Alladi Ramakrishnan Hall

Conductors and minimal discriminants of hyperelliptic curves

V. Padmavati

MIT

The Ogg-Saito formula relates the Artin conductor of the minimal proper
regular model of an elliptic curve over a local field (which is a
certain numerical invariant that can be computed from the Galois action
on étale cohomology groups) with the valuation of the discriminant of a
minimal Weierstrass equation for the curve. The definition of the Artin
conductor naturally extends to models of curves of higher genus as well.
Lower bounds on the Artin conductor of a regular model can be used to
give upper bounds on the number of components of the special fiber of
the model, and this has applications to the study of rational points
(Chabauty's method). We show that the Artin conductor for the minimal
proper regular model of a hyperelliptic curve with rational Weierstrass
points, over a local field with perfect residue field (and residue
characteristic either zero or large enough compared to the genus) is
bounded above by the valuation of the discriminant of an integral
Weierstrass equation for the curve.