A Local Hilbert--P\'olya Realisation for Elliptic Curve $L$-Functions

TL;DR

This paper develops a novel operator-theoretic framework encoding local factors of L-functions, providing new insights into their structure and connections to elliptic curves and spectral measures.

Contribution

It introduces a class of causal operator pencils that exactly encode local Euler factors of L-functions, establishing a new spectral perspective on their arithmetic properties.

Findings

For g=0, recovers Euler factors of the Riemann zeta function.

For g=1, proves a universal Euler matching theorem for elliptic curves.

Provides new proofs of the CM Sato-Tate distribution and demonstrates an interpolation obstruction.

Abstract

We introduce a class of J-self-adjoint causal operator pencils whose spectral determinants exactly encode the local Euler factors of L-functions. Driven by a fractional causal kernel z^{-1/2}, these operators manifest a rigid arithmetic encoding hierarchy governed by the geometric genus g of their spectral curves. For g=0, a unique pencil recovers the Euler factors of the Riemann zeta(s). For g=1, we prove a universal Euler matching theorem: every 2x2 causal pencil canonically encodes the local factors of an elliptic curve E/Q, with the operator invariants mapping dominantly onto the elliptic moduli space. We resolve the arithmetic obstructions of quadratic twists and inert primes via the topological reality of the operator basepoints. For g=infinity, discrete encoding capacity provably collapses into continuous transcendental spectral measures. As applications, we provide new operator-theoretic proofs of the CM Sato-Tate distribution and establish an unconditional interpolation obstruction, proving that global L-functions are structurally inaccessible to any single local operator. Finally, we assemble the global restricted tensor product of these local Krein spaces. We demonstrate that the Hilbert-P\'olya realisation for zeta(s) reduces strictly to a single explicit convergence hypothesis on the global resolvent trace, which we state as a precise conjecture.