Arithmetic differential operators on a semistable model of ${\mathbb P}^1$

Deepam Patel (1) , Tobias Schmidt (2, 3) , Matthias Strauch (4)

1 Department of mathematics Purdue University
2 Institut für Mathematik [Berlin]
3 IRMAR - Institut de Recherche Mathématique de Rennes
4 Department of Mathematics, Indiana University

In this paper we study sheaves of logarithmic arithmetic differential operators on a particular semistable model of the projective line. The main result here is that the first cohomology group of these sheaves is non-torsion. We also consider a refinement of the order filtration on the sheaf of level zero (before taking the p-adic completion). The associated graded sheaf, which we explicitly determine, explains to some extent the occurrence of the cohomology classes in degree one.

Domaines: Théorie des représentations [math.RT]; Théorie des nombres [math.NT]

Deepam Patel, Tobias Schmidt, Matthias Strauch. Arithmetic differential operators on a semistable model of ${\mathbb P}^1$. Mathematische Zeitschrift, 2019, 293 (1-2), pp.319-338. ⟨hal-01937294⟩