Arithmetic of hyperelliptic curves over local fields
Tim Dokchitser, Vladimir Dokchitser, Céline Maistret, Adam Morgan
Abstract
Abstract We study hyperelliptic curves $$y^2 = f(x)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> over local fields of odd residue characteristic. We introduce the notion of a “cluster picture” associated to the curve, that describes the p -adic distances between the roots of f ( x ), and show that this elementary combinatorial object encodes the curve’s Galois representation, conductor, whether the curve is semistable, and if so, the special fibre of its minimal regular model, the discriminant of its minimal Weierstrass equation and other invariants.