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Serre’s Conjecture

In: Modular Forms and Fermat’s Last Theorem

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  • Bas Edixhoven

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.

Suggested Citation

  • Bas Edixhoven, 1997. "Serre’s Conjecture," Springer Books, in: Gary Cornell & Joseph H. Silverman & Glenn Stevens (ed.), Modular Forms and Fermat’s Last Theorem, pages 209-242, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-1974-3_7
    DOI: 10.1007/978-1-4612-1974-3_7
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