## Motivic Iwasawa theory.(English)Zbl 0741.11042

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, 421-456 (1989).
[For the entire collection see Zbl 0721.00006.]
The author aims here at a general theory of p-adic L-functions for varieties over number fields which parallels the theory of complex $$L$$- functions. He develops a cohomological (étale and $$\ell$$-adic) formalism which in some sense may be viewed as a motivic Iwasawa theory. As the constructions are very technical and lengthy, we can only give a not too precise overview:
Let $$X$$ be a projective smooth geometrically connected variety over a number field $$k$$. Let $$k_ \infty/k$$ be the cyclotomic $$\mathbb{Z}_ p$$- extension, $$\Gamma=Gal(k_ \infty/k)$$, $${\mathcal O}_ \infty$$ the ring of integers of $$k_ \infty$$. By analogy with the function field case, and also with the case where $$X$$ is a curve (Mazur, Coates/Wiles), to construct $$p$$-adic $$L$$-functions attached to the “motif” (this is just a way of speaking) $$M:=H^ i(X)$$, the author proceeds as follows:
1) First he “extends” the torsion sheaf $$F(n):=H^ i(\bar X,\mathbb{Q}_ p/\mathbb{Z}_ p(n))$$ on $$k_{{\acute e}t}$$, with $$\Gamma$$-action, to a suitable complex of sheaves on $$({\mathcal O}_ \infty)_{{\acute e}t}$$. Two approaches are possible:
– “Greenberg’s approach” [R. Greenberg, ibid., 97-137 (1989; Zbl 0739.11045)], which provides us with a complex denoted by $${\mathcal G}_ p(M(n))$$.
– The approach “à la Lichtenbaum”, based on an axiomatic – and as yet hypothetical for $$n\neq 2$$ – theory of arithmetic cohomology [S. Lichtenbaum, Invent. Math. 88, 183-215 (1987; Zbl 0615.14004)], which provides us with a complex denoted by $${\mathcal H}_ p(M(n))$$.
The relations between $${\mathcal G}_ p$$ and $${\mathcal H}_ p$$ are not entirely clarified (in the “non ordinary” case, the two approaches seem to lead to different $$\Gamma$$-modules).
2) The Pontryagin duals of the cohomology groups of $${\mathcal G}_ p$$ and $${\mathcal H}_ p$$ are noetherian compact modules over the Iwasawa algebra $$\Lambda={\mathbb{Z}}_ p[[\Gamma]]$$.
Suppose moreover that $$X$$ has ordinary good reduction at $$p$$. Then:
Conjecture: a) If $$i$$ is odd, $$H^ 2({\mathcal O}_ \infty,{\mathcal G}_ p(M(i+1)/2))) \hat{}$$ is $$\Lambda$$-torsion.
b) If $$i$$ is even, $$H^ 1({\mathcal O}_ \infty,{\mathcal G}_ p(M(i/2))) \hat{}$$ and $$H^ 2({\mathcal O}_ \infty,{\mathcal G}_ p(M(i/2)+1))) \hat{}$$ are $$\Lambda$$-torsion.
It follows from this conjecture that $\hbox{rank }H^ 1({\mathcal O}_ \infty,{\mathcal G}_ p(M(i+1-n))) \hat{}= \hbox{rank}_ \Lambda H^ 2({\mathcal O}_ \infty,{\mathcal G}_ p(M(n)) \hat{}=0$ if $$n={i+1\over 2}$$ or $${i\over 2}+1$$.
3) Assume that $$X$$ has ordinary good reduction at $$p$$ and that rank $$H^ \nu({\mathcal O}_ \infty,{\mathcal G}_ p(M(i+1-n))) \hat{}=0$$ for $$\nu=1,2$$. The author defines “the” $$p$$-adic $$L$$-function of the “motif” $$M(n)$$ to be: $L_ p(M(n),s)=\prod^ 2_{\nu=0}P\Bigl({\mathcal K}(\gamma)^{- s}; H^{\nu}\bigl({\mathcal O}_{\infty},{\mathcal G}_ p(M(i+1-n))\bigr) \hat{}\Bigr)^{(-1)^{\nu+1}},$ where $$\Gamma=\langle \gamma\rangle$$, $${\mathcal K}$$ is the cyclotomic character and, for any $$\Lambda$$-torsion module $$H$$, $$P(T;H):=p^{\mu(H)}\hbox{det}(1-\gamma^{- 1}T;H\otimes_{{\mathbb{Z}}_ p}{\mathbb{Q}}_ p)$$.
The author then discusses four major questions about these $$p$$-adic $$L$$- functions: dependence on the twist; location of poles and zeros; functional equation; $$p$$-adic regulators.

### MSC:

 11R23 Iwasawa theory 11S40 Zeta functions and $$L$$-functions 14F99 (Co)homology theory in algebraic geometry

### Citations:

Zbl 0721.00006; Zbl 0615.14004; Zbl 0739.11045