Quantitative N\'eron theory for torsion bundles

Let R be a discrete valuation ring with algebraically closed residue field, and consider a smooth curve CK over the field of fractions K. For any positive integer r prime to the residual characteristic, we consider the finite K-group scheme Pic_{CK}[r] of r-torsion line bundles on CK. We determine when there exists a finite R-group scheme, which is a model of Pic_{CK}[r] over R; in other words, we establish when the N\'eron model of Pic_{CK}[r] is finite. To this effect, one needs to analyse the points of the N\'eron model over R, which, in general, do not represent r-torsion line bundles on a semistable reduction of CK. Instead, we recast the notion of models on a stack-theoretic base: there, we find finite N\'eron models, which represent r-torsion line bundles on a stack-theoretic semistable reduction of CK. This allows us to quantify the lack of finiteness of the classical N\'eron models and finally to provide an efficient criterion for it.