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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
Software:
PARI/GP; ecdata
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References:
[1] Benois D., ”La conjecture de Bloch et Kato pour les motifs Q(m) sur un corps abélien.” (2000)
[2] Bernardi D., C. R. Acad. Sci. Paris Sér. I Math. 317 (3) pp 227– (1993)
[3] DOI: 10.1007/BF01390344 · Zbl 0358.10016
[4] Bloch S., The Grothendieck Festschrift pp 333– (1990)
[5] Colmez P., Astérisque pp 21– (2000)
[6] Cremona J., ”Algorithms for Modular Elliptic Curves.” (1997) · Zbl 0872.14041
[7] [GDR Medicis] GDR Medicis. Available from World Wide Web (http://www.medicis.polytechnique.fr
[8] DOI: 10.5802/aif.763 · Zbl 0403.12006
[9] DOI: 10.1007/BFb0093453
[10] Kato K., ”p-adic Hodge Theory and Values of Zeta Functions of Modular Forms.” (2001)
[11] Kurihara M., ”On the Tate-Shafarevich Groups Over Cyclotomic Fields of an Elliptic Curve with Supersingular Reduction.” (2000) · Zbl 1033.11028
[12] Manin Yu. I., Izv. Akad. Nauk SSSR, Ser. Mat. 36 (1972)
[13] DOI: 10.1007/BF01389815 · Zbl 0245.14015
[14] DOI: 10.1215/S0012-7094-91-06229-0 · Zbl 0735.14020
[15] DOI: 10.1007/BF01388731 · Zbl 0699.14028
[16] PARI/GP. 2003. [PARI/GP 03], logiciel, version 2.1.5, Bordeaux. Available from World Wide Web (http ://www.parigphome.de)
[17] [Perrin-Riou] Available from World Wide Web (http://www.math.u-psud.fr/ bpr
[18] DOI: 10.1007/BF01234420 · Zbl 0715.11030
[19] DOI: 10.1007/BF01232022 · Zbl 0781.14013
[20] DOI: 10.5802/aif.1362 · Zbl 0840.11024
[21] DOI: 10.1007/BF01231755 · Zbl 0838.11071
[22] Perrin-Riou B., p-adic L-Functions and p-Adic Representations. (2000) · Zbl 0988.11055
[23] Pollack R., PhD diss., in: ”On the p-adic L Function of a Modular Form at a Supersingular Prime.” (2001)
[24] DOI: 10.1017/CBO9780511662010.009
[25] DOI: 10.1007/BF01405086 · Zbl 0235.14012
[26] Stein, W. 2003. [Stein 03], Available from World Wide Web (http://modular.fas.harvard.edu/Tables/
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