Congruences of models of elliptic curves

TL;DR

This paper investigates how the special fibers of elliptic curve models over discrete valuation rings are determined by infinitesimal fibers and Galois actions, providing a deeper understanding of their congruences and reductions.

Contribution

It establishes that the special fibers of minimal Weierstrass and regular models are determined by infinitesimal fibers and Galois group actions for sufficiently large m.

Findings

Special fibers are determined by infinitesimal fibers and Galois actions.

The determination depends on the minimal discriminant and different of the extension.

Results apply to models over discrete valuation rings with perfect residue fields.

Abstract

Let O_K be a discrete valuation ring with field of fractions K and perfect residue field. Let E be an elliptic curve over K, let L/K be a finite Galois extension and let O_L be the integral closure of O_K in L. Denote by X' the minimal regular model of E_L over O_L. We show that the special fibers of the minimal Weierstrass model and the minimal regular model of E over O_K are determined by the infinitesimal fiber X'_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).