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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
ecdata; PARI/GP
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References:
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