Serre's Conjectures

Bryden Cais

These notes are an introduction to the precise statement of Serre's conjecture concerning continuous odd irreducible two-dimensional mod p Galois representations. We spend some time outlining the geometric theory of (Katz) modular forms, following Gross' article "A Tameness criterion..." to a large extent. We give a complete proof of the fact that the kernel of the "q-expansion map" on the algebra of modular forms of given level N is generated by A-1, where A is the Hasse invariant (we expand on Gross' argument by proving a critical lemma of classical geometric flavor). We then give a detailed account of Serre's prescription for the "modular type" of a Galois representation. Finally, we elaborate on some examples of Mestre, quoted in Serre's paper "Sur les representations modulaires...", concerning icosahedral mod 2 representations. In the course of our exposition, we provide several examples. I wrote a MAGMA script to calculate these examples, which can be obtained for use from the "Programs" section of my publications list.

These notes were revised November 26, 2009 and some errors were fixed. If you have an older version, please destroy it.