Seminar on number theory: abelian l-adic representations

On Monday, 16-18h (the first meeting will be April 15, 2024, not April 8, 2024)

Room: WSC-S-U-3.01

In this seminar, we will study abelian l-adic representations, in particular those arising from Tate modules attached to CM elliptic curves (topic of last semester's seminar). We will consider arithmetic/analytic properties of L-functions attached to (a compatible family of) l-adic representations, local algebraicity and open image problem of such representations.


Prerequiste: basic notions of algebraic number theory and elliptic curves


Tentative program

  1. Introduction and distribution of talks (April 15, 2024)
  2. [S] I.1.1-1.2.4, l-adic representations
  3. [S] I.2.5-I.A.3, L-functions and Cebotarev density theorem
  4. [S] II.1.1-II.2.2, torus T and groups T_m, S_m
  5. [S] II.2.3-II.2.5, l-adic representations valued in S_m and its representations
  6. [S] II.3.1-II.3.4, structure of T_mand applications
  7. [S] III.1.1-III.1.2, locally algebraic representations (local case)
  8. [S] III.2.1-III.2.4, locally algebraic representations (global case)
  9. [S] III.3.1-III.3.4, local algebraicity for the case of composites of quadratic fields
  10. [S] IV.1.1-IV.2.1, Irreducibility theorem for Galois representation G_l on Tate module of elliptic curves
  11. [S] IV.2.2-IV.2.3, Open image theorem for G_l
  12. [S] IV.3.1-IV.3.4, Variations of G_l with l

References:

  1. [S]Serre, Abelian l-adic representations and elliptic curves, McGill University lecture notes, (1968).
  2. Cassels-Frölich, Algebraic Number Theory, Academic Press (1967).