2025/02/12 by Lilybelle Cowland Kellock, Kellock, Lilybelle Cowland, Elisa Lorenzo +1
Computer Science · Engineering · Mathematics · #14G20 #14H45 #14Q05 #Advanced Numerical Analysis Techniques #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT) #Primary: 11G20 #Secondary: 14D10
paper · pdf · doi:10.48550/arxiv.2502.08487
openalex publication_date 2025/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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(jE)≥ 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/Kunr can be recovered from ord(jE). 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/Kunr 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.