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Serre’s Conjecture

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Modular Forms and Fermat’s Last Theorem

Abstract

The aim of the first section is to state Serre’s conjecture and to tell what is presently known about it, without proof. We start by recalling what modular forms are. Then we recall the result, due to Deligne, that to a mod p modular form one can associate a mod p Galois representation. After that we state Serre’s conjecture and what we know about it. In Section 2 we will see which cases of it are actually needed in order to prove, following Wiles, that all semi-stable elliptic curves over ℚ are modular. In the last two sections we will sketch the proofs in those cases. These notes follow, to some extent, the lectures given by Dick Gross during the conference.

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Authors
  1. Bas Edixhoven
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Editors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT, 06268, USA

    Gary Cornell

  2. Department of Mathematics, Brown University, Providence, RI, 02912, USA

    Joseph H. Silverman

  3. Department of Mathematics, Boston University, Boston, MA, 02215, USA

    Glenn Stevens

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Edixhoven, B. (1997). Serre’s Conjecture. In: Cornell, G., Silverman, J.H., Stevens, G. (eds) Modular Forms and Fermat’s Last Theorem. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1974-3_7

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  • DOI: https://doi.org/10.1007/978-1-4612-1974-3_7

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-98998-3

  • Online ISBN: 978-1-4612-1974-3

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