Major Conjectures and Cosmological Tests J; The Cosmological Falsifiability of the Birch and Swinnerton-Dyer Conjecture for Elliptic Curves

This paper proposes a physical theoretical framework that connects the Birch
and Swinnerton-Dyer (BSD) conjecture in number theory with observations of the
large-scale structure of the universe. Based on the two axioms of information con
servation and computability, the analytic rank r of an elliptic curve—an arithmetic
property—is mapped to a specific form of modulation signal in the primordial power
spectrum of the universe, and it is predicted that this signal will leave a detectable
”rank modulation” imprint in the four-point correlation function of the late-time
cosmic matter distribution. Theoretically, this modulation exhibits an amplitude
proportional to r in the comoving wavenumber range k ∼ 0.1–0.4hMpc−1, with
a 1/ln(k) envelope and fixed-period oscillations. Through numerical simulation
experiments, this study demonstrates that under conditions close to real observa
tions, the modulation signal for r = 1 can achieve a statistical significance of ap
proximately 3.9σ. The paper further argues for the falsifiability of this theoretical
framework, pointing out that observations from next-generation cosmology surveys
(e.g., SKA-2) can provide a decisive test for this ”arithmetic geometry-cosmology”
correspondence model, thereby opening a new empirical path for exploring the fun
damental role of mathematical structures in the physical universe.