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On the parity of ranks of Selmer groups. (English) Zbl 0973.11066
Let \(f=\sum_{n\geq 1}a_n(f)q^n\) be a normalized newform of weight \(k_0\geq 2\) for \(\Gamma_0(N)\). Let \(F\) be the number field generated by the coefficients of \(f\) and \(\mathfrak{p}\) a prime of \(F\) lying over \(p\). Let \(V(f):G_{\mathbb Q}\to\text{GL}_2(F_{\mathfrak{p}})\) be the 2-dimensional representation associated to \(F\) of the absolute Galois group \(G_{\mathbb Q}=\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\). Let \(V_{k_0}=V(f)(k_0/2)\) be its Tate twist (which is self-dual). Let \(L_{\infty}(f,s)=\sum_{n\geq 1}a_n(f)n^{-s}\) be the associated complex \(L\)-function. Bloch and Kato defined {} a generalized “Selmer group” \(H^1_f(\mathbb Q,V_{k_0})\subset H^1(\mathbb Q,V_{k_0})\) and conjectured that \(\text{ord}_{s=k_0/2}L_{\infty}(f,s)\) would be equal to \(\dim_{F_{\mathfrak{p}}}H^1_f(\mathbb Q,V_{k_0})\). The parity conjecture just requires that these two numbers have the same parity.
In this paper the authors assume that \(p>3\) and \(f\) is ordinary at \(p\). By Hida’s theory, this means that there exists an integer \(c\geq 0\) such that for every integer \(k\geq 2\) satisfying \(k\equiv k_0\pmod{(p-1)p^c}\), then there is an ordinary newform \(f_k\) of weight \(k\) on \(\Gamma_0(N)\) such that \(f_{k_0}=f\) and \(k\equiv k'\pmod{(p-1)p^{n+c}}\) implies \(f_k\equiv f_{k'}\pmod{p^n}\). Let \(\varepsilon_k=1\), if \(p\mid\mid N\), \(k=2\), \(a_p(f)=1\), and \(\varepsilon_k=0\), otherwise.
The authors consider Selmer groups \(H^1_f(\mathbb Q,V_k)\) associated to Galois representations \(V_k=V(f_k)\) \((k/2)\) and “extended Selmer groups” sitting inside the exact sequences \[ 0\to(F_{\mathfrak{p}})^{\varepsilon_k}\to {\widetilde{H}}^1_f(\mathbb Q,V_k)\to H^1_f(\mathbb Q,V_k)\to 0. \] The paper’s main result states that if \(p>3\), \(f\) is ordinary at \(p\), \(k_0\equiv 2\pmod{(p-1)}\) and \(V(f)\) has an irreducible residual representation, then there exists an integer \(n\geq c\) such that \(\dim_{F_{\mathfrak{p}}}{\widetilde{H}}^1_f(\mathbb Q,V_k)\equiv\dim_{F_{\mathfrak{p}}}{\widetilde{H}}^1_f(\mathbb Q, V_{k_0})\pmod{2}\), whenever \(k\geq 2\) and \(k\equiv k_0\pmod{2(p-1)p^n}\).
The next result relates the latter one to Greenberg’s conjecture. There exists a two variable \(p\)-adic \(L\)-function \(L_p(k,s)\) defined for \(s\in\mathbb Z_p\) and \(k\in k_0+p^c\mathbb Z_p\) such that \(L_p(k,k-s)=w_pL_p(k,s)\), and if \(k\equiv k_0\pmod{2(p-1)p^c}\), \(k\geq 2\) integer, then \(L_p(k,s)=C_kL_p(f_k,s)\) for some \(C_k\neq 0\), where \(L_p(f_k,s)\) denotes the \(p\)-adic \(L\)-function associated to \(f_k\). The value \(w_p(f_k)=w_{\infty}(f_k)(-1)^{\varepsilon_k}=w_p=(-1)^{e_p}\) does not depend on \(k\) and \(e_p\equiv e_{\infty}+\varepsilon_{k_0}\pmod{2}\). Greenberg’s conjecture states that the generic order of vanishing of \(L_p(k,s)\) on the line \(s=k/2\) is zero or one. The second result of the paper says that under the assumptions of the first theorem, Greenberg’s conjecture implies \(\dim_{F_{\mathfrak{p}}}H^1_f(\mathbb Q,V_{k_0})\equiv e_p-\varepsilon_{k_0}\equiv e_{\infty}\pmod{2}\).
Finally, in the special case where \(k_0=2\) and \(F=\mathbb Q\), \(f\) corresponds to a modular elliptic curve \(E\) defined over \(\mathbb Q\). The Selmer group associated to \(V_{k_0}=T_p(E)\otimes_{\mathbb Z_p}\mathbb Q_p\) coincides with the usual Selmer group with \(\mathbb Q_p\) coefficients and sits inside the exact sequence \[ 0\to E(\mathbb Q)\otimes\mathbb Q_p\to H^1_f(\mathbb Q,V_{k_0})\to T_p(\text{ Ш}(E/\mathbb Q))\otimes_{\mathbb Z_p}\mathbb Q_p\to 0. \] The authors prove that if \(E/\mathbb Q\) have ordinary reduction at some prime \(p>3\), the \(p\)-torsion \(E_p(\overline{\mathbb Q})\) is an irreducible \(\mathbb F_p[G_{\mathbb Q}]\)-module and the Greenberg’s conjecture holds for the two dimensional \(p\)-adic \(L\)-function of \(E\), then \[ \dim_{\mathbb Q}(E(\mathbb Q)\otimes\mathbb Q)+\text{cork}_{\mathbb Z_p}\text{ Ш}(E/\mathbb Q)\equiv\text{ord}_{s=1}L_{\infty}(E,s)\pmod{2}. \]

MSC:
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11F33 Congruences for modular and \(p\)-adic modular forms
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F80 Galois representations
11G05 Elliptic curves over global fields
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