Lectures on Serre's conjectures. (Q2768079)

From the preface: ``We shall begin by discussing some examples of mod \(\ell\) representations of \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\). We'll try to motivate Serre's conjectures by referring first to the case of representations that are unramified outside \(\ell\); these should come from cusp forms on the full modular group \(\text{SL}(2;\mathbb Z)\). In another direction, one might think about representations coming from \(\ell\)-division points on elliptic curves, or more generally from \(\ell\)-division points on abelian varieties of ``\(\text{GL}_2\)-type.'' Amazingly, Serre's conjectures imply that all odd irreducible two-dimensional mod \(\ell\) representations of \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\) may be realized in spaces of \(\ell\)-division points on such abelian varieties. The weak Serre conjecture states that all such representations come from modular forms, andNEWLINEthen it takes only a bit of technique to show that one can take the modular formsNEWLINEto have weight two (if one allows powers of \(\ell\) in the level). NEWLINENEWLINESince little work has been done toward proving the weak Serre conjecture, these notes will focus on the bridge between the weak and the strong conjectures.NEWLINENEWLINEThis paper emerged out of a series of lectures that were delivered by the firstNEWLINEauthor at the 1999 IAS/Park City Mathematics Institute. The second author created the text based on the lectures and added examples, diagrams, an exerciseNEWLINEsection, and the index. One appendix is given by Kevin Buzzard: a mod \(\ell\)NEWLINEmultiplicity one result, the other by Brian Conrad describes a construction of Shimura.''NEWLINENEWLINEA highly recommendable introductory paper to this fascinating subject with useful exercises and a list of 118 references.NEWLINENEWLINEFor the entire collection see [Zbl 0974.00034].