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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 \(\hat S_{V^*_ p/T^*_ p}(\mathbb{Q}_ \infty)\).

11R23 Iwasawa theory
11S25 Galois cohomology
11S40 Zeta functions and \(L\)-functions