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The modularity of some Q-curves

Published online by Cambridge University Press:  04 December 2007

BOYD B. ROBERTS  and
Lawrence C. WASHINGTON
BOYD B. ROBERTS
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742 e-mail: [email protected]
Lawrence C. WASHINGTON
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742 e-mail: [email protected]
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Abstract

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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).

Keywords

elliptic curves formsQ-curves
Type
Research Article
Information
Compositio Mathematica , Volume 111 , Issue 1 , March 1998 , pp. 35 - 49
Copyright
1998 Kluwer Academic Publishers