An Analogue of the Brauer–Siegel Theorem for Abelian Varieties in Positive Characteristic

2016 ◽  
Vol 16 (1) ◽  
pp. 45-93 ◽  
Author(s):  
Marc Hindry ◽  
Amílcar Pacheco
Keyword(s):  
Positive Characteristic ◽  
Abelian Varieties ◽  
Siegel Theorem

scholarly journals Dynamics on abelian varieties in positive characteristic

2018 ◽  
Vol 12 (9) ◽  
pp. 2185-2235 ◽  
Author(s):  
Jakub Byszewski ◽  
Gunther Cornelissen
Keyword(s):  
Positive Characteristic ◽  
Abelian Varieties

scholarly journals DUALITY FOR COHOMOLOGY OF CURVES WITH COEFFICIENTS IN ABELIAN VARIETIES

2018 ◽  
Vol 240 ◽  
pp. 42-149 ◽  
Author(s):  
TAKASHI SUZUKI
Keyword(s):  
Function Field ◽  
Positive Characteristic ◽  
Abelian Varieties ◽  
Base Field ◽  
Tate Pairing ◽  
Global Function ◽  
Crystalline Cohomology ◽  
Local Duality ◽  
Height Pairing ◽  
Finite Base

In this paper, we formulate and prove a duality for cohomology of curves over perfect fields of positive characteristic with coefficients in Néron models of abelian varieties. This is a global function field version of the author’s previous work on local duality and Grothendieck’s duality conjecture. It generalizes the perfectness of the Cassels–Tate pairing in the finite base field case. The proof uses the local duality mentioned above, Artin–Milne’s global finite flat duality, the nondegeneracy of the height pairing and finiteness of crystalline cohomology. All these ingredients are organized under the formalism of the rational étale site developed earlier.

scholarly journals Questions de corps de définition pour les variétés abéliennes en caractéristique positive

2008 ◽  
Vol 7 (4) ◽  
pp. 623-639 ◽  
Author(s):  
Franck Benoist ◽  
Françoise Delon
Keyword(s):  
Algebraic Geometry ◽  
Abelian Variety ◽  
Positive Characteristic ◽  
Abelian Varieties ◽  
Direct Proof ◽  
Point Of View ◽  
Theoretical Point ◽  
Definition Of

AbstractDichotomies in various conjectures from algebraic geometry are in fact occurrences of the dichotomy among Zariski structures. This is what Hrushovski showed and which enabled him to solve, positively, the geometric Mordell–Lang conjecture in positive characteristic. Are we able now to avoid this use of Zariski structures? Pillay and Ziegler have given a direct proof that works for semi-abelian varieties they called ‘very thin’, which include the ordinary abelian varieties. But it does not apply in all generality: we describe here an abelian variety which is not very thin. More generally, we consider from a model-theoretical point of view several questions about the fields of definition of semi-abelian varieties.

scholarly journals The Fields of Moduli for Polarized Abelian Varieties and for Curves

1972 ◽  
Vol 48 ◽  
pp. 37-55 ◽  
Author(s):  
Shoji Koizumi
Keyword(s):  
Algebraic Geometry ◽  
Characteristic Zero ◽  
Essential Role ◽  
Positive Characteristic ◽  
Abelian Varieties ◽  
Complex Multiplications ◽  
Definition Of ◽  
Fields Of Moduli

In the study of moduli of polarized abelian varieties and of curves as well as in the theory of complex multiplications, the notion of fields of moduli for structures plays an essential role. This notion was first introduced by Matsusaka [7] for polarized varieties with some pleasing properties and later was given a more comprehensible treatment by Shimura [10] in the case of polarized abelian varieties or polarized abelian varieties with some further structures. Both authors discussed fields of moduli not only in algebraic geometry of characteristic zero but also in that of positive characteristic, but in the latter case the definition of fields of moduli seems somewhat artificial and there have been no essential applications of them so far.

scholarly journals On the μ-invariants of abelian varieties over function fields of positive characteristic

2021 ◽  
Vol 15 (4) ◽  
pp. 863-907
Author(s):  
King-Fai Lai ◽  
Ignazio Longhi ◽  
Takashi Suzuki ◽  
Ki-Seng Tan ◽  
Fabien Trihan
Keyword(s):  
Positive Characteristic ◽  
Function Fields ◽  
Abelian Varieties
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