# zbMATH — the first resource for mathematics

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
##### Keywords:
Selmer groups; parity of ranks
Full Text: