Global Weierstrass equations of hyperelliptic curves

Abstract

Given a hyperelliptic curve C of genus g over a number field K and a Weierstrass model C of C over the ring of integers O K (i.e. the hyperelliptic involution of C extends to C and the quotient is a smooth model of P 1 K over O K ), we give necessary and sometimes sufficient conditions for C to be defined by a global Weierstrass equation. In particular, if C has everywhere good reduction, we prove that it is defined by a global Weierstrass equation with invertible discriminant if the class number h K is prime to 2(2g + 1), confirming a conjecture of M. Sadek.

Global Weierstrass equations of hyperelliptic curves

“…For a given (C, w 0 ), the minimal pointed Weierstrass equation exists and is unique up to the transformations described in Lemma 4.1 below. See also [5] and [4], Corollary 5.2. The next lemma is stated in [5], Remark after Definition 2.1.…”

“…by [4], Lemma 5.1, we have v(∆) < v(∆ 1 ) (with the notation of op. cit., d = 2v(c −1 u) > 0 and the point p is the pole of x in W k which is a smooth point).…”

“…Let p 0 ∈ W (k). Let W (p 0 ) be the model defined by Equation (8) in Lemma 2.3 (4). See also a more geometrical description in [3], Définition 12, p. 4592.…”