58 The Trace Formula

Section 58.1: Introduction
Section 58.2: The trace formula
- Equation 58.2.0.1
Section 58.3: Frobenii
- Exercise 58.3.1
- Lemma 58.3.2
- Theorem 58.3.3: The Baffling Theorem
- Definition 58.3.4
- Lemma 58.3.5
- Remark 58.3.6
- Lemma 58.3.7
- Definition 58.3.8
- Theorem 58.3.9
- Definition 58.3.10
  - Equation 58.3.10.1
- Remark 58.3.11
Section 58.4: Traces
- Definition 58.4.1
- Exercise 58.4.2
Section 58.5: Why derived categories?
Section 58.6: Derived categories
- Remark 58.6.1
- Lemma 58.6.2
- Lemma 58.6.3
- Definition 58.6.4
- Remark 58.6.5
Section 58.7: Filtered derived category
- Definition 58.7.1
- Lemma 58.7.2
Section 58.8: Filtered derived functors
- Definition 58.8.1
- Proposition 58.8.2
Section 58.9: Application of filtered complexes
Section 58.10: Perfectness
- Definition 58.10.1
- Proposition 58.10.2
Section 58.11: Filtrations and perfect complexes
- Lemma 58.11.1: Additivity
- Lemma 58.11.2
Section 58.12: Characterizing perfect objects
- Definition 58.12.1
- Lemma 58.12.2
- Remark 58.12.3
Section 58.13: Cohomology of nice complexes
- Proposition 58.13.1
Section 58.14: Lefschetz numbers
- Definition 58.14.1
- Definition 58.14.2
- Theorem 58.14.3: Lefschetz Trace Formula
  - Equation 58.14.3.1
- Theorem 58.14.4: Weil
- Example 58.14.5
- Lemma 58.14.6
Section 58.15: Preliminaries and sorites
- Lemma 58.15.1
- Definition 58.15.2
- Lemma 58.15.3
- Lemma 58.15.4
- Lemma 58.15.5
- Lemma 58.15.6
- Lemma 58.15.7
- Lemma 58.15.8
- Lemma 58.15.9
- Remark 58.15.10
Section 58.16: Proof of the trace formula
- Theorem 58.16.1
  - Equation 58.16.1.1
- Remark 58.16.2
- Remark 58.16.3
Section 58.17: Applications
Section 58.18: On l-adic sheaves
- Definition 58.18.1
- Lemma 58.18.2
- Lemma 58.18.3
- Example 58.18.4
- Remark 58.18.5
- Definition 58.18.6
- Remark 58.18.7
- Definition 58.18.8
Section 58.19: L-functions
- Definition 58.19.1
- Remark 58.19.2
- Definition 58.19.3
Section 58.20: Cohomological interpretation
- Theorem 58.20.1: Finite Coefficients
- Theorem 58.20.2: Adic sheaves
- Remark 58.20.3
- Theorem 58.20.4
- Theorem 58.20.5
- Lemma 58.20.6
Section 58.21: List of things which we should add above
Section 58.22: Examples of L-functions
Section 58.23: Constant sheaves
- Lemma 58.23.1
- Exercise 58.23.2
Section 58.24: The Legendre family
Section 58.25: Exponential sums
Section 58.26: Trace formula in terms of fundamental groups
Section 58.27: Fundamental groups
- Definition 58.27.1
- Theorem 58.27.2: Grothendieck
- Proposition 58.27.3
- Theorem 58.27.4: Deligne, Weil II
Section 58.28: Profinite groups, cohomology and homology
Section 58.29: Cohomology of curves, revisited
- Lemma 58.29.1
- Remark 58.29.2
- Proposition 58.29.3
- Remark 58.29.4
Section 58.30: Abstract trace formula
- Remark 58.30.1
- Example 58.30.2
Section 58.31: Automorphic forms and sheaves
- Definition 58.31.1
- Theorem 58.31.2
- Theorem 58.31.3
- Remark 58.31.4
- Proposition 58.31.5
- Lemma 58.31.6
- Lemma 58.31.7
- Theorem 58.31.8
- Theorem 58.31.9
- Theorem 58.31.10
Section 58.32: Counting points
Section 58.33: Precise form of Chebotarev
Section 58.34: How many primes decompose completely?
- Proposition 58.34.1
Section 58.35: How many points are there really?