Field of definition and Galois orbits for the Macbeath-Hurwitz curves

2000 ◽  
Vol 74 (5) ◽  
pp. 342-349 ◽  
Author(s):  
M. Streit
Keyword(s):  
Field Of Definition ◽  
Galois Orbits

scholarly journals Fields of Definition for Representations of Associative Algebras

2018 ◽  
Vol 62 (1) ◽  
pp. 291-304
Author(s):  
Dave Benson ◽  
Zinovy Reichstein
Keyword(s):  
Finite Extension ◽  
Representation Type ◽  
Essential Dimension ◽  
Extension Field ◽  
Associative Algebras ◽  
Separable Extension ◽  
Finite Representation ◽  
Field Of Definition ◽  
Finite Dimensional ◽  
A Finite Group

AbstractWe examine situations, where representations of a finite-dimensionalF-algebraAdefined over a separable extension fieldK/F, have a unique minimal field of definition. Here the base fieldFis assumed to be a field of dimension ≼1. In particular,Fcould be a finite field ork(t) ork((t)), wherekis algebraically closed. We show that a unique minimal field of definition exists if (a)K/Fis an algebraic extension or (b)Ais of finite representation type. Moreover, in these situations the minimal field of definition is a finite extension ofF. This is not the case ifAis of infinite representation type orFfails to be of dimension ≼1. As a consequence, we compute the essential dimension of the functor of representations of a finite group, generalizing a theorem of Karpenko, Pevtsova and the second author.

scholarly journals The Walker Abel–Jacobi map descends

2021 ◽  
Author(s):  
Jeffrey D. Achter ◽  
Sebastian Casalaina-Martin ◽  
Charles Vial
Keyword(s):  
Abelian Variety ◽  
Projective Manifold ◽  
Field Of Definition ◽  
Definition Of ◽  
Complex Projective ◽  
Intermediate Jacobian ◽  
Cycle Classes ◽  
Coniveau Filtration

AbstractFor a complex projective manifold, Walker has defined a regular homomorphism lifting Griffiths’ Abel–Jacobi map on algebraically trivial cycle classes to a complex abelian variety, which admits a finite homomorphism to the Griffiths intermediate Jacobian. Recently Suzuki gave an alternate, Hodge-theoretic, construction of this Walker Abel–Jacobi map. We provide a third construction based on a general lifting property for surjective regular homomorphisms, and prove that the Walker Abel–Jacobi map descends canonically to any field of definition of the complex projective manifold. In addition, we determine the image of the l-adic Bloch map restricted to algebraically trivial cycle classes in terms of the coniveau filtration.

scholarly journals Finite simple groups with short Galois orbits on conjugacy classes

2020 ◽  
Vol 544 ◽  
pp. 151-169
Author(s):  
Victor Bovdi ◽  
Thomas Breuer ◽  
Attila Maróti
Keyword(s):  
Conjugacy Classes ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Galois Orbits

scholarly journals Average of the first invariant factor of the reductions of abelian varieties of CM type

2016 ◽  
Vol 12 (02) ◽  
pp. 445-463 ◽  
Author(s):  
Sungjin Kim
Keyword(s):  
Abelian Group ◽  
Asymptotic Formula ◽  
Prime Ideal ◽  
Elliptic Curves ◽  
Short Range ◽  
Abelian Varieties ◽  
Prime Ideals ◽  
Good Reduction ◽  
Invariant Factor ◽  
Field Of Definition

For a field of definition [Formula: see text] of an abelian variety [Formula: see text] and prime ideal [Formula: see text] of [Formula: see text] which is of a good reduction for [Formula: see text], the structure of [Formula: see text] as abelian group is: [Formula: see text] where [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text]. We are interested in finding an asymptotic formula for the number of prime ideals [Formula: see text] with [Formula: see text], [Formula: see text] has a good reduction at [Formula: see text], [Formula: see text]. We succeed in proving this under the assumption of the Generalized Riemann Hypothesis (GRH). Unconditionally, we achieve a short range asymptotic for abelian varieties of CM type, and the full cyclicity theorem for elliptic curves over a number field containing the CM field.

scholarly journals DISTINGUISHED MODELS OF INTERMEDIATE JACOBIANS

2018 ◽  
Vol 19 (3) ◽  
pp. 891-918 ◽  
Author(s):  
Jeffrey D. Achter ◽  
Sebastian Casalaina-Martin ◽  
Charles Vial
Keyword(s):  
Abelian Variety ◽  
Hilbert Scheme ◽  
Projective Variety ◽  
Galois Representation ◽  
Complete Intersections ◽  
Base Field ◽  
Field Of Definition ◽  
Level 1 ◽  
Albanese Variety ◽  
Coniveau Filtration

We show that the image of the Abel–Jacobi map admits functorially a model over the field of definition, with the property that the Abel–Jacobi map is equivariant with respect to this model. The cohomology of this abelian variety over the base field is isomorphic as a Galois representation to the deepest part of the coniveau filtration of the cohomology of the projective variety. Moreover, we show that this model over the base field is dominated by the Albanese variety of a product of components of the Hilbert scheme of the projective variety, and thus we answer a question of Mazur. We also recover a result of Deligne on complete intersections of Hodge level 1.

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