Galois actions on Néron models of Jacobians (Q983144)

The semistable reduction theorem guarantees that if \(K\) is the field of fractions of a complete DVR \(R\) with algebraically closed residue field \(k\), and \(X\) is a smooth projective curve over \(K\) of positive genus, then \(X\) admits a semistable model after a finite separable field extension \(L/K\). The extension may have to be ramified, even wildly ramified. It is not at all clear how to find \(L\), but at least if tame ramification is sufficient some information may be extracted by passing to Jacobians and looking at the Néron model. Suppose that \(K'/K\) is a finite separable tamely ramified field extension of degree~\(n\), and let \(R'\) be the integral closure of \(R\) in \(K'\). The Jacobian of \(X_{K'}\) has Néron model \({\mathcal J}'\) (over \(S'={\text{Spec}} R'\), and the Néron model \({\mathcal J}\) of the Jacobian of \(X\) can be described as the Galois-invariant part of the restriction of scalars (Weil restriction) from \(S'\) to \(S={\text{Spec}}R\). This gives a decreasing filtration of the reduction \({\mathcal J}_k\) by closed subgroup schemes \[ {\mathcal J}_k={\mathcal F}^0_n\supseteq\ldots\supseteq {\mathcal F}_n^n=0. \] All this is due to \textit{B. Edixhoven} [Compos. Math. 81, No.3, 291--306 (1992; Zbl 0759.14033)], who also suggested allowing \(n\) to vary and putting \({\mathcal F}^{i/n}={\mathcal F}^i_n\): this yields a filtration indexed by \(a\in {\mathbb Z}_{(p)}\cap [0,1]\), where \(p\) is the residue characteristic. One of the main points of this paper is to study the jumps in this filtration: by Edixhoven's results this is essentially the same as computing the characters of the action of the Galois group on the tangent space \(T_{{\mathcal J}'_k,0}\). Because of the description of Néron models of Jacobians in terms of the relative Picard functor, this is in turn equivalent to describing the Galois module structure of \(H^1({\mathcal Z}_k,{\mathcal O}_{{\mathcal Z}_k})\) for a suitable regular model \({\mathcal Z}\) of \(X_{K'}\). The length of the paper is accounted for by the considerable technical difficulties in making this all work effectively. One must establish the relations among filtrations, Néron models and Galois actions and show that suitable models exist, and that the filtration does not depend on the choices made. Some geometric assumptions on \(X\) (sufficient to avoid wild ramification) are needed. The regular model is obtained by taking a model \({\mathcal X}\) for \(X\), pulling back to \(S'\) and taking the minimal resolution, and this must be done explicitly in order to be able to compute the Galois action. That is possible because the singularities are (tame) cyclic quotient singularities, whose minimal resolution may be combinatorially described in terms of Hirzebruch-Jung continued fractions. A further technical difficulty is that in positive characteristic the characters of the group action cannot be computed from traces alone: instead one must lift to the Witt vectors and use Brauer characters. The main result of the paper is indeed that this programme succeeds. It does also yield useful and computable information. The jumps turn out to be independent of \(p\), and to occur at indices with denominator \(\tilde n\), the lcm of the multiplicities of the principal (nondestabilising) components of \({\mathcal X}_k\) (this number is prime to \(p\) because of the assumptions about tameness). The jumps can be and are computed for \(g(X)=1\) and \(g(X)=2\), where one has a list of possible central fibres.