Curves, dynamical systems, and weighted point counting.

Area of Science:

• Algebraic Geometry
• Number Theory
• Finite Fields

Background:

• Counting points on algebraic curves over finite fields is fundamental.
• Tate's theorem states that equal zeta functions (point counts over extensions) imply isogenous Jacobians, not unique curves.
• The isospectrality or arithmetic equivalence problem seeks to determine curves by spectral data.

Purpose of the Study:

• To demonstrate that Dirichlet L-series, not just zeta functions, uniquely determine algebraic curves over finite fields.
• To establish an analogue of the isospectrality problem for curves over finite fields.
• To show that weighted point counting determines a curve.

Main Methods:

• Comparing Dirichlet L-series of two algebraic curves over a finite field k.
• Utilizing isomorphisms of their Dirichlet character groups.
• Relating curve properties to dynamical systems in class field theory.

Main Results:

• Equality of all Dirichlet L-series implies algebraic curves are isomorphic up to Frobenius twists.
• Weighted point counting via L-series determines a curve, unlike simple point counting (zeta function).
• The study provides a precise definition of spectral data (eigenvalues of Frobenius acting on cohomology) that determines a curve.

Conclusions:

• Dirichlet L-series provide a more refined invariant than zeta functions for algebraic curves over finite fields.
• The arithmetic equivalence problem is solved for curves over finite fields using L-series and spectral data.
• The research connects algebraic geometry, number theory, and dynamical systems through weighted point counting.