We introduce an operator-theoretic framework for extracting arithmetic invariants of elliptic curves over the rationals via self-adjoint operators. To a semistable optimal elliptic curve E we associate a canonically defined Schrodinger-type operator on the real line, constructed from the E-isotypic component of the homology of the modular curve. The potential is determined by the intersection pairing and encodes the modular degree. We prove that for curves of analytic rank zero the spectral gap converges, in the strong-coupling limit, to the modular degree — yielding a new analytic realisation independent of the modular parametrisation. For curves of positive rank the spectral gap vanishes identically, providing a spectral criterion distinguishing rank-zero from positive-rank curves. Numerical experiments on curves from the LMFDB confirm the theoretical predictions with relative error below 0.001. The accompanying open-source implementation is available at https://doi.org/10.5281/zenodo.19165246. This is Part I of a series towards a spectral interpretation of the Birch and Swinnerton-Dyer conjecture.
Andrew Timakov (Mon,) studied this question.