Duke Mathematical Journal

On the modularity of ℚ-curves

Jordan S. Ellenberg and Chris Skinner

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Abstract

A ℚ-curve is an elliptic curve over a number field K which is geometrically isogenous to each of its Galois conjugates. K. Ribet [17] asked whether every ℚ-curve is modular, and he showed that a positive answer would follow from J.-P. Serre's conjecture on mod p Galois representations. We answer Ribet's question in the affirmative, subject to certain local conditions at 3.

Article information

Source
Duke Math. J., Volume 109, Number 1 (2001), 97-122.

Dates
First available in Project Euclid: 5 August 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1091737223

Digital Object Identifier
doi:10.1215/S0012-7094-01-10914-9

Mathematical Reviews number (MathSciNet)
MR1844206

Zentralblatt MATH identifier
1009.11038

Subjects
Primary: 11G05: Elliptic curves over global fields [See also 14H52]
Secondary: 11F80: Galois representations 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35] 14G25: Global ground fields 14H52: Elliptic curves [See also 11G05, 11G07, 14Kxx]

Citation

Ellenberg, Jordan S.; Skinner, Chris. On the modularity of ℚ-curves. Duke Math. J. 109 (2001), no. 1, 97--122. doi:10.1215/S0012-7094-01-10914-9. https://projecteuclid.org/euclid.dmj/1091737223


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