[Paper Review] Effective Height Upper Bounds on Algebraic Tori

Philipp Habegger|arXiv (Cornell University)|Jan 16, 2012

Polynomial and algebraic computation28 references3 citations

TL;DR

This paper establishes effective upper bounds on the absolute logarithmic Weil height of algebraic points on algebraic tori, particularly in the context of unlikely intersections in $\mathbf{G}_m^n$. By refining height bounds from Bombieri–Zannier and extending them effectively, it proves finiteness results for points on curves lying in codimension-2 subgroups, resolving a conjecture of Bombieri–Masser–Zannier in low dimensions and providing foundational tools for Pink’s conjecture in Shimura varieties.

ABSTRACT

The main emphasis will be on height upper bounds in the algebraic torus G^{n}_{m}. By height we will mean the absolute logarithmic Weil height. Section 3.2 contains a precise definition of this and other more general height functions. The first appendix gives a short overview of known results in the abelian case. The second appendix contains a few height bounds in Shimura varieties.

Motivation & Objective

• To establish effective, uniform upper bounds on the absolute logarithmic Weil height of points on algebraic tori lying in algebraic subgroups of codimension at least 2.
• To resolve the question of whether the set of such points on a curve not contained in a proper coset is finite, particularly in the toric setting.
• To extend effective height bounds from subvarieties of dimension ≤1 to higher-dimensional subvarieties via a detour to surfaces in $\mathbf{G}_m^3$.
• To provide a foundation for bounded height conjectures in Shimura varieties, especially in $Y(1)^2$, by analyzing singular moduli and special subvarieties.
• To investigate the implications of these height bounds for Pink’s conjecture on unlikely intersections in Shimura varieties, particularly for curves not contained in proper special subvarieties.

Proposed method

• Uses effective versions of Bombieri–Zannier height bounds for subvarieties of $\mathbf{G}_m^n$ of dimension at most 1, with explicit dependence on degree and height of the subvariety.
• Applies a geometric construction to lift a curve $C o \mathbf{G}_m^n$ with $n \leq 5$ to a surface $S \subset \mathbf{G}_m^3$, enabling the use of known height bounds on surfaces.
• Employs the theory of singular moduli and modular curves $Y_0(N)$ to analyze height growth in $Y(1)^2$, particularly for points lying on special subvarieties.
• Combines results from analytic number theory—such as polynomial and logarithmic bounds on the height of singular moduli—under GRH and unconditionally.
• Introduces the concept of minimal special subvarieties ${\mathcal{S}}(p)$ for a point $p$ in $Y(1)^2$, and uses their degree $\deg_{\mathbf{Q}} S$ as a measure of complexity.
• Proposes a weakly bounded height conjecture in $Y(1)^2$ stating that height is $O(\log \deg_{\mathbf{Q}} S)$, and a super-weak version with $O((\deg_{\mathbf{Q}} S)^\epsilon)$, to circumvent reliance on GRH.

Experimental results

Research questions

• RQ1 Is the set of points on an irreducible curve $C \subset \mathbf{G}_m^n$ defined over $\overline{\mathbf{Q}}$ and not contained in a proper coset, which lie in algebraic subgroups of codimension at least 2, finite?
• RQ2 Can effective height upper bounds be established for points on subvarieties of $\mathbf{G}_m^n$ lying in algebraic subgroups of codimension equal to the dimension of the subvariety?
• RQ3 What is the growth rate of the height of singular moduli $j$ in terms of their discriminant $\Delta$, and how does this affect height bounds in Shimura varieties?
• RQ4 Can a bounded height conjecture be formulated for curves in $Y(1)^2$ that are not special, with height bounded in terms of the degree of the curve over $\mathbf{Q}$?
• RQ5 What are the implications of effective height bounds for the finiteness of points on curves in $Y(1)^n$ lying in special subvarieties of codimension at least 2?

Key findings

• For $n \leq 5$, the set of points on an irreducible curve $C \subset \mathbf{G}_m^n$ not contained in a proper coset and lying in an algebraic subgroup of codimension at least 2 is finite.
• The proof is effective: if the Bombieri–Zannier height bound for subvarieties of dimension ≤1 is effective, then the finiteness result is effective, with explicit bounds in terms of degree and height of the curve.
• The height of a singular moduli $j$ with fundamental discriminant $\Delta$ is bounded above by $c|\Delta|^{\epsilon}$ for any $\epsilon > 0$, unconditionally, and by $c \log |\Delta|$ under GRH.
• A weakly bounded height conjecture in $Y(1)^2$ states that for non-special curves, the height of points $p$ with $\dim \mathcal{S}(p) \leq 1$ is $O(\log \deg_{\mathbf{Q}} S)$, and this holds unconditionally for points with $\mathcal{S}(p) = Y_0(N)$.
• A super-weak bounded height conjecture with $O((\deg_{\mathbf{Q}} S)^\epsilon)$ growth implies the finiteness of points on non-special curves in $Y(1)^n$ lying in special subvarieties of codimension ≥2.
• The effective height bounds established in this paper provide a key tool for proving finiteness results in the context of Pink’s conjecture, particularly when combined with Pila’s o-minimal methods.