## 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)

### Citations:

Zbl 0948.14017; Zbl 0789.14015
Full Text:

### References:

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