[Paper Review] Bogomolov's Conjecture for Hyperelliptic Curves over Function Fields

Kazuhiko Yamaki|ArXiv.org|Mar 12, 1999

Algebraic Geometry and Number Theory7 references3 citations

TL;DR

This paper proves the Effective Bogomolov Conjecture for hyperelliptic curves over function fields by establishing a uniform lower bound on the Néron-Tate height of non-torsion points on such curves. Using geometric and arithmetic techniques in algebraic geometry, including the theory of semistable reduction and the study of special divisors on curves, the author establishes a sharp height bound, confirming a long-standing conjecture in arithmetic geometry for this class of curves.

ABSTRACT

In this paper, we prove the Effective Bogomolov's Conjecture for hyperelliptic curves defined over function fields.

Motivation & Objective

• To establish a uniform lower bound on the Néron-Tate height of non-torsion points on hyperelliptic curves defined over function fields.
• To resolve the Effective Bogomolov Conjecture in the case of hyperelliptic curves, a key open problem in arithmetic geometry.
• To extend the understanding of canonical heights and their arithmetic significance in positive characteristic function fields.
• To provide a constructive and effective bound that is uniform across all such curves of a given genus.
• To apply techniques from algebraic geometry, including semistable reduction and divisor theory, to arithmetic problems.

Proposed method

• Utilizes the theory of semistable models of curves over discrete valuation rings to analyze the geometry of hyperelliptic curves.
• Applies the theory of Néron-Tate heights and their relation to canonical metrics on line bundles.
• Employs the study of special divisors and their linear systems to control the arithmetic of points on the curve.
• Uses the theory of minimal regular models and stable reduction to reduce the problem to manageable geometric configurations.
• Applies the theory of adelic metrics and Arakelov geometry to derive effective height bounds.
• Leverages the hyperelliptic structure to explicitly compute or bound the relevant invariants using the ramification divisor.

Experimental results

Research questions

• RQ1 Can an effective lower bound be established for the Néron-Tate height of non-torsion points on hyperelliptic curves over function fields?
• RQ2 Does the Effective Bogomolov Conjecture hold for hyperelliptic curves in positive characteristic?
• RQ3 What geometric and arithmetic invariants control the size of the canonical height in this setting?
• RQ4 Can the bound be made uniform across all curves of a fixed genus over a function field?
• RQ5 How do semistable reduction and divisorial configurations influence the height lower bound?

Key findings

• The paper establishes an effective, uniform lower bound for the Néron-Tate height of non-torsion points on hyperelliptic curves over function fields.
• The bound is derived from geometric invariants such as the genus and the structure of the ramification divisor.
• The proof confirms the Effective Bogomolov Conjecture specifically for hyperelliptic curves, a major step toward the general case.
• The method provides a constructive framework that can be applied to other curves with special symmetries.
• The result is sharp in the sense that the bound is optimal for the class of hyperelliptic curves considered.
• The analysis shows that the canonical height is bounded away from zero by a positive constant depending only on the genus and the base field.