Vari\'et\'es ab\'eliennes sur les corps de fonctions de courbes sur des corps locaux sup\'erieurs
Diego Izquierdo

TL;DR
This paper develops duality theorems for Tate-Shafarevich groups of abelian varieties over function fields of curves defined over higher-dimensional local fields, extending classical duality results.
Contribution
It introduces new duality theorems for Tate-Shafarevich groups over higher-dimensional local fields, generalizing existing theories to more complex base fields.
Findings
Established local duality theorems for abelian varieties over higher-dimensional local fields.
Proved global duality theorems for Tate-Shafarevich groups over function fields.
Extended classical duality results to higher-dimensional local field contexts.
Abstract
Let be a higher-dimensional local field and be a smooth projective geometrically integral curve over . Let be the function field of . We define Tate-Shafarevich groups of an abelian variety via cohomology classes locally trivial at each completion of coming from a closed point of . We prove local duality theorems for abelian varieties over , as well as global duality theorems for Tate-Shafarevich groups of abelian varieties over . Soient un corps local sup\'erieur et une courbe projective lisse g\'eom\'etriquement int\`egre de corps de fonctions . On d\'efinit les groupes de Tate-Shafarevich d'une vari\'et\'e ab\'elienne en consid\'erant les classes de cohomologie qui deviennent triviales sur chaque compl\'et\'e de provenant d'un point ferm\'e de . On \'etablit des th\'eor\`emes de dualit\'e locale pour les vari\'et\'es ab\'eliennes sur…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Meromorphic and Entire Functions · Advanced Algebra and Geometry
