Let O_K be a complete discrete valuation ring with algebraically closed residue field of positive characteristic and field of fractions K. Let X_K be a torsor under an elliptic curve A_K over K, X the proper minimal regular model of X_K over S := Spec(O_K), and J the identity component of the Neron model of Pic^0_{X_K/K}. We study the canonical morphism q : Pic^0_{X/S}\to J which extends the natural isomorphism on generic fibres. We show that q is pro-algebraic in nature with a construction that recalls Serre's work on local class field theory. Furthermore, we interpret our results in relation to Shafarevich's duality theory for torsors under abelian varieties.
On torsors under elliptic curves and Serre's pro-algebraic structures
BERTAPELLE, ALESSANDRA;
2014
Abstract
Let O_K be a complete discrete valuation ring with algebraically closed residue field of positive characteristic and field of fractions K. Let X_K be a torsor under an elliptic curve A_K over K, X the proper minimal regular model of X_K over S := Spec(O_K), and J the identity component of the Neron model of Pic^0_{X_K/K}. We study the canonical morphism q : Pic^0_{X/S}\to J which extends the natural isomorphism on generic fibres. We show that q is pro-algebraic in nature with a construction that recalls Serre's work on local class field theory. Furthermore, we interpret our results in relation to Shafarevich's duality theory for torsors under abelian varieties.Pubblicazioni consigliate
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