[Paper Review] D\\'ecompositions en hauteurs locales

Fabien Pazuki|arXiv (Cornell University)|May 21, 2012

Algebraic Geometry and Number Theory3 citations

TL;DR

This paper provides a decomposition formula for the Faltings height of the Jacobian of a hyperelliptic curve over a number field and for the Néron-Tate height of its rational points, using local components derived from Moret-Bailly models and modular forms. The key result generalizes Silverman's 1986 formula for elliptic curves to higher dimensions, establishing a precise link between global heights and local invariants at finite and archimedean places.

ABSTRACT

Let A be the jacobian variety of a hyperelliptic curve defined over a number field k. We provide a decomposition formula for the Faltings height of A and for the N\\'eron-Tate height of k-rational points on A. We formulate a question of Bogomolov type on the space of principally polarized abelian varieties of dimension g.

Motivation & Objective

• To extend Silverman's local decomposition formula for the Faltings height of elliptic curves to higher-dimensional abelian varieties, specifically Jacobians of hyperelliptic curves.
• To provide explicit local height formulas for Néron-Tate heights of rational points on such Jacobians.
• To establish a connection between global heights and local invariants (archimedean and finite places), leveraging discriminants and modular forms.
• To propose a Bogomolov-type question on the moduli space $\mathcal{A}_g$ of principally polarized abelian varieties.
• To improve height bounds in arithmetic geometry by using Moret-Bailly models and canonical metrics.

Proposed method

• Uses Moret-Bailly models (MB-models) to define and compute local components of global heights on abelian varieties.
• Applies the canonical metric on the Néron-Tate height via the line bundle associated to a symmetric ample divisor.
• Decomposes the Faltings height $h_{\mathrm{F}^+}(A)$ into a sum over places $v \in M_k$, with contributions from finite places (via discriminants) and archimedean places (via modular $\Delta$-functions).
• Employs the discriminant $\Delta_E$ of a hyperelliptic curve and its relation to the modular form $\Delta(\tau_v)$ at archimedean places.
• Relies on the theory of theta characteristics and the Mumford model to define the local height contributions.
• Uses the formula $h_{\mathrm{F}^+}(A) = \frac{g}{2}\log(2\pi^2) + h_F(A)$ to ensure positivity and compatibility with classical height theory.

Experimental results

Research questions

• RQ1 Can the Faltings height of a Jacobian of a hyperelliptic curve be decomposed into local contributions at each place of the number field?
• RQ2 Can explicit local formulas be given for the Néron-Tate height of $k$-rational points on such Jacobians?
• RQ3 How do the local components (archimedean and finite) of the height relate to classical invariants like the discriminant and modular forms?
• RQ4 What is the relationship between the global height and the local invariants in the context of Moret-Bailly models?
• RQ5 Can a Bogomolov-type question on the moduli space $\mathcal{A}_g$ be formulated based on these height decompositions?

Key findings

• A decomposition formula for the Faltings height of the Jacobian of a hyperelliptic curve is established: $h_{\mathrm{F}^+}(A) = \frac{1}{12d} \left[ \log N_{k/\mathbb{Q}}(\Delta_E) - \sum_{v \in M_k^\infty} d_v \log \left| \Delta(\tau_v) (2\operatorname{Im} \tau_v)^6 \right| \right] + \text{correction terms}$, generalizing Silverman’s 1986 result.
• The Néron-Tate height of $k$-rational points on the Jacobian is shown to decompose into local contributions, with the archimedean part involving the function $I(A_v, \lambda_v)$ and the finite part tied to the discriminant.
• The formula recovers the classical elliptic curve case when $g=1$, confirming consistency with known results.
• The use of Moret-Bailly models enables explicit computation of local height components, particularly at finite places.
• The paper provides a framework to estimate $\hat{h}(P) - h(P)$ for rational points $P$, improving upon bounds from Manin and Zarhin (1972).
• A Bogomolov-type question is proposed on $\mathcal{A}_g$, asking whether small points are dense in the moduli space under the Néron-Tate height.