Open Access

Ramification in the division fields of elliptic curves with potential supersingular reduction

Research in Number Theory20162:8

DOI: 10.1007/s40993-016-0040-z

Received: 18 August 2015

Accepted: 16 February 2016

Published: 15 April 2016

Abstract

Let d≥1 be fixed. Let F be a number field of degree d, and let E/F be an elliptic curve. Let E(F)tors be the torsion subgroup of E(F). In 1996, Merel proved the uniform boundedness conjecture, i.e., there is a constant B(d), which depends on d but not on the chosen field F or on the curve E/F, such that the size of E(F)tors is bounded by B(d). Moreover, Merel gave a bound (exponential in d) for the largest prime that may be a divisor of the order of E(F)tors. In 1996, Parent proved a bound (also exponential in d) for the largest p-power order of a torsion point that may appear in E(F)tors. It has been conjectured, however, that there is a bound for the size of E(F)tors that is polynomial in d. In this article we show that if E/F has potential supersingular reduction at a prime ideal above p, then there is a linear bound for the largest p-power order of a torsion point defined over F, which in fact is linear in the ramification index of the prime of supersingular reduction.

Mathematics Subject Classification: Primary: 11G05, Secondary: 14H52