Indices of double coverings of genus 1 over \(p\)-adic fields
Let \(K\) be a local non-dyadic field of characteristic zero and \(O_K\) the valuation ring in \(K\). Consider the curve \(C_h\) defined by the equation \(Y^2=h(X)\), where \(h(X)\) is a polynomial of \(K[X]\) with no multiple roots. The index of \(C_h\), \(I(C_h)\), is by definition the greatest common divisor of the degrees of all \(K\)-rational divisors on \(C_h\). Let \(\pi\) be a uniformising element of \(K\) and \(\alpha\) a unit of \(O_K\) which is not a square in \(K\). In [Manuscr. Math. 96, 317--333 (1998; Zbl 1015.11023)] the authors determined the index of the elliptic curves \(C_{\varepsilon f}\), where \(\varepsilon\in \{1,\alpha,\pi, \alpha\pi\}\) and \(f(X)\) is a monic polynomial of \(O_K[X]\) which is not of the form \(f(X)= f_1(X) f_2(X)\), where \(f_1(X), f_2(X)\) are irreducible monic polynomials of \(O_K[X]\). In this paper the authors determine the index of the elliptic curves \(C_{\varepsilon f}\) for such polynomials \(f(X)\).
scientific article; zbMATH DE number 1491219