Control theorem and functional equation of Selmer groups over p-adic Lie extensions

Abstract

Let $K_\infty$ be a p-adic Lie extension of a number field K which fits into the setting of non-commutative Iwasawa theory formulated by Coates–Fukaya–Kato–Sujatha–Venjakob. For the first main result, we will prove the control theorem of Selmer group associated to a motive, which generalizes previous results by the second author and Greenberg. As an application of this control theorem, we prove the functional equation of the dual Selmer groups, which generalizes previous results by Greenberg, Perrin-Riou and Zábrádi. Especially, we generalize the result of Zábrádi for elliptic curves to general motives. Note that our proof is different from the proof of Zábrádi even in the case of elliptic curves. We also discuss the functional equation for the analytic p-adic L-functions and check the compatibility with the functional equation of the dual Selmer groups.

Appendix A: Conjectural existence of analytic p-adic L-functions

In this section, we state a conjectural existence of analytic p-adic L-function for a given Galois representation V. For technical reasons, we restrict ourselves to the situation $K=\mathbb {Q}$ in the analytic side. Also, in order to talk about the algebraic part of special values of Hasse–Weil L-functions, we fix embeddings $\overline{\mathbb {Q}}\hookrightarrow \mathbb {C}$ and $\overline{\mathbb {Q}}\hookrightarrow \overline{\mathbb {Q}}_p$ simultaneously.

Let V be a p-adic Galois representation of $G_\mathbb {Q}$ which satisfies the condition (Geom) stated in Introduction. Then the L-function L(s, V) given by the following Euler product is convergent on $\mathrm {Re}(s) > \dfrac{w}{2} +1$:

$$\begin{aligned} L(s,V) = \prod _{q \not =p } \mathrm {det} (1-\mathrm {Frob}^{-1}_q q^{-s} ;V^{I_q })^{-1} \times \mathrm {det} (1-\varphi p^{-s} ; D_{\mathrm {pst}} (V)^{^{I_p}})^{-1} \end{aligned}$$

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where w is the weight of motive whose p-adic étale realization gives V, $D_{\mathrm {pst}} (V)$ is a potentially stable filtered module of Fontaine on which the operator $\varphi$ is acting. It is known that each Euler factor is a polynomial whose coefficients are in $\overline{\mathbb {Q}}$ and the Euler product is absolutely convergent on $\mathrm {Re}(s) >1+\dfrac{w}{2}$ due to Deligne.

We recall the following well-known conjecture which is a folklore.

Conjecture A.1

Let $V \cong \mathcal {K}^d$ be a p-adic Galois representation of $G_\mathbb {Q}$ which satisfies the condition (Geom) stated in Introduction. Then the following statements hold.

• (1) The L-function L(s, V) is meromorphically continued to the whole $\mathbb {C}$-plane with at most finitely many poles. Further, if V does not contain any direct summand which is isomorphic to Tate twists of the trivial representation $\mathbb {Q}_p$, the L-function L(s, V) is holomorphic on the whole $\mathbb {C}$-plane.

• (2) We have the following functional equation:

  $$\begin{aligned} L(s, V) = a( V )^{\frac{w+1}{2} -s} \epsilon (V ) L(1-s ,V^*(1) ) \end{aligned}$$

  where a(V) (resp. $\epsilon (V)$) is the Artin conductor (resp. epsilon factor) for V respectively, which we do not explain here.

Conjecture A.2

(Deligne) Let $V\cong \mathcal {K}^{2d}$ be a p-adic Galois representation of $G_\mathbb {Q}$ of even dimension which satisfies the condition (Geom) stated in Introduction. For simplicity, we assume that V does not contain any Artin representations as its subquotient. Further we assume the following conditions

• (i) The motive which corresponds to V is critical in the sense of Deligne [9].

• (ii) Conjecture A.1 holds true for V.

Then there exist two constants $\Omega _+ (V) , \Omega _- (V)$ such that, for any Artin representation W of $G_{\mathbb {Q}}$ with values in $\mathcal {K}$ such that the motive which corresponds to $V \otimes W$ is critical in the sense of Deligne and Conjecture A.1 holds true for $V\otimes W$, we have

$$\begin{aligned} \dfrac{L(0 ,V\otimes _{\mathcal {K}} W)}{\Omega _+(V)^{{dd_+ (W)} } \Omega _-(V)^{{d d_- (W)}}} \in \overline{\mathbb {Q}} \end{aligned}$$

where $d_+ (W)$ (resp. $d_- (W)$) is the dimension of the eigenspace of complex conjugate with eigenvalue $+1$ (resp. $-1$).

Based on the conjectural existence of L-values and periods, the conjectural existence of analytic p-adic L-function was formulated by [5, Conj. 5.7] (for elliptic curves) and [13, Thm 4.2.26] (for general motives).

Conjecture A.3

Let $V\cong \mathcal {K}^{2d}$ be a p-adic Galois representation of $G_\mathbb {Q}$ of even dimension which satisfies the condition (Geom) stated in Introduction. For simplicity, we assume that V does not contain any Artin representations as its subquotient. Let ${\mathcal {O}}$ be the ring of integers of a finite extension of ${\mathbb {Q}}_p$ and define

$$\begin{aligned} S = \left\{ f \in \Lambda _{\mathcal {O}}(G) \ \bigg | \ \frac{\Lambda _{\mathcal {O}}(G)}{\Lambda _{\mathcal {O}}(G)f} \text { is a finitely generated } \Lambda _{\mathcal {O}}(H) \text {-module} \right\} . \end{aligned}$$

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Set $S^* = \underset{n \ge 0}{\cup }p^n S$. Then S and $S^*$ are left and right Ore set in $\Lambda _{\mathcal {O}}(G)$ [5, §2, §3]. In particular the localization $\Lambda _{\mathcal {O}}(G)_{S^*}$ is well-defined and we have an exact sequence of K-groups

$$\begin{aligned} K_1(\Lambda _{\mathcal {O}}(G)) \longrightarrow K_1(\Lambda _{\mathcal {O}}(G)_{S^*}) {\mathop {\longrightarrow }\limits ^{\delta _G}} K_0(\Lambda _{\mathcal {O}}(G), \Lambda _{\mathcal {O}}(G)_{S^*})= K_0({\mathfrak {M}}_H(G)) \longrightarrow 0.\nonumber \\ \end{aligned}$$

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Given $M \in {\mathfrak {M}}_H(G)$, we denote by [M] its class in $K_0({\mathfrak {M}}_H(G))$ and the preimage of [M] in $K_1(\Lambda _{\mathcal {O}}(G)_{S^*})$ by $\xi _M$. Note that, under the assumption that G has no element of order p, the surjectivity of $\delta _G$ was proved in [5, Proposition 3.4].

Also given an element $\xi \in K_1(\Lambda _{\mathcal {O}}(G)_{S^*})$ and an Artin representation $\eta$ of G, [5, §3] associates a canonical evaluation map which gives rise to an element $\eta (\xi ) \in {\bar{{\mathbb {Q}}}}_p \cup \{\infty \}$.

Further we assume the following conditions

• (i) The motive which corresponds to V is critical in the sense of Deligne.

• (ii) Conjecture A.1 and Conjecture A.2 hold true for V.

• (iii) The representation V is crystalline at p in the sense of Fontaine.

• (iv) The representation V is of Panchishkin type at p in the sense that V has a $G_{\mathbb {Q}_p}$-stable filtration $\mathrm {F}^+_p V \subset V$ of dimension d such that the Hodge-Tate weights of $\mathrm {F}^+_p V$ (resp. $V/\mathrm {F}^+_p V$) are all negative (resp. non-negative).

For any non-trivial Artin representation $\eta$ of G and for any prime number q, we define the Euler factor at q as follows:

$$\begin{aligned} P_q (\eta , X) = {\left\{ \begin{array}{ll} \mathrm {det} (1-\mathrm {Frob}_q^{-1} X \mid (W_\eta )^{I_q}) &{}\quad q\not =p , \\ \mathrm {det} (1-\varphi X \mid D_{\mathrm {crys}}(W_\eta )) &{}\quad q =p. \\ \end{array}\right. } \end{aligned}$$

Then, we have an analytic function $\mathcal {L}_p (V) \in K_1 (\Lambda _{\mathcal {O}} (G)_{S^*} )$, where ${\mathcal {O}}$ is the ring of integers of a certain finite extension of ${\mathbb {Q}}_p$, such that

$$\begin{aligned}&\eta ^*(\mathcal {L}_p (V )) \nonumber \\&\quad = \epsilon _p (W^*_\eta )^{-d} ( \alpha ^{(1)} \ldots \alpha ^{(d)} )^{-C_p(\eta ) }\times \prod ^d_{i=1} \frac{P_p(\eta ^*, (\alpha ^{(i)} )^{-1} )}{P_p(\eta , (\beta ^{(i)})^{-1})}\times \dfrac{L_{P}(0 ,V\otimes _{\mathcal {K}} W_\eta )}{\Omega _+(V)^{{dd_+ (W_\eta )} } \Omega _-(V)^{{d d_- (W_\eta )}}}\nonumber \\ \end{aligned}$$

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for any non-trivial Artin representation $\eta$ of G, where $\epsilon _p (W^*_\eta )$ is the local epsilon factor at p for $W^*_\eta$, $C_p(\eta )$ is the p-order of the conductor of $\eta$, $\alpha ^{(1)} ,\ldots ,\alpha ^{(d)}$ (resp. $\beta ^{(1)} ,\ldots ,\beta ^{(d)}$) is the eigenvalues of $\varphi$-operator acting on $D_{\mathrm {cris}} (F^+_p V)$ (resp. $D_{\mathrm {cris}} (V/F^+_p V)$). Here $\eta ^*$ denotes the contragadient representation of $\eta$. Also P denotes the set of primes q of ${\mathbb {Q}}$ such that the image of $I_{\mathbb {Q}_q}$ in G is infinite and $L_{P} (s ,V\otimes _{\mathcal {K}} W^*_\eta )$ means the L-function which is obtained by removing the Euler factors at every prime $q \in P$ from the L-function $L (s ,V\otimes _{\mathcal {K}} W^*_\eta )$.

We can also state the main conjecture in our setting assuming Conjectures A.1, A.2 and A.3.

Conjecture A.4

(Iwasawa Main Conjecture) Assume Conjectures A.1, A.2 and A.3. Then via the natural map from $K_1 (\Lambda _{\mathcal {O}} (G)_{S^*} ) {\mathop {\longrightarrow }\limits ^{\delta }} K_0({\mathfrak {M}}_H(G))$, the image of $\mathcal {L}_p (V)$ in $K_0({\mathfrak {M}}_H(G))$, $\delta (\mathcal {L}_p (V))$ has the property,

$$\begin{aligned} \delta ({\mathcal {L}}_p (V)) =[{\mathrm {Sel}}^{\mathrm {BK}}_{A} (K_\infty )^\vee ] \in ~ K_0({\mathfrak {M}}_H(G)). \end{aligned}$$