Spectral gap and modular degree of elliptic curves. Part II: The spectral rank criterion and Selmer theory

We develop an operator-theoretic approach to the arithmetic of elliptic curves, continuing the framework of arithmetic spectroscopy introduced in Part I. In this setting, we introduce a Selmer operator whose kernel coincides with the Bloch–Kato Selmer group, providing an algebraic formulation of rank detection in terms of zero modes of a positive operator. We reduce the problem of detecting positive rank from spectral data to a purely arithmetic problem: a comparison between the analytic deformation operator and the Selmer operator, together with the classical Tate–Shafarevich obstruction. The Reduction Principle (Theorem 6.8) shows that the failure of the spectral criterion is completely accounted for by the Tate–Shafarevich obstruction. We prove the rank-detection criterion unconditionally for all curves of analytic rank at most one via Gross–Zagier and Kolyvagin, and for all CM curves via Coates–Wiles and Rubin. These two results constitute two independent locks on the criterion, each without appeal to unproven hypotheses. A central advance of this work is that the Bridge Hypothesis is no longer symmetric: we prove unconditionally that arithmetic degeneracy implies spectral degeneracy in both regimes, reducing the Bridge to a one-sided open problem.On the computational side, we present a numerical method based on spectral asymptotics in which a first-order coefficient in the expansion of the spectral gap provides a stable empirical signal distinguishing rank-zero curves from those of positive rank. All computations are fully reproducible via an open-source implementation. Finally, we formulate a programme for recovering the Néron–Tate regulator from the geometry of flat spectral directions, which will be developed in Part III. This work is Part II of the 'Arithmetic Spectroscopy' series. Part I is available at DOI: 10.5281/zenodo.19183116.