One half log discriminant and division polynomials

TL;DR

This paper provides an arithmetic proof relating the valuation of minimal discriminants of elliptic and hyperelliptic curves to torsion intersection properties, extending previous geometric results.

Contribution

It introduces an arithmetic approach using division polynomials and generalizes the valuation formula to hyperelliptic curves.

Findings

Valuation of minimal discriminant expressed via torsion intersections

Extension of results from elliptic to hyperelliptic curves

Arithmetic proof based on division polynomials

Abstract

L. Szpiro and T. Tucker recently proved that under mild conditions, the valuation of the minimal discriminant of an elliptic curve with semistable reduction over a discrete valuation ring can be expressed in terms of intersections between n-torsion and 2-torsion, where n tends to infinity. The argument of Szpiro and Tucker is geometric in nature. We give a proof based on the arithmetic of division polynomials, and generalize the result to the case of hyperelliptic curves.