A General Central Singularity Formula and Its Arithmetic Consequences

This paper introduces a General Central Singularity Formula (GCSF) governing thefull central behavior of automorphic L-functions associated with elliptic curvesover the rational numbers. Using the relative trace formula for GL(2) together with a one-parameter familyof test functions satisfying prescribed moment vanishing conditions, centralsingular coefficients are extracted via differentiation. The order of vanishingand the leading coefficient of L(E,s) at the central point s=1 arise as intrinsicsingular invariants. On the geometric side, these coefficients are identified with Arakelov heightpairings of higher special cycles, generalizing classical Heegner points. Thisframework unifies the results of Waldspurger, Gross–Zagier, and Kolyvagin aslow-order instances of a single structural principle and naturally extends toarbitrary analytic rank. In the complex multiplication case, the full formula is establishedunconditionally. For general elliptic curves, the formula is expected topropagate under deformation and stability principles of the relative traceformula. From this perspective, the Birch–Swinnerton-Dyer conjecture appears as adirect consequence of the General Central Singularity Formula.