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

Perrin-Riou, Bernadette

Exp. Math. 12, No. 2, 155-186 (2003).

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.

Reviewer: Franz Lemmermeyer (Bilkent)

MSC:

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

Keywords:

elliptic curves; Tate-Shafarevich groups; cyclotomic \(\mathbb Z_p\)-extensions; supersingular reduction; Iwasawa theory

Citations:

Zbl 0245.14015

Software:

ecdata; PARI/GP

References:

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