EFFECTIVE UNIFORM BOUNDS FOR THE ARAKELOV GREEN FUNCTION ON X 0 (N ): A REFINED HYBRID APPROACH

We establish effective, pointwise uniform bounds for the Arakelov Green function g µ on modular curves X 0 (N) for square-free levels N as N → ∞. To overcome the analytic obstruction posed by the shrinking volume normalization V −1 N , we employ a hybrid strategy separating spectral and geometric regimes. For large times (t ≥ 1), we utilize the Deshouillers-Iwaniec spectral large sieve methodology to control the average density of the continuous spectrum. Exploiting the unitarity of the scattering matrix, we derive a uniform pointwise bound of O(N −1+ε), ensuring that spectral mass concentration does not overwhelm the volume factor. For short times (t ≤ 1), we resolve the problem of varying cusp widths by leveraging the transitive action of Atkin-Lehner involu-tions. We prove that every cusp is isometric to the standard cusp at ∞ via an element of the normalizer, thereby enforcing a "Universal In-tegrality Gap" derived from the discreteness of SL 2 (Z). This provides a level-independent geometric barrier against non-parabolic trajectories. Finally, we demonstrate the asymptotic stability of the constant term, showing that the leading logarithmic singularities of the scattering determinant exactly cancel the volume regularization, leaving a minimal arithmetic residue of O(log log N).