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.

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.

Reviewer: T.Nguyen Quang Do (Besançon)

##### MSC:

11R23 | Iwasawa theory |

11S40 | Zeta functions and \(L\)-functions |

14F99 | (Co)homology theory in algebraic geometry |