scholarly journals Selmer groups of abelian varieties in extensions of function fields

2008 ◽  
Vol 261 (4) ◽  
pp. 787-804 ◽  
Author(s):  
Amílcar Pacheco
Keyword(s):  
Function Fields ◽  
Abelian Varieties ◽  
Selmer Groups

scholarly journals On the Selmer groups of abelian varieties over function fields of characteristic p > 0

2009 ◽  
Vol 146 (1) ◽  
pp. 23-43 ◽  
Author(s):  
TADASHI OCHIAI ◽  
FABIEN TRIHAN
Keyword(s):  
Number Field ◽  
Function Field ◽  
Function Fields ◽  
Abelian Varieties ◽  
Iwasawa Theory ◽  
Selmer Groups ◽  
Characteristic P ◽  
Field Case ◽  
New Phenomena ◽  
Geometric Analogue

AbstractWe study a (p-adic) geometric analogue for abelian varieties over a function field of characteristic p of the cyclotomic Iwasawa theory and the non-commutative Iwasawa theory for abelian varieties over a number field initiated by Mazur and Coates respectively. We will prove some analogue of the principal results obtained in the case over a number field and we study new phenomena which did not happen in the case of number field case. We also propose a conjecture (Conjecture 1.6) which might be considered as a counterpart of the principal conjecture in the case over a number field.

scholarly journals On the arithmetic of abelian varieties

2020 ◽  
Vol 2020 (762) ◽  
pp. 1-33
Author(s):  
Mohamed Saïdi ◽  
Akio Tamagawa
Keyword(s):  
Function Fields ◽  
Abelian Varieties ◽  
Galois Cohomology ◽  
Rational Points ◽  
Selmer Groups ◽  
Selmer Group ◽  
Cohomology Groups ◽  
Finitely Generated ◽  
Natural Objects ◽  
Weil Group

AbstractWe prove some new results on the arithmetic of abelian varieties over function fields of one variable over finitely generated (infinite) fields. Among other things, we introduce certain new natural objects “discrete Selmer groups” and “discrete Shafarevich–Tate groups”, and prove that they are finitely generated {\mathbb{Z}}-modules. Further, we prove that in the isotrivial case, the discrete Shafarevich–Tate group vanishes and the discrete Selmer group coincides with the Mordell–Weil group. One of the key ingredients to prove these results is a new specialisation theorem for first Galois cohomology groups, which generalises Néron’s specialisation theorem for rational points of abelian varieties.

Non-communtative Iwasawa main conjecture

2020 ◽  
Vol 16 (09) ◽  
pp. 2041-2094
Author(s):  
Malte Witte
Keyword(s):  
Functional Equation ◽  
Galois Group ◽  
Function Field ◽  
Function Fields ◽  
Abelian Varieties ◽  
Main Conjecture

We formulate and prove an analogue of the non-commutative Iwasawa Main Conjecture for [Formula: see text]-adic representations of the Galois group of a function field of characteristic [Formula: see text]. We also prove a functional equation for the resulting non-commutative [Formula: see text]-functions. As corollaries, we obtain non-commutative generalizations of the main conjecture for Picard-[Formula: see text]-motives of Greither and Popescu and a main conjecture for abelian varieties over function fields in precise analogy to the [Formula: see text] main conjecture of Coates, Fukaya, Kato, Sujatha and Venjakob.

scholarly journals The Iwasawa Main Conjecture for constant ordinary abelian varieties over function fields

2016 ◽  
Vol 112 (6) ◽  
pp. 1040-1058 ◽  
Author(s):  
King Fai Lai ◽  
Ignazio Longhi ◽  
Ki-Seng Tan ◽  
Fabien Trihan
Keyword(s):  
Function Fields ◽  
Abelian Varieties ◽  
Main Conjecture
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