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On a noncommutative Iwasawa main conjecture for varieties over finite fields. (English) Zbl 1290.14016

The noncommutative Iwasawa main conjecture over totally real fields has been shown recently by J. Ritter and A. Weiss [J. Am. Math. Soc. 24, No. 4, 1015–1050 (2011; Zbl 1228.11165)] and by M. Kakde [Invent. Math. 193, No. 3, 539–626 (2013; Zbl 1300.11112)] independently. Here the author formulates and proves an analogue for \(\ell\)-adic Lie extensions of a separated scheme \(X\) of finite type over a finite field \(\mathbb F_q\) of characteristic prime to \(\ell\). Assume first that \(X\) is geometrically connected and let \(G\) be a factor group of the fundamental group of \(X\) such that \(G\) is the semi-direct product of a compact \(\ell\)-adic Lie group \(H\) and of \(\Gamma = \mathrm{Gal}(\mathbb F_(q^\infty)/\mathbb F_q) = \mathbb Z_\ell\). Write \(\Lambda_G\) for the complete algebra \(\mathbb Z_\ell[[G]]\). Let \(S = \{f \in \Lambda_G : \Lambda_G/\Lambda_Gf\text{ is of finite type over }\Lambda_H\}\) be the canonical Ore set. Recall the exact localisation sequence of algebraic K-groups \(K_1(\Lambda_G) \to K_1((\Lambda_G)_S) \to^d K_0(\Lambda_G,(\Lambda_G)_S) \to 0\). Let us turn \(\Lambda_G\) into a smooth \(\Lambda_G\)-sheaf \(\mathcal M(G)\) on \(X\) via the contragredient action of the fundamental group of \(X\) on \(\Lambda_G\). Every continuous \(\mathbb Z_\ell\)-representation \(\rho\) of \(G\) induces a homomorphism \(\rho\) of \(K_1(\Lambda_G)\) into the units \(Q(\Lambda_G)^{\times}\) of the field of fractions of \(\Lambda_G\). On the other hand, \(\rho\) gives rise to a flat and smooth \(\mathbb Z_\ell\)-sheaf \(\mathcal M(\rho)\) on \(X\). Let \(\mathcal F\) be the compact cohomology of a flat constructible \(\mathbb Z_\ell\)-sheaf \(\mathcal F\) on \(X\). The Grothendieck trace formula expressing the \(L\)-function \(L(\mathcal F,T)\) as the alternate product of the determinants \(\det(1-T\phi : H_c^i (\bar{X}, \mathcal F))\), where \(\phi\) is the geometric Frobenius, implies that \(L(\mathcal F,T)\) is in fact a rational function.
The author’s main result then reads:
1)
\(R\Gamma_c(X, \mathcal M(G)\otimes\mathcal F)\) is a perfect complex of \(\Lambda_G\)-modules whose cohomology is S-torsion. Moreover, \(id-\phi\) is a quasi-automorphism of the complex \(R\Gamma_c(\bar{X}, \mathcal M(G)_S\otimes\mathcal F)\), and hence gives rise to an element \(\mathcal L_G (X/\mathbb F_q, \mathcal F) = [id-\phi]^{-1}\) in \(K_1(\Lambda_G)_S)\).
2)
\(d(\mathcal L_G (X/\mathbb F_q, \mathcal F)) =[R\Gamma_c (X,\mathcal M(G)\otimes\mathcal F)]^{-1}\).
3)
If the representation is continuous, then \(\rho(\mathcal L_G (X/\mathbb F_q, \mathcal F)) = L(\mathcal M(\rho)\otimes\mathcal F, [\phi]^{-1})\) in \(Q(\Lambda_G)^{\times}\).
This theorem implies the expected interpolation property of \(\mathcal L_G (X/\mathbb F_q, \mathcal F)\) with respect to special values of \(L\)-functions. In the final section of the paper, the limitation to geometrically connected schemes is overcome by allowing \(G\) (which is then only required to be a virtual pro-\(\ell\)-group) to be the covering group of any suitable principal covering of \(X\). Note that using the results of this paper and of M. Emerton and M. Kisin [Ann. Math. (2) 153, No. 2, 329–354 (2001; Zbl 1076.14027)], D. Burns, “On main conjectures of geometric Iwasawa theory and related conjectures” Preprint (2011)] has recently constructed, in the case when \(\ell\) divides \(q\), a modification of \(\mathcal L_G (X/\mathbb F_q, \mathcal F)\) which has the right interpolation property. Also in the case when \(\ell = p\), Trihan and Vauclair have announced proofs of general main conjectures for varieties over function fields.

MSC:

14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11R23 Iwasawa theory
14G15 Finite ground fields in algebraic geometry
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References:

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