Formal groups of elliptic curves with potential good supersingular reduction

Álvaro Lozano-Robledo

Vol. 261 (2013), No. 1, 145–164

DOI: 10.2140/pjm.2013.261.145

Abstract

Let L be a number field and let E∕L be an elliptic curve with potentially supersingular reduction at a prime ideal ℘ of L above a rational prime p. In this article we describe a formula for the slopes of the Newton polygon associated to the multiplication-by-p map in the formal group of E, depending only on the congruence class of p mod 12, the ℘-adic valuation of the discriminant of a model for E over L, and the valuation of the j-invariant of E. The formula is applied to prove a divisibility formula for the ramification indices in the field of definition of a p-torsion point.

Keywords

elliptic curves, supersingular, formal group, torsion points

Mathematical Subject Classification 2010

Primary: 11G05, 11G07

Secondary: 14H52, 14L05

Milestones

Received: 23 May 2012

Revised: 9 August 2012

Accepted: 27 August 2012

Published: 28 February 2013