Extending torsors over regular models of curves

Abstract

Let R be a discrete valuation ring with field of fractions K and residue field k of characteristic p> 0. Given a commutative finite group scheme G over K and a smooth projective curve C over K with a rational point, we study the extension of pointed fppf G-torsors over C to pointed torsors over some R-regular model C of C. We first study this problem in the category of log schemes: given a finite flat R-group scheme G, we prove that the data of a pointed G-log torsor over C is equivalent to that of a morphism GD→PicC/Rlog, where GD is the Cartier dual of G and PicC/Rlog the log Picard functor. After that, we give a sufficient condition for such a log extension to exist, and then we compute the obstruction for the existence of an extension in the category of usual schemes. In a second part, we generalize a result of Chiodo (Manuscr Math 129(3):337–368, 2009) which gives a criterion for the r-torsion subgroup of the Néron model of J to be a finite flat group scheme, and we combine it with the results of the first part. Finally, we give a detailed example of extension of torsors when C is a hyperelliptic curve defined over Q, which illustrates our techniques.