Invariants recovering the reduction type of a hyperelliptic curve
Year: 2025
Type: article
Abstract: Tate's algorithm tells us that for an elliptic curve E over a local field K of residue characteristic ≥5, E / K has potentially good reduction if and only if ord ( j E ) ≥ 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 unr can be recovered from ord ( j E ) . We generalise these results to hyperelliptic curves of genus g ≥ 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 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 ≥ 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.
Source: Journal of Symbolic Computation
Institution: University College London
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,160
OpenAlex ID: W4416718151