Title:
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Sato-Tate distributions and Galois endomorphism modules in genus 2
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Author:
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Fité, Francesc; Kedlaya, Kiran; Rotger Cerdà, Víctor; Sutherland, Andrew
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Other authors:
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Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada II; Universitat Politècnica de Catalunya. TN - Grup de Recerca en Teoria de Nombres |
Abstract:
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For an abelian surface A over a number eld k, we study the limit-
ing distribution of the normalized Euler factors of the L-function of A.
This distribution is expected to correspond to taking characteristic poly-
nomials of a uniform random matrix in some closed subgroup of USp(4);
this Sato-Tate group may be obtained from the Galois action on any Tate
module of A. We show that the Sato-Tate group is limited to a particular
list of 55 groups up to conjugacy. We then classify A according to the
Galois module structure on the R-algebra generated by endomorphisms of
AQ (the Galois type), and establish a matching with the classi cation of
Sato-Tate groups; this shows that there are at most 52 groups up to con-
jugacy which occur as Sato-Tate groups for suitable A and k, of which 34
can occur for k = Q. Finally, we exhibit examples of Jacobians of hyperel-
liptic curves exhibiting each Galois type (over Q whenever possible), and
observe numerical agreement with the expected Sato-Tate distribution by
comparing moment statistics. |
Abstract:
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Peer Reviewed |
Subject(s):
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-Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria::Geometria algebraica -Curves -Abelian surfaces -Endomorphism algebras -Galois type -Sato-Tate distributions -Corbes algebraiques -14H Curves |
Rights:
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Attribution-NonCommercial-NoDerivs 3.0 Spain
http://creativecommons.org/licenses/by-nc-nd/3.0/es/ |
Document type:
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Article - Published version Article |
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