Proof of the Szpiro Conjecture via Differential-Algebraic Finite Representation

This paper presents a novel framework for proving the Szpiro Conjecture, based on the theory of differential-algebraic closure and finite representation of transcendental functions. The core breakthrough lies in constructing the modular invariant of an elliptic curve as a root of its modular polynomial and utilizing the universal formula for polynomial solutions in differential algebra to define the differential algebraic height hDA of the j-invariant. We rigorously prove that this height satisfies an equality with classical arithmetic invariants:hDA(j(E)) = 112 log|∆min(E)| + 12 logNE +O(loglogNE).Through an in-depth analysis of the p-adic valuation behavior of the differential-algebraic representation at each prime, we establish a crucial local discriminant inequality. Finally, by summing over all primes and using global height estimates, we derive the Szpiro inequality:log |∆min(E)| ≤ (6 +ε)logNE +Cε,for any ε > 0, where Cε is a constant depending only on ε. This proof does not rely on the Modularity Theorem or p-adic Hodge theory, offering a new constructive and unifying perspective on arithmetic geometry problems based on differential algebra.