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Research in Number Theory

, 5:33 | Cite as

Cyclic covers and Ihara’s question

  • Christopher RasmussenEmail author
  • Akio Tamagawa
Research
  • 14 Downloads

Abstract

Let \(\ell \) be a rational prime. Given a superelliptic curve C / k of \(\ell \)-power degree, we describe the field generated by the \(\ell \)-power torsion of the Jacobian variety in terms of the branch set and reduction type of C (and hence, in terms of data determined by a suitable affine model of C). If the Jacobian is good away from \(\ell \) and the branch set is defined over a pro-\(\ell \) extension of \(k({\varvec{\upmu }}_{\ell ^\infty })\) unramified away from \(\ell \), then the \(\ell \)-power torsion of the Jacobian is rational over the maximal such extension. By decomposing the covering into a chain of successive cyclic \(\ell \)-coverings, the mod \(\ell \) Galois representation attached to the Jacobian is decomposed into a block upper triangular form. The blocks on the diagonal of this form are further decomposed in terms of the Tate twists of certain subgroups \(W_s\) of the quotients of the Jacobians of consecutive coverings. The result is a natural extension of earlier work by Anderson and Ihara, who demonstrated that a stricter condition on the branch locus guarantees the \(\ell \)-power torsion of the Jacobian is rational over the fixed field of the kernel of the canonical pro-\(\ell \) outer Galois representation attached to an open subset of \({\mathbb {P}}^1\).

Notes

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan

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