A Continuous Spectral Framework for Elliptic Curves via the Moroz Transform – An Initial Proposal

This work introduces a continuous oscillatory integral associated with elliptic curves, termed theMoroz Transform, and investigate its structural relationship with classical arithmetic invariants such as theFrobenius trace and the Sato–Tate distribution. While the transform does not directly reproduce thecoefficients 𝑎𝑝 of the L-function, it generates two natural invariants — amplitude and phase — which encodegeometric and spectral information. Through numerical examples, this paper shows that the phase exhibits adistribution comparable to the Sato–Tate angle, suggesting that the Moroz Transform captures a continuousanalogue of Frobenius spectral data.