Control theorems for l-adic Lie extensions of global function fields

• Andrea Bandini ^[1] ; Maria Valentino ^[2]
  1. [1] University of Parma

     University of Parma

     Parma, Italia

     
  2. [2] University of Calabria

     University of Calabria

     Cosenza, Italia

     
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 14, Nº 4, 2015, págs. 1065-1092
• Idioma: inglés
• Enlaces
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• Resumen

  • Let F be a global function field of characteristic p>0, K/F an `-adic Lie extension unramified outside a finite set of places S and A/F an abelian variety. We study SelA(K)_` (the Pontrjagin dual of the Selmer group) and (under some mild hypotheses) prove that it is a finitely generated Z`[[Gal(K/F)]]- module via generalizations of Mazur’s Control Theorem. If Gal(K/F) has no elements of order ` and contains a closed normal subgroup H such that Gal(K/F)/H ' Z`, we are able to give sufficient conditions for SelA(K)_` to be finitely generated as Z`[[H]]-module and, consequently, a torsion Z`[[Gal(K/F)]]-module. We deal with both cases ` 6= p and ` = p.