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Overcoherent arithmetic \(\mathcal D\)-modules. Application to \(L\)-functions. (\(\mathcal D \)-modules arithmétiques surcohérents. Application aux fonctions \(L\).) (French) Zbl 1129.14030

This article is one piece of the author’s program to construct a category of \(p\)-adic coefficients, attached to a scheme over a finite field \(k\), which is stable by the six Grothendieck operations: direct image (resp. exc. direct image), inverse image (resp. exc. inverse image), (external) tensor product, duality and local cohomological functors.
In this review \(D^{\dagger}\) will denote the sheaf of arithmetic \(D\)-modules of Berthelot and \(V\) is a discrete valuation ring such that \(k\) is the residue field of \(V\). We will work with modules which are endowed with a Frobenius structure denoted by \(F\), that means \(D^{\dagger}\)-modules \(M\) endowed with a \(D^{\dagger}\)-linear isomorphism \(F^*M\simeq M\). Let us notice that some interesting results are valid in this article for arithmetic \(D\)-modules which are not endowed with such a structure.
Conjecturally, the category of holonomic \(D^{\dagger}\)-modules of P. Berthelot [“D-modules arithmétiques. II. Descente par Frobenius.” Mém. Soc. Math. Fr., Nouv. Sér. 81 (2000; Zbl 0948.14017)] endowed with a Frobenius structure should be stable by the \(6\) operations but this result is not yet known. The approach here consists into defining a new subcategory of the derived category of arithmetic \(D^{\dagger}\)-modules with overconvergent singularities along a divisor, by forcing some property of stability by image inverse functors that holonomic \(F\)-\(D^{\dagger}\)-modules should conjecturally verify. This category is called the category of overcoherent \(F\)-\(D^{\dagger}\)-modules. In the case of a curve, it is known that an overcoherent \(D^{\dagger}\)-module is holonomic. In general, it is a consequence of Berthelot’s conjectures that the category introduced here is the same as the category of holonomic \(F\)-\(D^{\dagger}\)-modules defined by  Berthelot. As an example unit-root \(F\)-isocrystals are overcoherent.
One application of this construction is that D. Caro can define a category of \(F\)-\(D^{\dagger}_U\)-overcoherent modules for a separated scheme over \(k\). Locally, if \(U\) is obtained as \(T\backslash Z\) where \(T\) and \(Z\) are divisors of a smooth scheme \(X\), this category is obtained as the category of overcoherent \(F\)-\(D^{\dagger}\)-modules \(E\), over a smooth lifting of \(X\) over \(\text{Spf\,}V\), such that \(E\) has support in \(Z\) and such that the localization functor \(R\Gamma^{\dagger}_T E\) is \(0\). This category depends only on \(U\), and is stable by direct images and by exceptional inverse images.
In the last part, the author defines \(L\)-functions for duals of an overcoherent \(F\)-\(D^{\dagger}\)-module over a separated scheme \(U\) and proves a cohomological formula. This generalizes a previous formula of J.-Y. Étesse and B. Le Stum [Math. Ann. 296, No. 3, 557–576 (1993; Zbl 0789.14015)].

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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References:

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