Serre-l-adic-Representations

This is a project for TeXing Serre's monograph "Abelian l-adic representations and elliptic curves" (Addison-Wesley, 1989) into modern LaTeX.

Status

• Chapter I. l-adic Representations
  • 1 The notion of an l-adic representation
    • 1.1 Definition
    • 1.2 Examples
  • 2 l-adic representations of number fields
    • 2.1 Preliminaries
    • 2.2 Cebotarev's density theorem
    • 2.3 Rational l-adic representations
    • 2.4 Representations with values in a linear algebraic group
    • 2.5 L-functions attached to rational representations
  • A. Equipartition and L-functions
    • A.1 Equipartition
    • A.2 The connection with L-functions
    • A.3 Proof of theorem 1
• Chapter II. The Groups Sm
  • 1 Preliminaries
    • 1.1 The torus T
    • 1.2 Cutting down T
    • 1.3 Enlarging groups
  • 2 Construction of Tm and Sm
    • 2.1 Ideles and idele-classes
    • 2.2 The groups Tm and Sm
    • 2.3 The canonical l-adic representation with values in Sm
    • 2.4 Linear representations of Sm
    • 2.5 l-adic representations associated to a linear representation of Sm
    • 2.6 Alternative construction
    • 2.7 The real case
    • 2.8 An example: complex multiplication of abelian varieties
  • 3 Structure of Tm and applications
    • 3.1 Structure of X(Tm)
    • 3.2 The morphism j
    • 3.3 Structure of Tm
    • 3.4 How to compute Frobeniuses
  • A. Killing arithmetic groups in tori
    • A.1 Arithmetic groups in tori
    • A.2 Killing arithmetic subgroups
• Chapter III. Locally Algebraic Abelian Representations
  • 1 The local case
    • 1.1 Definitions
    • 1.2 Alternative definition of "locally algebraic" via Hodge-Tate modules
  • 2 The global case
    • 2.1 Definitions
    • 2.2 Modulus of a locally algebraic abelian representation
    • 2.3 Back to Sm
    • 2.4 A mild generalization
    • 2.5 The function field case
  • 3 The case of a composite of quadratic fields
    • 3.1 Statement of the result
    • 3.2 A criterion for local algebraicity
    • 3.3 An auxiliary result on tori
    • 3.4 Proof of the theorem
  • A. Hodge-Tate decompositions and locally algebraic representations
    • A.1 Invariance of Hodge-Tate decompositions
    • A.2 Admissible characters
    • A.3 A criterion for local triviality
    • A.4 The character xi
    • A.5 Characters associated with Hodge-Tate decompositions
    • A.6 Locally compact case
    • A.7 Tate's theorem
• Chapter IV. l-adic Representations Attached to Elliptic Curves
  • 1 Preliminaries
    • 1.1 Elliptic curves
    • 1.2 Good reduction
    • 1.3 Properties of Vl related to good reduction
    • 1.4 Safarevic's theorem
  • 2 The Galois modules attached to E
    • 2.1 The irreducibility theorem
    • 2.2 Determination of the Lie algebra of Gl
    • 2.3 The isogeny theorem
  • 3 Variation of Gl and Gl with l
    • 3.1 Preliminaries
    • 3.2 The case of a non integral j
    • 3.3 Numerical example
    • 3.4 Proof of the main lemma of 3.1
  • A. Local results
    • A.1 The case v(j) < 0
      • A.1.1 The elliptic curves of Tate
      • A.1.2 An exact sequence
      • A.1.3 Determination of gl and il
      • A.1.4 Application to isogenies
      • A.1.5 Existence of transvections in the inertia group
    • A.2 The case v(j) >= 0
      • A.2.1 The case l /= p
      • A.2.2 The case l = p with good reduction of height 2
      • A.2.3 Auxiliary results on abelian varieties
      • A.2.4 The case l = p with good reduction of height 1

Credits

This was done by José Cuevas Barrientos and Rocío Sepúlveda Manzo (magicalharuka123). We do not claim any originality of the text, aside from slightly updating the notation (see "Editors' notes"). The content of the document is original of Jean-Pierre Serre.