On the parity of ranks of Selmer groups. II (Q2708180)

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 NEWLINENEWLINE\[NEWLINE0\to E(F)\otimes\mathbb Q_p/\mathbb Z_p\to S_p(E/F)\to\text Ш(E/F)[p^{\infty}]\to 0,NEWLINE\]NEWLINE NEWLINEwhere \(S_p(E/F)=\varinjlim_nS(E/F,p^n)\). 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.NEWLINENEWLINENEWLINELet \(K\) 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 \(\text{corank}_{\mathbb Z_p}S_p(E/K)\equiv 1\pmod 2\).NEWLINENEWLINEThe reader can find references for earlier results in Part I [the author and \textit{A. Plater}, Asian J. Math. 4, No. 2, 437--497 (2000; Zbl 0973.11066)].