Formal groups in genus 2. (English) Zbl 0702.14025

Grant, David

J. Reine Angew. Math. 411, 96-121 (1990).

For any curve of genus two with a rational Weierstrass point defined over a field of characteristic other \(than\quad 2,\) we find equations defining a projective embedding for its Jacobian, and determine its group law. When the curve is defined over a ring complete under a non-archimedean valuation, this allows us to find explicit parameters for the formal group on the kernel of reduction of the Jacobian modulo the maximal ideal of the ring. The calculations are facilitated by first considering curves defined over the complex numbers and employing theta functions.

Reviewer: D.Grant

MSC:

14H45	Special algebraic curves and curves of low genus
14K25	Theta functions and abelian varieties
14H40	Jacobians, Prym varieties
14H55	Riemann surfaces; Weierstrass points; gap sequences
14L05	Formal groups, \(p\)-divisible groups

Keywords:

curve of genus two; rational Weierstrass point; embedding; Jacobian; formal group; theta functions