Probabilistic Szpiro, Baby Szpiro, and Explicit Szpiro from Mochizuki's Corollary 3.12
Abstract
In Dupuy2020a we gave some explicit formulas for the "indeterminacies" Ind1,Ind2,Ind3 in Mochizuki's Inequality as well as a new presentation of initial theta data. In the present paper we use these explicit formulas, together with our probabilistic formulation of [Corollary 3.12]IUT3 to derive variants of Szpiro's inequality (in the spirit of IUT4). In particular, for an elliptic curve in initial theta data we show how to derive uniform Szpiro (with explicit numerical constants). The inequalities we get will be strictly weaker than [Theorem 1.10]IUT4 but the proofs are more transparent, modifiable, and user friendly. All of these inequalities are derived from an probabilistic version of [Corollary 3.12]IUT3 formulated in Dupuy2020a based on the notion of random measurable sets.