63 The Trace Formula

Section 63.1: Introduction
Section 63.2: The trace formula
- Equation 63.2.0.1
Section 63.3: Frobenii
- Exercise 63.3.1
- Lemma 63.3.2
- Theorem 63.3.3: The Baffling Theorem
- Definition 63.3.4
- Lemma 63.3.5
- Remark 63.3.6
- Lemma 63.3.7
- Definition 63.3.8
- Theorem 63.3.9
- Definition 63.3.10
  - Equation 63.3.10.1
- Remark 63.3.11
Section 63.4: Traces
- Definition 63.4.1
- Exercise 63.4.2
Section 63.5: Why derived categories?
Section 63.6: Derived categories
- Remark 63.6.1
- Lemma 63.6.2
- Lemma 63.6.3
- Definition 63.6.4
- Remark 63.6.5
Section 63.7: Filtered derived category
- Definition 63.7.1
- Lemma 63.7.2
Section 63.8: Filtered derived functors
- Definition 63.8.1
- Proposition 63.8.2
Section 63.9: Application of filtered complexes
Section 63.10: Perfectness
- Definition 63.10.1
- Proposition 63.10.2
Section 63.11: Filtrations and perfect complexes
- Lemma 63.11.1: Additivity
- Lemma 63.11.2
Section 63.12: Characterizing perfect objects
- Definition 63.12.1
- Lemma 63.12.2
- Remark 63.12.3
Section 63.13: Cohomology of nice complexes
- Proposition 63.13.1
Section 63.14: Lefschetz numbers
- Definition 63.14.1
- Definition 63.14.2
- Theorem 63.14.3: Lefschetz Trace Formula
  - Equation 63.14.3.1
- Theorem 63.14.4: Weil
- Example 63.14.5
- Lemma 63.14.6
Section 63.15: Preliminaries and sorites
- Lemma 63.15.1
- Definition 63.15.2
- Lemma 63.15.3
- Lemma 63.15.4
- Lemma 63.15.5
- Lemma 63.15.6
- Lemma 63.15.7
- Lemma 63.15.8
- Lemma 63.15.9
- Remark 63.15.10
Section 63.16: Proof of the trace formula
- Theorem 63.16.1
  - Equation 63.16.1.1
- Remark 63.16.2
- Remark 63.16.3
Section 63.17: Applications
Section 63.18: On l-adic sheaves
- Definition 63.18.1
- Lemma 63.18.2
- Lemma 63.18.3
- Example 63.18.4
- Remark 63.18.5
- Definition 63.18.6
- Remark 63.18.7
- Definition 63.18.8
Section 63.19: L-functions
- Definition 63.19.1
- Remark 63.19.2
- Definition 63.19.3
Section 63.20: Cohomological interpretation
- Theorem 63.20.1: Finite Coefficients
- Theorem 63.20.2: Adic sheaves
- Remark 63.20.3
- Theorem 63.20.4
- Theorem 63.20.5
- Lemma 63.20.6
Section 63.21: List of things which we should add above
Section 63.22: Examples of L-functions
Section 63.23: Constant sheaves
- Lemma 63.23.1
- Exercise 63.23.2
Section 63.24: The Legendre family
Section 63.25: Exponential sums
Section 63.26: Trace formula in terms of fundamental groups
Section 63.27: Fundamental groups
- Definition 63.27.1
- Theorem 63.27.2: Grothendieck
- Proposition 63.27.3
- Theorem 63.27.4: Deligne, Weil II
Section 63.28: Profinite groups, cohomology and homology
Section 63.29: Cohomology of curves, revisited
- Lemma 63.29.1
- Remark 63.29.2
- Proposition 63.29.3
- Remark 63.29.4
Section 63.30: Abstract trace formula
- Remark 63.30.1
- Example 63.30.2
Section 63.31: Automorphic forms and sheaves
- Definition 63.31.1
- Theorem 63.31.2
- Theorem 63.31.3
- Remark 63.31.4
- Proposition 63.31.5
- Lemma 63.31.6
- Lemma 63.31.7
- Theorem 63.31.8
- Theorem 63.31.9
- Theorem 63.31.10
Section 63.32: Counting points
Section 63.33: Precise form of Chebotarev
Section 63.34: How many primes decompose completely?
- Proposition 63.34.1
Section 63.35: How many points are there really?