[Paper Review] Points de hauteur bornee sur les varietes de drapeaux en caracteristique finie

Emmanuel Peyre|arXiv (Cornell University)|Mar 5, 2003

Advanced Algebra and Geometry4 citations

TL;DR

This paper establishes an asymptotic formula for the number of rational points of bounded height on generalized flag varieties over global function fields of positive characteristic, using Eisenstein series techniques developed by Morris. It proves that the residue of the height zeta function has a Tamagawa-type constant, analogous to the classical Manin conjecture over number fields, with explicit geometric and arithmetic factors derived from the flag variety's structure and adelic metrics.

ABSTRACT

The aim of this paper is to apply the work of Morris on Eisenstein series over global function fields to the study of the asymptotic behavior of the points of bounded height on a generalized flag variety defined as the quotient of a semi-simple algebraic group by a reduced parabolic subgroup over such a field. The formula obtained for the height zeta function has an interpretation similar to the one known over a number field.

Motivation & Objective

• To extend the Manin conjecture framework for rational points of bounded height to global fields of positive characteristic.
• To establish an asymptotic formula for the counting function of rational points on generalized flag varieties over function fields.
• To interpret the leading constant in the asymptotic formula using geometric and arithmetic invariants, analogous to the classical Tamagawa measure.
• To provide a complete interpretation of the constant in the residue of the height zeta function, filling a gap left by prior work.
• To generalize Tamagawa measure theory to adelic metrics on flag varieties in positive characteristic, using tools from automform theory and algebraic groups.

Proposed method

• Adapts the theory of adelic metrics and height functions from number fields to global function fields, using the product formula and normalized valuations.
• Constructs a height zeta function over the adelic space of the flag variety, with height defined via line bundles and metrics on the base field.
• Applies Eisenstein series theory over function fields, developed by Morris, to analyze the analytic behavior of the zeta function.
• Computes the order of the pole at the top of the cone of effective divisors, showing it is simple and related to the Picard group.
• Derives the residue of the zeta function as a product of local densities and a global constant involving the Tamagawa measure.
• Expresses the leading constant as a product of an arithmetic factor α(V) and a geometric factor β(V), with β(V) related to the volume of the flag variety under the Tamagawa measure.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the number of rational points of bounded height on a generalized flag variety over a global function field?
• RQ2 How does the residue of the height zeta function on such varieties relate to geometric and arithmetic invariants in positive characteristic?
• RQ3 Can the classical Manin conjecture's prediction for the leading constant be extended to function fields using Eisenstein series?
• RQ4 What is the role of the Tamagawa measure in the function field setting for flag varieties?
• RQ5 How do the local densities and global constants in the asymptotic formula compare to the number field case?

Key findings

• The height zeta function for a generalized flag variety over a global function field has a simple pole at the top of the effective cone, with residue capturing the asymptotic growth rate.
• The residue is expressed as a product of a global constant α(V) and a geometric factor β(V), with β(V) equal to the Tamagawa measure of the flag variety.
• The constant α(V) is shown to be rational and related to the class group and unit group of the base field.
• The formula for the residue matches the structure of the classical Manin conjecture, confirming a function field analogue of the expected asymptotic behavior.
• The method successfully interprets the leading constant in terms of geometric and arithmetic invariants, resolving a gap left by prior work such as [LY].
• The result confirms that the Tamagawa measure on the flag variety, defined via adelic integration, gives the correct normalization for the asymptotic count.