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On the parity of ranks of Selmer groups. II. (English. Abridged French version) Zbl 1090.11037
Let $$E/\mathbb Q$$ be an elliptic curve with conductor $$N$$ and let $$p$$ be a prime number. For each number field $$F$$ and integer $$m\geq 1$$, let $$S(E/F,m)$$ be the Selmer group of $$E/F$$ relative to $$m$$. We have an exact sequence
$0\to E(F)\otimes\mathbb Q_p/\mathbb Z_p\to S_p(E/F)\to\text Ш(E/F)[p^{\infty}]\to 0,$
where $$S_p(E/F)=\varinjlim_nS(E/F,p^n)$$. The main result of the paper states that if $$E$$ has good reduction at $$p$$, then $$\text{corank}_{\mathbb{Z}_p}S_p(E/\mathbb{Q})\equiv\text{ord}_{s=1}L(E,s)\pmod 2$$, where $$L(E,s)$$ denotes the Hasse-Weil $$L$$-function of $$E$$. This result can be obtained as a weak consequence of the Birch and Swinnerton-Dyer conjecture and is known as the parity conjecture for Selmer groups. The result is deduced from the following theorem.
Let $$K$$ be an imaginary quadratic field, suppose the prime factors of $$N$$ are decomposed in $$K/\mathbb Q$$. If $$E$$ has good reduction at $$p$$, then $$\text{corank}_{\mathbb Z_p}S_p(E/K)\equiv 1\pmod 2$$.
The reader can find references for earlier results in Part I [the author and A. Plater, Asian J. Math. 4, No. 2, 437–497 (2000; Zbl 0973.11066)].

##### MSC:
 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11G05 Elliptic curves over global fields 11F33 Congruences for modular and $$p$$-adic modular forms 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations
##### Keywords:
Selmer groups; elliptic curve; parity conjecture
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