Quantitative Néron theory for torsion bundles

Abstract

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

Extending torsors over regular models of curves

“…We finish this article by generalizing a result by Chiodo [9,Propositions 7.4.1 and 7.5.1] which provides a criterion for the existence of a finite étale R-model of J[r], when r is prime to p. In fact, his result is a finiteness criterion for J [r], because when r is prime to p, J [r] is étale, hence is the natural candidate for being the Néron model of J[r]. This no longer holds when p divides r; nevertheless, we will show that the same finiteness criterion for J [r] holds.…”

“…Then according to (7), all the q i are integers. Hence, according to (8), for any degree zero divisor c K on C, the Néron symbol [c K , (D t ) K ] is an integer, hence c k , t k = 0 by (9). It follows that c k , t k = 0 for all possible choices of c k .…”

Smith normal form and Laplacians

“…Computations were done by Grant Fiddyment in the case of the 15 strongly regular graphs with parameters (25, 12, 5, 6) (see [36] for the list of graphs); in this case, nc = 150, n−1 4 = 6, and the eigenvalues are integral: 15 and 10. As predicted by 2.3, Φ(G) contains subgroups isomorphic to (Z/10Z) 11 , (Z/15Z) 11 , and (Z/6Z) 12 . The prime-to-5 part of all groups is (Z/6Z) 12 .…”

“…As predicted by 2.3, Φ(G) contains subgroups isomorphic to (Z/10Z) 11 , (Z/15Z) 11 , and (Z/6Z) 12 . The prime-to-5 part of all groups is (Z/6Z) 12 . There are ten such graphs where the 5-part of the group is…”

On Néron models of moduli spaces of theta characteristics

“…They have been employed in arithmetic and geometry and recently also in the moduli theory of curves (see [C2,Ch,B]). …”