Galois representations

A fifteen-lesson course on profinite Galois groups, elliptic-curve Tate modules, local Weil–Deligne representations, compatible systems and Galois representations attached to modular forms. Includes editable lesson sources, exercises with solutions and reproducible figures. The mathematical statements specify their hypotheses and the additional geometric inputs used in conditional arguments.

1. Profinite groups and ℓ-adic representations
2. Frobenius elements and determination by traces
3. The ℓ-adic Tate module of an elliptic curve
4. Representations of Weil groups
5. Conductors of Weil-group representations
6. Local L-factors and epsilon factors
7. Weil–Deligne representations and Grothendieck’s monodromy theorem
8. Elliptic curves over local fields and their Weil–Deligne representations
9. Compatible systems and global L-functions
10. Modular curves over Q and the Eichler–Shimura congruence
11. Galois representations of weight-two newforms
12. Deligne's construction in higher weight and the Ramanujan–Petersson bound
13. Weight one: the theorem of Deligne and Serre
14. p-adic Hodge theory and the Fontaine–Mazur conjecture
15. Residual representations and Serre's modularity theorem

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