CM points on shimura curves and p-adic binary quadratic forms

Piermarco Milione*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

Abstract

We prove that the set of CM points on the Shimura curve associated to an Eichler order inside an indefinite quaternion Q-algebra is in bijection with the set of certain classes of p-adic binary quadratic forms, where p is a prime dividing the discriminant of the quaternion algebra. The classes of p-adic binary quadratic forms are obtained by the action of a discrete and cocompact subgroup of PGL2(Qp) arising from the p-adic uniformization of the Shimura curve. We finally compute families of p-adic binary quadratic forms associated to an infinite family of Shimura curves studied by Amorós and Milione (2018). This extends some results of Alsina–Bayer (2004) to the p-adic context.
Original languageEnglish
Pages (from-to)237-256
Number of pages20
JournalActa Arithmetica
Volume183
Issue number3
DOIs
Publication statusPublished - 1 Jan 2018
MoE publication typeA1 Journal article-refereed

Keywords

  • Binary quadratic forms
  • CM points
  • P-adic uniformization
  • Quaternion algebras
  • Shimura curves

Fingerprint Dive into the research topics of 'CM points on shimura curves and p-adic binary quadratic forms'. Together they form a unique fingerprint.

Cite this