF01231755 · Zbl 0838.11071  Perrin-Riou B., p-adic L-Functions and p-Adic Representations. (2000) · Zbl 0988.11055  Pollack R., PhD diss., in: ”On the p-adic L Function of a Modular Form at a Supersingular Prime.” (2001)  DOI: 10.1017/CBO9780511662010.009  DOI: 10.1007/BF01405086 · Zbl 0235.14012  Stein, W. 2003. [Stein 03], Available from World Wide Web (http://modular.fas.harvard.edu/Tables/
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.