scholarly journals Equivariant Tamagawa number conjecture for abelian varieties over global fields of positive characteristic

2018 ◽  
pp. 1
Author(s):  
Fabien Trihan ◽  
David Vauclair
Keyword(s):  
Positive Characteristic ◽  
Abelian Varieties ◽  
Global Fields ◽  
Tamagawa Number Conjecture

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 GENERALIZED EXPLICIT DESCENT AND ITS APPLICATION TO CURVES OF GENUS 3

2016 ◽  
Vol 4 ◽  
Author(s):  
NILS BRUIN ◽  
BJORN POONEN ◽  
MICHAEL STOLL
Keyword(s):  
Abelian Varieties ◽  
Selmer Groups ◽  
Explicit Computation ◽  
Selmer Group ◽  
Common Generalization ◽  
Global Fields

We introduce a common generalization of essentially all known methods for explicit computation of Selmer groups, which are used to bound the ranks of abelian varieties over global fields. We also simplify and extend the proofs relating what is computed to the cohomologically defined Selmer groups. Selmer group computations have been practical for many Jacobians of curves over $\mathbb{Q}$ of genus up to 2 since the 1990s, but our approach is the first to be practical for general curves of genus 3. We show that our approach succeeds on some genus 3 examples defined by polynomials with small coefficients.

scholarly journals Local-global principles for Weil–Châtelet divisibility in positive characteristic

2017 ◽  
Vol 163 (2) ◽  
pp. 357-367 ◽  
Author(s):  
BRENDAN CREUTZ ◽  
JOSÉ FELIPE VOLOCH
Keyword(s):  
Elliptic Curve ◽  
Rational Point ◽  
Positive Characteristic ◽  
Number Fields ◽  
Global Field ◽  
Global Fields ◽  
Characteristic 2 ◽  
Global Principles ◽  
Global Principle

AbstractWe extend existing results characterizing Weil-Châtelet divisibility of locally trivial torsors over number fields to global fields of positive characteristic. Building on work of González-Avilés and Tan, we characterize when local-global divisibility holds in such contexts, providing examples showing that these results are optimal. We give an example of an elliptic curve over a global field of characteristic 2 containing a rational point which is locally divisible by 8, but is not divisible by 8 as well as examples showing that the analogous local-global principle for divisibility in the Weil-Châtelet group can also fail.

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