Invariants recovering the reduction type of a hyperelliptic curve
Lilybelle Cowland Kellock, Elisa Lorenzo
TL;DR
The work generalizes Tate's reading of elliptic-curve reduction types to hyperelliptic curves of genus $g\geq 2$ over local fields of odd residue characteristic by defining an explicit finite family of absolute invariants whose valuations encode the reduction data. Central to the approach are cluster pictures, the stable model tree $T_C$, and the BY tree, with absolute invariants Inv$_{T,I,n}$ constructed to extract distances between-cluster structures; in the semistable case these valuations recover the dual graph of the minimal regular model after passing to $K^{\text{unr}}$, and in the non-semistable case they determine the dual graph of the potential stable model. The paper proves a no-go result showing that, for $g\geq 2$, no invariant-list valuation suffices to recover the dual graph in the non-semistable setting, while providing an explicit genus-2 construction and an algorithm to recover the dual graph from invariants (and, in genus 2, relating these invariants to Igusa invariants). In addition to establishing the theoretical framework, the authors give concrete genus-2 invariants and show how to compute reduction types from Weierstrass data without requiring root extraction, highlighting practical computational paths via Igusa invariants.
Abstract
Tate's algorithm tells us that for an elliptic curve $E$ over a local field $K$ of residue characteristic $\geq 5$, $E/K$ has potentially good reduction if and only if $\text{ord}(j_E)\geq 0$. It also tells us that when $E/K$ is semistable the dual graph of the special fibre of the minimal regular model of $E/K^{\text{unr}}$ can be recovered from $\text{ord}(j_E)$. We generalise these results to hyperelliptic curves of genus $g\geq 2$ over local fields of odd residue characteristic $K$ by defining a list of absolute invariants that determine the potential stable model of a genus $g$ hyperelliptic curve $C$. They also determine the dual graph of the special fibre of the minimal regular model of $C/K^{\text{unr}}$ if $C/K$ is semistable. This list depends only on the genus of $C$, and the absolute invariants can be written in terms of the coefficients of a Weierstrass equation for $C$. 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 $g \geq 2$, there is no list of invariants whose valuations determine the dual graph of the special fibre of the minimal regular model of a genus $g$ hyperelliptic curve $C$ over a local field $K$ of odd residue characteristic when $C$ is not assumed to be semistable.