Compositio Mathematica

Table of Contents - 1998 - Volume 111, Issue 01  


The modularity of some Q-curves

BOYD B. ROBERTS a1 and Lawrence C. WASHINGTON a1
a1 Department of Mathematics, University of Maryland, College Park, MD 20742 e-mail: lcw@math.umd.edu

Article author query
roberts bb   [Google Scholar] 
washington lc   [Google Scholar] 
 

Abstract

A Q-curve is an elliptic curve, defined over a number field, that is isogenous to each of its Galois conjugates. Ribet showed that Serre's conjectures imply that such curves should be modular. Let E be an elliptic curve defined over a quadratic field such that E is 3-isogenous to its Galois conjugate. We give an algorithm for proving any such E is modular and give an explicit example involving a quotient of $J_o$ (169). As a by-product, we obtain a pair of 19-isogenous elliptic curves, and relate this to the existence of a rational point of order 19 on $J_1$ (13).


Key Words: elliptic curves forms; Q-curves.