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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
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