IndisputableMonolith.NumberTheory.HilbertPolyaFunctionField
This module defines the Frobenius angle θ for elliptic curves over finite fields together with the Hasse bound and associated operators in the Hilbert-Polya function-field setting. Number theorists studying Riemann-hypothesis analogues via function fields cite these objects. The module consists of successive definitions and short lemmas that establish symmetry and positivity properties of the hpOperator.
claimFor an elliptic curve $E/mathbb{F}_q$ with Frobenius trace $a$, the angle $theta$ is defined by $cos theta = a/(2 sqrt{q})$ whenever $|a| leq 2 sqrt{q}$ (Hasse-Weil bound).
background
The module belongs to the NumberTheory domain and imports Mathlib together with IndisputableMonolith.Cost. Its central definition, given in the module documentation, is the Frobenius angle $theta$ of an elliptic curve $E/mathbb{F}_q$ satisfying $cos theta = a/(2 sqrt{q})$ precisely when the argument lies in $[-1,1]$. Sibling declarations introduce hasseBound, the symmetric operator hpOperator, and the composite construction hilbert_polya_elliptic_curve.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the basic objects required by the Hilbert-Polya approach in function fields. It has no listed downstream uses in the current dependency graph.
scope and limits
- Does not treat elliptic curves over number fields.
- Does not prove any form of the Riemann hypothesis.
- Does not compute explicit spectra of the hpOperator.
- Does not address the global field case.