On torsors under elliptic curves and Serre's pro-algebraic structures

TL;DR

This paper investigates the pro-algebraic structure of the canonical morphism from the Picard scheme of a torsor under an elliptic curve to the Néron model, connecting it with Serre's local class field theory and Shafarevich's duality.

Contribution

It demonstrates that the morphism is pro-algebraic and provides a new interpretation relating to Serre's work and Shafarevich's duality theory.

Findings

The morphism q is pro-algebraic in nature.

A construction recalling Serre's local class field theory is provided.

Results relate to Shafarevich's duality for torsors under abelian varieties.

Abstract

Let $K$ be a local field with algebraically closed residue field and $X_K$ a torsor under an elliptic curve $J_K$ over $K$. Let $X$ be a proper minimal regular model of $X_K$ over the ring of integers of $K$ and $J$ the identity component of the N\'eron model of $J_K$. We study the canonical morphism $q\colon \mathrm{Pic}^{0}_{X/S}\to J$ which extends the biduality 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.