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Iwasawa theory for $$p$$-adic representations. (English) Zbl 0739.11045
Algebraic number theory - in honor of K. Iwasawa, Proc. Workshop Iwasawa Theory Spec. Values $$L$$-Funct., Berkeley/CA (USA) 1987, Adv. Stud. Pure Math. 17, 97-137 (1989).
[For the entire collection see Zbl 0721.00006.]
The author proposes an Iwasawa theoretic realization of motives [for a more geometric approach, see P. Schneider’s “Motivic Iwasawa theory”, Adv. Stud. Pure Math. 17, 421-456 (1989; Zbl 0741.11042)] and attached $$p$$-adic $$L$$-functions, by what he calls a “compatible system” of $$\ell$$-adic representations $$V=\{V_ \ell\}$$ over $$\mathbb{Q}$$. Since the precise definitions are technical and lengthy, we’ll only sketch (and a little bit approximatively) the main lines: $$V_ \ell$$ is a $$\mathbb{Q}_ \ell$$- vector space of finite dimension $$d=d_ V$$, on which $$G_ \mathbb{Q}$$ (and also $$G_{\mathbb{Q}_ p}$$, for any prime $$q$$) acts. Take a prime $$p$$ which is “ordinary” for $$V$$, in the sense that $$V_ p$$ admits a filtration of $$\mathbb{Q}_ p$$-subspaces involving some inertia conditions. Choose a $$G_ \mathbb{Q}$$-lattices $$T_ p$$ contained in $$V_ p$$, called the “Tate module”. Let $$A_ p=V_ p/T_ p$$ and consider the “Selmer group”: $$S_{A_ p}(\mathbb{Q}_ \infty)=\{\sigma\in H^ 1(\mathbb{Q}_ \infty,A_ p)$$; $$\sigma$$ locally trivial at all places of $$\mathbb{Q}_ \infty\}$$ (here, “locally trivial” means with respect to inertia). The main object is the $$\Lambda$$-module $$\hat S_{A_ p}(\mathbb{Q}_ \infty)$$, where $$(\hat{\;})$$ denotes the Pontryagin dual and $$\Lambda$$ the usual Iwasawa algebra. In the classical (= cyclotomic and elliptic) cases, one recovers the usual Iwasawa modules. Here, $$\hat S_{A_ p}(\mathbb{Q}_ \infty)$$ is expected to give rise to $$p$$-adic $$L$$-functions in the following way: For every prime $$q$$, for $$\ell\neq q$$, denote Frob$$(p)$$ the arithmetic Frobenius and define $$E_ q(T)=\hbox{det}(I-\hbox{Frob}(q)T\mid(V_ \ell)_{unr})$$. With certain assumptions on $$E_ q(T)$$, and for Re$$(s)\gg0$$, define the complex $$L$$-function: $$L_ V(s)=\prod_ q E_ q(q^{-s})^{-1}$$. There should be a functional equation relating $$L_ V(2-s)$$ and $$L_{V^*}(s)$$ (where $$( )^*$$ denotes duals with values in $$\mathbb{Q}_ \ell(1)$$), via some factor $$\Gamma_ V(s)$$. Call $$r_ V$$ the order of the pole for $$\Gamma_ V(s)$$ at $$s=1$$.
Conjecture 1: $$\hat S_{A_ p}(\mathbb{Q}_ \infty)$$ has $$\Lambda$$-rank equal to $$r_ V$$.
In the particular case where $$r_ V=r_{V^*}=0$$, there should exist a $$p$$-adic $$L$$-function $$L_ p(\varphi,V)$$, for $$\varphi\in\hat\Gamma$$, the properties of which imply the existence of a certain $$\lambda_ V\in\Lambda$$, conjecturally described by
Conjecture 2: $$\hat S_{A_ p}(\mathbb{Q}_ \infty)$$ has characteristic ideal $$(\lambda_ V)$$.
In the usual (= cyclotomic and elliptic) cases, the above conjectures are classical results and conjectures of Iwasawa theory. Here, using Poitou- Tate theorems on Galois cohomology, the author proves a weak version of conjecture 1, and also a reflection theorem relating $$\hat S_{V_ p/T_ p}(\mathbb{Q}_ \infty)$$ and a twisted version of $$\hat S_{V^*_ p/T^*_ p}(\mathbb{Q}_ \infty)$$.

##### MSC:
 11R23 Iwasawa theory 11S25 Galois cohomology 11S40 Zeta functions and $$L$$-functions