manuscripta mathematica

, Volume 96, Issue 3, pp 317–333

Indices of hyperelliptic curves over p-adic fields

  • J. Van Geel
  • V. I. Yanchevskii

DOI: 10.1007/s002290050070

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


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 Y2=h(X). We discuss the index of C if h(X)=ɛf(X), where ɛ is either a non-square unit or a uniformising element in Ok 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.

Mathematics Subject Classification (1991):11D88, 11S25, 12G05, 14G20, 14H25

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • J. Van Geel
    • 1
  • V. I. Yanchevskii
    • 2
  1. 1. University of Gent, Department of Pure Mathematics and Computer Algebra, Galglaan 2, B-9000 Gent, Belgium.¶e-mail:
  2. 2.Institute of Mathematics, Academy of Science of Belarus, Surganov str. 11, 220072 Minsk, Belarus.¶e-mail: v.yanchevskii@im.bas-net.byBY