##
**\(L\)-functions associated with arithmetical \({\mathcal D}\)-modules. The case of curves.
(Fonctions \(L\) associées aux \({\mathcal D}\)-modules arithmétiques. Cas des courbes.)**
*(French)*
Zbl 1167.14012

Let \(K \) be a complete discretely valued field with valuation ring \({\mathcal V} \) and finite residue field \(k={\mathbb F}_q \) and \(Y \) a separated smooth \(k \)-scheme. To any overconvergent \(F \)-isocrystal \(E \) on \(Y \), one naturally associated to it an \(L \)-function \(L(E,Y,t) \) [see J.-Y. Etesse and B. Le Stum, Math. Ann. 296, 557–576 (1993; Zbl 0789.14015)]. In loc. cit., the authors also prove a cohomological formula expressing this \(L \)-function in terms of the Frobenius action on the rigid cohomology associated to the dual of \(E \).

In the paper under review, the author generalizes the construction to the case of a certain class of holonomic \(F \)-\({\mathcal D} \)-modules – a generalization of overconvergent \(F \)-isocrystals. Let \({\mathcal X} \) be a proper smooth \({\mathcal V} \)-scheme with reduction \(X \), \(T \) a divisor on \(X \) and \(Y \) its complement in \(X \). In this situation, P. Berthelot defined the sheaf \({\mathcal D}^{\dagger}_{\mathcal X}(^{\dagger} T)_{\mathbb Q} \) of overconvergent differential operators of finite niveau and the appropriate derived category \(F \)-\(D^b_{hol}({\mathcal D}^{\dagger}_{{\mathcal X},{\mathbb Q}}(^{\dagger} T) )\) of complexes of modules over this sheaf together with a Frobenius action such that the cohomology sheaves are holonomic \(F \)-\({\mathcal D}^{\dagger}_{{\mathcal X}, {\mathbb Q}} \)- modules.

Regarding the natural generalization of the definition of the \(L \)-function, a subtle issue arises with the latter category, namely the open question if holonomicity implies for the cohomology of the fibres at a closed point \(y \in Y \) to be finite dimensional \(K \)-vector spaces. D. Caro introduces the notion of pseudo-holonomicity which includes this requirement. Conjecturally, the resulting category should coincide with the holonomic one. It is known, that this conjecture follows if the conjecture of Berthelot on the stability of holonomicity by extraordinary inverse images holds true.

For a pseudo-holonomic complex \({\mathcal E} \), the author gives a natural definition of the \(L \)-function as well as a candidate for a cohomological variant, denoted by \(P({\mathcal Y}, {\mathcal E}, t) \). It is conjectured that these coincide: \(L({\mathcal Y}, {\mathcal E}, t)=P({\mathcal Y}, {\mathcal E}, t) \).

The main result of the paper gives an affirmative answer to this conjecture in the case of curves (Theorem 3.4.1). Additionally, the author proves that the stronger condition of pseudo-holonomicity is implied by the usual holonomicity in the case of curves also (Corollaire 2.3.4). The proof of the main result relies on an intermediate theorem (2.2.17) which allows to reduce to the case of an \(F \)-isocrystal overconvergent along a suitable divisor \(T \).

In the last chapter, the author applies the methods developed before in order to prove a conjecture of Berthelot in the case of curves, stating that a coherent \(F \)-\({\mathcal D}^{\dagger}_{{\mathcal X}, {\mathbb Q}}(^{\dagger}T) \)-module which is holonomic on the complement \({\mathcal Y} \) of \(T \) is holonomic as an \(F \)-\({\mathcal D}^{\dagger}_{{\mathcal X}, {\mathbb Q}} \)-module.

In the paper under review, the author generalizes the construction to the case of a certain class of holonomic \(F \)-\({\mathcal D} \)-modules – a generalization of overconvergent \(F \)-isocrystals. Let \({\mathcal X} \) be a proper smooth \({\mathcal V} \)-scheme with reduction \(X \), \(T \) a divisor on \(X \) and \(Y \) its complement in \(X \). In this situation, P. Berthelot defined the sheaf \({\mathcal D}^{\dagger}_{\mathcal X}(^{\dagger} T)_{\mathbb Q} \) of overconvergent differential operators of finite niveau and the appropriate derived category \(F \)-\(D^b_{hol}({\mathcal D}^{\dagger}_{{\mathcal X},{\mathbb Q}}(^{\dagger} T) )\) of complexes of modules over this sheaf together with a Frobenius action such that the cohomology sheaves are holonomic \(F \)-\({\mathcal D}^{\dagger}_{{\mathcal X}, {\mathbb Q}} \)- modules.

Regarding the natural generalization of the definition of the \(L \)-function, a subtle issue arises with the latter category, namely the open question if holonomicity implies for the cohomology of the fibres at a closed point \(y \in Y \) to be finite dimensional \(K \)-vector spaces. D. Caro introduces the notion of pseudo-holonomicity which includes this requirement. Conjecturally, the resulting category should coincide with the holonomic one. It is known, that this conjecture follows if the conjecture of Berthelot on the stability of holonomicity by extraordinary inverse images holds true.

For a pseudo-holonomic complex \({\mathcal E} \), the author gives a natural definition of the \(L \)-function as well as a candidate for a cohomological variant, denoted by \(P({\mathcal Y}, {\mathcal E}, t) \). It is conjectured that these coincide: \(L({\mathcal Y}, {\mathcal E}, t)=P({\mathcal Y}, {\mathcal E}, t) \).

The main result of the paper gives an affirmative answer to this conjecture in the case of curves (Theorem 3.4.1). Additionally, the author proves that the stronger condition of pseudo-holonomicity is implied by the usual holonomicity in the case of curves also (Corollaire 2.3.4). The proof of the main result relies on an intermediate theorem (2.2.17) which allows to reduce to the case of an \(F \)-isocrystal overconvergent along a suitable divisor \(T \).

In the last chapter, the author applies the methods developed before in order to prove a conjecture of Berthelot in the case of curves, stating that a coherent \(F \)-\({\mathcal D}^{\dagger}_{{\mathcal X}, {\mathbb Q}}(^{\dagger}T) \)-module which is holonomic on the complement \({\mathcal Y} \) of \(T \) is holonomic as an \(F \)-\({\mathcal D}^{\dagger}_{{\mathcal X}, {\mathbb Q}} \)-module.

Reviewer: Marco Hien (Regensburg)

### MSC:

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

14F30 | \(p\)-adic cohomology, crystalline cohomology |

13N10 | Commutative rings of differential operators and their modules |