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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.

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G05 Elliptic curves over global fields
11R23 Iwasawa theory
PARI/GP; ecdata
Full Text: DOI Euclid EuDML
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