Probabilistic Szpiro, Baby Szpiro, and Explicit Szpiro from Mochizuki's Corollary 3.12

英文摘要：

In \cite{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 \cite[Corollary 3.12]{IUT3} to derive variants of Szpiro's inequality (in the spirit of \cite{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 \cite[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 \cite[Corollary 3.12]{IUT3} formulated in \cite{Dupuy2020a} based on the notion of random measurable sets.