Congruences of models of elliptic curves

Abstract

Let 𝒪K be discrete valuation ring with a field of fractions K and a perfect residue field. Let E be an elliptic curve over K, let L/K be a finite Galois extension and let 𝒪L be the integral closure of 𝒪K in L. Denote by χ′ the minimal regular model of EL over 𝒪L. We show that the special fibers of the minimal Weierstrass model and the minimal regular model of E over 𝒪K are determined by the infinitesimal fiber χ′m together with the action of Gal(L/K), when m is big enough (depending on the minimal discriminant of E and the different of L/K).