[Paper Review] Heights, ranks and regulators of abelian varieties

Fabien Pazuki|arXiv (Cornell University)|Jun 16, 2015

Algebraic Geometry and Number Theory24 references4 citations

TL;DR

This paper establishes an unconditional explicit upper bound on the Mordell-Weil rank of abelian varieties over number fields by linking the Faltings height to the injectivity diameter and bad reduction primes. Under the Lang–Silverman height conjecture, it proves a Northcott property for the regulator on polarized abelian varieties with dense rational points, removing the simplicity assumption in the principally polarized case via a refined conjecture.

ABSTRACT

We lower bound the Faltings height of an abelian variety over a number field by the sum of its injectivity diameter and the norm of its bad reduction primes. It leads to an unconditional bound on the rank of Mordell-Weil groups. Assuming the height conjecture of Lang and Silverman, we then obtain a Northcott property for the regulator on the set of simple abelian varieties defined over a fixed number field, of fixed dimension $g$, bounded rank and with dense rational points over a number field. We remove the simplicity assumption in the principally polarized case by giving a refined version of the Lang-Silverman conjecture.

Motivation & Objective

• To establish an unconditional explicit upper bound on the Mordell-Weil rank of abelian varieties over number fields.
• To prove a Northcott property for the regulator on polarized abelian varieties under the Lang–Silverman height conjecture.
• To remove the simplicity assumption in the Northcott property for regulators in the principally polarized case.
• To refine the Lang–Silverman conjecture to apply to non-simple abelian varieties in the principally polarized setting.
• To extend the analogy between number fields and abelian varieties by linking regulators, heights, and torsion structures.

Proposed method

• Derives a lower bound on the Faltings height of an abelian variety in terms of the norm of bad reduction primes and the injectivity diameter of its complex uniformization.
• Applies the Matrix Lemma to relate the regulator to the product of successive minima and the height of the abelian variety.
• Uses explicit bounds from Bertini-type theorems and height reduction via quotients to control the constants in the height inequality.
• Employs induction on subvarieties to bound canonical heights of points when the endomorphism action is not dense.
• Introduces a refined version of the Lang–Silverman conjecture tailored to the principally polarized case, enabling removal of the simplicity assumption.
• Applies Minkowski’s successive minima inequality to the Mordell-Weil lattice to relate regulator to canonical heights and the Faltings height.

Experimental results

Research questions

• RQ1 Can an unconditional explicit upper bound on the Mordell-Weil rank be derived from height and bad reduction data?
• RQ2 Does the Lang–Silverman height conjecture imply a Northcott property for the regulator on polarized abelian varieties with dense rational points?
• RQ3 Can the simplicity assumption in such a Northcott property be removed in the principally polarized case?
• RQ4 What refined form of the Lang–Silverman conjecture enables this removal of the simplicity condition?
• RQ5 How do the analogies between number fields and abelian varieties (e.g., regulator, class number, torsion) extend to regulators and heights?

Key findings

• An unconditional lower bound on the Faltings height is established: $\mathop{h_{\mathrm{F}}^{+}}(A/K) \geq c \frac{1}{d}\log N^{0}_{A/K} + c_0$, with explicit constants $c = (12g)^{-12g^{12g^{4g}}}$ and $c_0 = -1/c$.
• For polarized abelian varieties, the bound is strengthened to include the injectivity diameter: $\mathop{h_{\mathrm{F}}^{+}}(A/K) \geq c_1 \frac{1}{d}\log N^{0}_{A/K} + c_2 \frac{1}{d}\sum_{v\in M_K^\infty} d_v \rho(A_v, L_v)^{-2} + c_3$, with $c_1 = (12g)^{-12g^{12g^{4g}}}/17$, $c_2 = 1/17$, $c_3 = -1/c_1$.
• Under the Lang–Silverman conjecture, the regulator is bounded from below by a power of the height, leading to a Northcott property for the regulator on the set of polarized abelian varieties with bounded dimension, rank, and Zariski-dense rational points.
• The simplicity assumption is removed in the principally polarized case by introducing a refined version of the Lang–Silverman conjecture, which controls the height in terms of the regulator and canonical height of points.
• The regulator is shown to be bounded from below by $c_{36}^{m - m_0} \max\{\mathop{h_{\mathrm{F}}^{+}}(A/K), 1\}^{m - m_0}$, where $m$ is the Mordell-Weil rank and $m_0$ is the maximal rank of an abelian subvariety.
• The paper establishes a complete dictionary between number fields and abelian varieties, linking regulators, heights, class numbers, torsion, and unit ranks.