The intrinsic subgroup of an elliptic curve and Mazur's torsion theorem
Takao Yamazaki, Yifan Yang, Hwajong Yoo, Myungjun Yu
TL;DR
This work defines a biadditive symmetric pairing on the torsion of a curve’s Picard group and introduces the intrinsic subgroup, revealing its potential as a new invariant tied to reduction types. Focusing on elliptic curves over Q, it uses explicit modular-function machinery (Dedekind eta, Hauptmoduls, Tate normal forms) to compute pairings and to parameterize torsion structures; it then encodes these constraints on new modular curves X_1(N)_M^± and X_1^0(N,2)_{ul M}^±, analyzing their rational points to classify which intrinsic-subgroup structures occur alongside Mazur’s torsion possibilities. The paper provides a complete classification of possible (E(Q)_{tors}, E(Q)_{tors}^{is}) pairs and proves nonexistence/nontriviality results via modular-curve rational points, with existence established through thin-set arguments in the remaining cases. A second construction using Bloch’s higher Chow groups links the pairing to higher-dimensional cases, yielding reduction-type constraints and a Tate-curve analysis, and showing the intrinsic-subgroup behavior reflects reduction data in a broad arithmetic-geometric setting.
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.