# Indices of hyperelliptic curves over *p*-adic fields

## Authors

DOI: 10.1007/s002290050070

- Cite this article as:
- Van Geel, J. & Yanchevskii, V. manuscripta math. (1998) 96: 317. doi:10.1007/s002290050070

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## Abstract:

Let *k* be a *p*-adic field of odd residue characteristic and let *C* be a hyperelliptic (or elliptic) curve defined by the affine equation *Y*^{2}=*h*(*X*). We discuss the index of *C* if *h*(*X*)=ɛ*f*(*X*), where ɛ is either a non-square unit or a uniformising element in *O*_{k} and *f*(*X*) a monic, irreducible polynomial with integral coefficients. If a root θ of *f* generates an extension *k*(θ) with ramification index a power of 2, we completely determine the index of *C* in terms of data associated to θ. Theorem 3.11 summarizes our results and provides an algorithm to calculate the index for such curves *C*.