61 The Trace Formula

• Section 61.1: Introduction
• Section 61.2: The trace formula
  • Equation 61.2.0.1
• Section 61.3: Frobenii
  • Exercise 61.3.1
  • Lemma 61.3.2
  • Theorem 61.3.3: The Baffling Theorem
  • Definition 61.3.4
  • Lemma 61.3.5
  • Remark 61.3.6
  • Lemma 61.3.7
  • Definition 61.3.8
  • Theorem 61.3.9
  • Definition 61.3.10
    • Equation 61.3.10.1
  • Remark 61.3.11
• Section 61.4: Traces
  • Definition 61.4.1
  • Exercise 61.4.2
• Section 61.5: Why derived categories?
• Section 61.6: Derived categories
  • Remark 61.6.1
  • Lemma 61.6.2
  • Lemma 61.6.3
  • Definition 61.6.4
  • Remark 61.6.5
• Section 61.7: Filtered derived category
  • Definition 61.7.1
  • Lemma 61.7.2
• Section 61.8: Filtered derived functors
  • Definition 61.8.1
  • Proposition 61.8.2
• Section 61.9: Application of filtered complexes
• Section 61.10: Perfectness
  • Definition 61.10.1
  • Proposition 61.10.2
• Section 61.11: Filtrations and perfect complexes
  • Lemma 61.11.1: Additivity
  • Lemma 61.11.2
• Section 61.12: Characterizing perfect objects
  • Definition 61.12.1
  • Lemma 61.12.2
  • Remark 61.12.3
• Section 61.13: Cohomology of nice complexes
  • Proposition 61.13.1
• Section 61.14: Lefschetz numbers
  • Definition 61.14.1
  • Definition 61.14.2
  • Theorem 61.14.3: Lefschetz Trace Formula
    • Equation 61.14.3.1
  • Theorem 61.14.4: Weil
  • Example 61.14.5
  • Lemma 61.14.6
• Section 61.15: Preliminaries and sorites
  • Lemma 61.15.1
  • Definition 61.15.2
  • Lemma 61.15.3
  • Lemma 61.15.4
  • Lemma 61.15.5
  • Lemma 61.15.6
  • Lemma 61.15.7
  • Lemma 61.15.8
  • Lemma 61.15.9
  • Remark 61.15.10
• Section 61.16: Proof of the trace formula
  • Theorem 61.16.1
    • Equation 61.16.1.1
  • Remark 61.16.2
  • Remark 61.16.3
• Section 61.17: Applications
• Section 61.18: On l-adic sheaves
  • Definition 61.18.1
  • Lemma 61.18.2
  • Lemma 61.18.3
  • Example 61.18.4
  • Remark 61.18.5
  • Definition 61.18.6
  • Remark 61.18.7
  • Definition 61.18.8
• Section 61.19: L-functions
  • Definition 61.19.1
  • Remark 61.19.2
  • Definition 61.19.3
• Section 61.20: Cohomological interpretation
  • Theorem 61.20.1: Finite Coefficients
  • Theorem 61.20.2: Adic sheaves
  • Remark 61.20.3
  • Theorem 61.20.4
  • Theorem 61.20.5
  • Lemma 61.20.6
• Section 61.21: List of things which we should add above
• Section 61.22: Examples of L-functions
• Section 61.23: Constant sheaves
  • Lemma 61.23.1
  • Exercise 61.23.2
• Section 61.24: The Legendre family
• Section 61.25: Exponential sums
• Section 61.26: Trace formula in terms of fundamental groups
• Section 61.27: Fundamental groups
  • Definition 61.27.1
  • Theorem 61.27.2: Grothendieck
  • Proposition 61.27.3
  • Theorem 61.27.4: Deligne, Weil II
• Section 61.28: Profinite groups, cohomology and homology
• Section 61.29: Cohomology of curves, revisited
  • Lemma 61.29.1
  • Remark 61.29.2
  • Proposition 61.29.3
  • Remark 61.29.4
• Section 61.30: Abstract trace formula
  • Remark 61.30.1
  • Example 61.30.2
• Section 61.31: Automorphic forms and sheaves
  • Definition 61.31.1
  • Theorem 61.31.2
  • Theorem 61.31.3
  • Remark 61.31.4
  • Proposition 61.31.5
  • Lemma 61.31.6
  • Lemma 61.31.7
  • Theorem 61.31.8
  • Theorem 61.31.9
  • Theorem 61.31.10
• Section 61.32: Counting points
• Section 61.33: Precise form of Chebotarev
• Section 61.34: How many primes decompose completely?
  • Proposition 61.34.1
• Section 61.35: How many points are there really?