The intrinsic subgroup of an elliptic curve and Mazur's torsion theorem

TL;DR

This paper introduces the intrinsic subgroup of a curve, a new concept derived from a pairing on the torsion of the Picard group, and refines Mazur's torsion theorem by classifying which torsion subgroups appear as intrinsic subgroups of elliptic curves over rationals.

Contribution

It defines the intrinsic subgroup of a curve via a symmetric pairing and classifies which of Mazur's 15 torsion groups can be realized as intrinsic subgroups.

Findings

Intrinsic subgroup provides insights into reduction types of curves.

Refinement of Mazur's torsion theorem for intrinsic subgroups.

Classification of possible intrinsic subgroups of elliptic curves over .

Abstract

We define and study a biadditive symmetric (not necessarily perfect) pairing on the torsion part $\mathrm{Pic}(X)_{\mathrm{tors}}$ of the Picard group of a smooth projective curve $X$ over a field $k$ with values in $k^\times \otimes \mathbb{Q}/\mathbb{Z}$. We call its kernel the intrinsic subgroup of $X$. It turns out that some information on the reduction type of $X$ can be read off from the intrinsic subgroup. Mazur's torsion theorem says that there are exactly 15 isomorphism classes of abelian groups that appear as the rational torsion points of an elliptic curve $X$ over $\mathbb{Q}$ (identified with $\mathrm{Pic}(X)_{\mathrm{tors}}$). We refine this result by determining which subgroups of those 15 groups appear as the intrinsic subgroups.