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1 results

  • Finiteness theorems for algebraic groups over function fields

  • Part of
  • Brian Conrad
  • Journal:
    Compositio Mathematica / Volume 148 / Issue 2 / March 2012
    Published online by Cambridge University Press:
    30 November 2011, pp. 555-639
    Print publication:
    March 2012
    • Article
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  • We prove the finiteness of class numbers and Tate–Shafarevich sets for all affine group schemes of finite type over global function fields, as well as the finiteness of Tamagawa numbers and Godement’s compactness criterion (and a local analogue) for all such groups that are smooth and connected. This builds on the known cases of solvable and semi-simple groups via systematic use of the recently developed structure theory and classification of pseudo-reductive groups.