The index of certain hyperelliptic curves over \(p\)-adic fields
Let \(K\) be a finite extension of the field of \(p\)-adic numbers \(\mathbb{Q}_p\) and \(O_K\) be the ring of integers in \(K\). Let \(C\) be a geometrically connected smooth projective curve over \(K\). The index \(I(C)\) of \(C\) is defined to be the greatest common divisor of the divisors on \(C\). In this paper, which is the sequel of [\textit{J. Van Geel} and \textit{V. I. Yanchevskii}, Manuscr. Math. 96, 317--333 (1998; Zbl 1015.11023) and Ann. Fac. Sci. Toulouse, VI. Sér., Math. 8, 155--172 (1999; Zbl 0986.11036)], the authors deal with the index of curves \(C\) defined by an affine equation of the form \(Y^2= \pi f(X)\), where \(\pi\) is a uniformizing element and \(f(X)\) a monic irreducible polynomial of \(O_K[X]\). They give necessary and sufficient conditions for \(I(C)\) to be 1 and 2 in cases where \(K\) is dyadic and non-dyadic, respectively.
scientific article; zbMATH DE number 2031028