scholarly journals On Deformations of 1-motives

2019 ◽  
Vol 62 (1) ◽  
pp. 11-22
Author(s):  
A. Bertapelle ◽  
N. Mazzari
Keyword(s):  
Abelian Variety ◽  
Deformation Theory ◽  
Positive Characteristic ◽  
Infinitesimal Deformation ◽  
Tate Group

AbstractAccording to a well-known theorem of Serre and Tate, the infinitesimal deformation theory of an abelian variety in positive characteristic is equivalent to the infinitesimal deformation theory of its Barsotti–Tate group. We extend this result to 1-motives.

scholarly journals Frobenius Distribution for Quotients of Fermat Curves of Prime Exponent

2016 ◽  
Vol 68 (2) ◽  
pp. 361-394
Author(s):  
Francesc Fité ◽  
Josep González ◽  
Joan-Carles Lario
Keyword(s):  
Abelian Variety ◽  
Complex Multiplication ◽  
Cyclotomic Field ◽  
Explicit Method ◽  
Fermat Curve ◽  
Tate Conjecture ◽  
Tate Group ◽  
Fermat Curves ◽  
Prime Exponent ◽  
Simple Abelian Variety

AbstractLet denote the Fermat curve over ℚ of prime exponent ℓ. The Jacobian Jac() of splits over ℚ as the product of Jacobians Jac(k), 1 ≤ k ≤ ℓ −2, where k are curves obtained as quotients of by certain subgroups of automorphisms of . It is well known that Jac(k) is the power of an absolutely simple abelian variety Bk with complex multiplication. We call degenerate those pairs (ℓ, k) for which Bk has degenerate CM type. For a non-degenerate pair (ℓ, k), we compute the Sato–Tate group of Jac(Ck), prove the generalized Sato–Tate Conjecture for it, and give an explicit method to compute the moments and measures of the involved distributions. Regardless of whether (ℓ, k) is degenerate, we also obtain Frobenius equidistribution results for primes of certain residue degrees in the ℓ-th cyclotomic field. Key to our results is a detailed study of the rank of certain generalized Demjanenko matrices.

scholarly journals Deformations of elliptic fiber bundles in positive characteristic

2013 ◽  
Vol 211 ◽  
pp. 79-108 ◽  
Author(s):  
Holger Partsch
Keyword(s):  
Deformation Theory ◽  
Positive Characteristic ◽  
Fiber Bundles ◽  
Elliptic Surfaces ◽  
Elliptic Fibrations ◽  
Elliptic Fiber

AbstractWe study the deformation theory of elliptic fiber bundles over curves in positive characteristics. As applications, we give examples of nonliftable elliptic surfaces in characteristics 2 and 3, which answer a question of Katsura and Ueno. Also, we construct a class of elliptic fibrations, whose liftability is equivalent to a conjecture of Oort concerning the liftability of automorphisms of curves. Finally, we classify deformations of bielliptic surfaces.

scholarly journals Deformations of elliptic fiber bundles in positive characteristic

2013 ◽  
Vol 211 ◽  
pp. 79-108
Author(s):  
Holger Partsch
Keyword(s):  
Deformation Theory ◽  
Positive Characteristic ◽  
Fiber Bundles ◽  
Elliptic Surfaces ◽  
Elliptic Fibrations ◽  
Elliptic Fiber

AbstractWe study the deformation theory of elliptic fiber bundles over curves in positive characteristics. As applications, we give examples of nonliftable elliptic surfaces in characteristics 2 and 3, which answer a question of Katsura and Ueno. Also, we construct a class of elliptic fibrations, whose liftability is equivalent to a conjecture of Oort concerning the liftability of automorphisms of curves. Finally, we classify deformations of bielliptic surfaces.

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.

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