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 Birchand Swinnerton-Dyer (BSD) conjecture in number theory with observations of thelarge-scale structure of the universe. Based on the two axioms of information conservation and computability, the analytic rank r of an elliptic curve—an arithmeticproperty—is mapped to a specific form of modulation signal in the primordial powerspectrum 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-timecosmic matter distribution. Theoretically, this modulation exhibits an amplitudeproportional to r in the comoving wavenumber range k ∼ 0.1–0.4hMpc−1, witha 1/ln(k) envelope and fixed-period oscillations. Through numerical simulationexperiments, this study demonstrates that under conditions close to real observations, the modulation signal for r = 1 can achieve a statistical significance of approximately 3.9σ. The paper further argues for the falsifiability of this theoreticalframework, 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 fundamental role of mathematical structures in the physical universe.