Invariants Recovering the Reduction Type of a Hyperelliptic Curve

Abstract

Tate's algorithm tells us that for an elliptic curve [[EQUATION]] over a local field [[EQUATION]] of residue characteristic [[EQUATION]], [[EQUATION]] has potentially good reduction if and only if [[EQUATION]]. It also tells us that when [[EQUATION]] is semistable the dual graph of the special fibre of the minimal regular model of [[EQUATION]] can be recovered from [[EQUATION]]. We generalise these results to hyperelliptic curves of genus [[EQUATION]] over local fields of odd residue characteristic [[EQUATION]] by defining a list of absolute invariants that determine the potential stable model of a genus [[EQUATION]] hyperelliptic curve [[EQUATION]]. They also determine the dual graph of the special fibre of the minimal regular model of [[EQUATION]] if [[EQUATION]] is semistable. This list depends only on the genus of [[EQUATION]], and the absolute invariants can be written in terms of the coefficients of a Weierstrass equation for [[EQUATION]]. We explicitly describe the method by which the valuations of the invariants recover the dual graphs. Additionally, we show by way of a counterexample that if [[EQUATION]], there is no list of invariants whose valuations determine the dual graph of the special fibre of the minimal regular model of a genus [[EQUATION]] hyperelliptic curve [[EQUATION]] over a local field [[EQUATION]] of odd residue characteristic when  is not assumed to be semistable.

Keywords: hyperelliptic curves, invariants, local fields, reduction types, special fibre, stable type, local arithmetic of curves, cluster picture