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ℚ-Curves and Galois Representations

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Modular Curves and Abelian Varieties

Part of the book series: Progress in Mathematics ((PM,volume 224))

Abstract

Let K be a number field, Galois over ℚ. A ℚ-curve over K is an elliptic curve over K which is isogenous to all its Galois conjugates. The current interest in ℚ-curves, it is fair to say, began with Ribet’s observation [27] that an elliptic curve over ℚ admitting a dominant morphism from X 1 (N) must be a ℚ-curve. It is then natural to conjecture that, in fact, all ℚ-curves are covered by modular curves. More generally, one might ask: from our rich storehouse of theorems about elliptic curves over ℚ, which ones generalize to ℚ-curves?

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Authors and Affiliations

  1. 808 Fine Hall, Princeton University, Princeton, USA

    Jordan S. Ellenberg

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  1. Jordan S. Ellenberg
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Editor information

Editors and Affiliations

  1. School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK

    John E. Cremona

  2. Facultat de Matemàtiques i Estadística, Universitat Politècnica de Catalunya, Pau Gargallo, 5, 08028, Barcelona, Spain

    Joan-Carles Lario  & Jordi Quer  & 

  3. Department of Mathematics, University of California, M/C 3840, Berkeley, CA, 94720-3840, USA

    Kenneth A. Ribet

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Ellenberg, J.S. (2004). ℚ-Curves and Galois Representations. In: Cremona, J.E., Lario, JC., Quer, J., Ribet, K.A. (eds) Modular Curves and Abelian Varieties. Progress in Mathematics, vol 224. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7919-4_7

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  • DOI: https://doi.org/10.1007/978-3-0348-7919-4_7

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9621-4

  • Online ISBN: 978-3-0348-7919-4

  • eBook Packages: Springer Book Archive

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