The Scholz theorem in function fields

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


The Scholz theorem in function fields states that the l-rank difference between the class groups of an imaginary quadratic function field and its associated real quadratic function field is either 0 or 1 for some prime l. Furthermore, Leopoldt's Spiegelungssatz (= the Reflection theorem) in function fields yields a comparison between the m-rank of some subgroup of the class group of an imaginary cyclic function field L1 and the m-rank of some subgroup of the class group of its associated real cyclic function field L2 for some prime number m; then their m-ranks also equal or differ by 1. In this paper we find an explicit necessary condition for their m-ranks (respectively l-ranks) to be the same in the case of cyclic function fields (respectively quadratic function fields). In particular, in the case of quadratic function fields, if l does not divide the regulator of L2, then their l-ranks are the same, equivalently if their l-ranks differ by 1, then l divides the regulator of L2.

Original languageEnglish
Pages (from-to)408-414
Number of pages7
JournalJournal of Number Theory
Issue number2
StatePublished - Feb 2007


  • Class groups
  • Cyclic function fields
  • Imaginary quadratic function fields
  • Rank
  • Real quadratic function fields
  • Regulator
  • Scholz theorem


Dive into the research topics of 'The Scholz theorem in function fields'. Together they form a unique fingerprint.

Cite this