Essential dimension and pro-finite group schemes

• Giulio Bresciani ^[1]
  1. [1] Freie Universität Berlin, Germany
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 22, Nº 4, 2021, págs. 1899-1936
• Idioma: inglés
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• Resumen

  • t. A. Vistoli observed that, if Grothendieck’s section conjecture is true and X is a smooth hyperbolic curve over a field finitely generated over Q, then ⇡1.X / should somehow have essential dimension 1. We prove that an infinite, pro-finite etale group scheme always has infinite essential dimension. We intro- ´ duce a variant of essential dimension, the fce dimension, fced G, of a pro-finite group scheme G, which naturally coincides with ed G if G is finite, but has a better behaviour in the pro-finite case. Grothendieck’s section conjecture implies fced ⇡1.X / D dim X D 1 for X as above. We prove that, if A is an abelian variety over a field finitely generated over Q, then fced ⇡1.A/ D fced TA D dim A.