An Operator Bridge Between Arithmetic Heights and Analytic Residues for Elliptic Curves
Internal ID: CGI-RSR-000004
Document Type: Research Paper
Publication Date: May 2026
Status: Public
Domains: Mathematics
Research Topics: Number theory / arithmetic geometry, Clay Millennium Problems, Birch and Swinnerton-Dyer Conjecture, BSD
Abstract
We present a structured program relating the arithmetic and analytic structures appearing in the Birch–Swinnerton–Dyer conjecture for elliptic curves over Q. The construction proceeds through two parallel bridges. The arithmetic bridge reconstructs the NÅLeron–Tate height pairing from a system of local coherence pairings obtained via defect descent at all places of the curve. Summation of the local energies produces a global quadratic operator whose determinant yields the classical regulator. The analytic bridge constructs a quadratic operator from the spectral structure of the L–function. A coherence operator associated with the modular form admits a kernel representation whose spectral behavior is controlled by a strict bilinear inequality. Using a Mellin–harmonic correspondence, the critical residue of the completed L–function induces a finite-dimensional Hermitian operator on the period space of the elliptic curve. We show that the analytic residue operator coincides with the arithmetic height operator up to the natural period normalization. The resulting identification relates the kernel and determinant of the spectral operator to the Mordell–Weil rank and regulator, yielding the Birch–Swinnerton–Dyer relations. The purpose of this work is to exhibit an operator framework connecting arithmetic height structures and spectral residue operators, which may provide a common mechanism underlying several critical phenomena in problems related to elliptic curves and nonlinear PDE.
Available Document
Citation:
Petersen, B. L. (2026). An Operator Bridge Between Arithmetic Heights and Analytic Residues for Elliptic Curves. Zenodo. https://doi.org/10.5281/zenodo.19969495
Source Code and Supporting Materials
N/A
Summary and Notes
In the Coherence Geometry viewpoint, arithmetic and analytic structures are interpreted as two projections of a common quadratic organization principle.
Rather than treating rank, regulator, and critical residues as unrelated phenomena, the approach seeks a shared operator whose:
- spectral structure encodes analytic critical behavior
- kernel records degeneracy directions
- determinant records global volume or height data
Related Work
None listed.