Galois actions on Néron models of Jacobians

Abstract

We study group actions on regular models of curves. If X is a smooth curve defined over the fraction field K of a complete discrete valuation ring R, every tamely ramified extension K ′ /K with Galois group G induces a G-action on the extension X K ′ of X to K ′ . In this paper we study the extension of this G-action to certain regular models of X K ′ . In particular, we are interested in the induced action on the cohomology groups of the structure sheaf of the special fiber of such a regular model. We obtain a formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups. Inspired by this global study, we also consider similar group actions on the cohomology of the structure sheaf of the exceptional locus of a tame cyclic quotient singularity, and obtain an explicit polynomial formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups.We apply these results to study a natural filtration of the special fiber of the Néron model of the Jacobian of X by closed, unipotent subgroup schemes. We show that the jumps in this filtration only depend on the fiber type of the special fiber of the minimal regular model with strict normal crossings for X over Spec(R), and in particular are independent of the residue characteristic. Furthermore, we obtain information about where these jumps occur. We also compute the jumps for each of the finitely many possible fiber type for curves of genus 1 and 2.

Néron Models and Base Change

“…Thus let d be an element of N ′ that is prime to λ(C ). Using Lemma 2.3.2 in Chapter 3 and the results in Section 4 of that chapter, it is easy to check that both the statement and proof of [Ha10b,3.4] extend directly to our situation. This means that the following properties hold.…”

“…We will see in Theorem 3.1.3 that the tame base change conductor of an elliptic curve only depends on the reduction type, even for wildly ramified curves. If p = 1, then c tame (E) = c(E) for every elliptic K-curve E (Proposition 1.3.10), so that we obtain the following table of values for the tame base change conductor (see also [Ed92,5.4.5] and [Ha10b, Example 2.2.3. Assume that k is an algebraically closed field of characteristic 2 and that R = W (k).…”

“…In particular, this is the case if C is tamely ramified, by Saito's criterion [Sa87,3.11]. In [Ha10b], the first author proved the following result. We will now show that the condition ( * ) can be omitted, which yields the following stronger theorem.…”

“…in Section 4 of Chapter 3, it is easy to describe the unique lifting of the Gal(K(d)/K)-action to Z y , and we can use the same arguments as in [Ha10b,8.2]. Proof.…”

A logarithmic interpretation of Edixhoven's jumps for Jacobians

“…In this paper we are mainly interested in the wildly ramified case. We will study the jumps of A under the assumption that A is the Jacobian of a K-curve C. In this case, it was already proven by the second author in [Ha10b] that the jumps are rational; see also [HN14,§5.3]. In fact, he proved a much stronger result, namely that the jumps of A only depend on the combinatorial reduction data of C, and not on the characteristic of k. A result of Winters [Wi74] guarantees that we can always find a curve D over the field of complex Laurent series C((t)) with the same reduction data, and thus the same jumps, as C. Since D is automatically tame, it follows that the jumps of C are rational.…”