Aspects of Iwasawa theory over function fields

  • Andrea Bandini

    Università degli Studi di Pisa, Italy
  • Francesc Bars

    Universitat Autònoma de Barcelona, Bellaterra (Barcelona), Spain
  • Ignazio Longhi

    National Taiwan University, Taipei, Taiwan
Aspects of Iwasawa theory over function fields cover

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We consider ZpN\mathbb Z_p^{\mathbb{N}}-extensions F\mathcal F of a global function field FF and study various aspects of Iwasawa theory with emphasis on the two main themes already (and still) developed in the number fields case as well. When dealing with the Selmer group of an abelian variety AA defined over FF, we provide all the ingredients to formulate an Iwasawa Main Conjecture relating the Fitting ideal and the pp-adic LL-function associated to AA and F\mathcal F. We do the same, with characteristic ideals and pp-adic LL-functions, in the case of class groups (using known results on characteristic ideals and Stickelberger elements for Zpd\mathbb Z_p^d-extensions). The final section provides more details for the cyclotomic ZpN\mathbb Z_p^{\mathbb{N}}-extension arising from the torsion of the Carlitz module: in particular, we relate cyclotomic units with Bernoulli–Carlitz numbers by a Coates–Wiles homomorphism.