The Euler characteristic of arithmetic \(\mathcal D\)-modules on curves (Q2785991)

The aim of this article is to give a formula to compute the Euler characteristic of an holonomic module with Frobenius structure over a formal smooth curve \(\mathcal{X}\) over the formal spectrum of a discrete complete valuation ring of inequal characteristics \((0,p)\).NEWLINENEWLINEThe author first proves that a holonomic module with Frobenius structure over a formal smooth curve is either of punctual type or is the middle extension of an overconvergent \(F\)-isocrystal over some suitable open subset of the curve \(\mathcal{U}\). In this case, he proves a formula, analogous to the one of \textit{N. M. Katz} (Theorem 2.9.9 of [Exponential sums and differential equations. Princeton, NJ: Princeton University Press (1990; Zbl 0731.14008)]) for middle extensions, involving the Euler characteristic of the curve (times the rank of the isocrystal), the irregularity of the isocrystal and a term totdrop taking into account the irregularity and the dimension of the solution space at the points outside \(\mathcal{U}\) of the isocrystal. To define the term totdrop and solution space at some point the author needs \textit{R. Crew}'s theory of arithmetic \(\mathcal{D}\)-modules over the unit disk [Math. Ann. 336, No. 2, 439--448 (2006; Zbl 1131.14018)].