viii CONTENT S

8. The Poisson summation formula 108

9. Tate zeta functions 119

9.1. Definition and meromorphic continuation 119

9.2. Functional equation and behavior at s = 1 124

10. Intertwining operators and matrix coefficients 128

10.1. Linear algebra 129

10.2. Representation theory 134

10.3. Orthogonality of matrix coefficients 140

11. The discrete series of GL2(R) 151

11.1. if-type decompositions 151

11.2. Representations of 0(2) 155

11.3. K-type decomposition of an induced representation 156

11.4. The (s, If)-modules (VX)K 162

11.5. Classification of irreducible (g, K)-modules for GL2(R) 165

11.6. A detailed look at the Lie algebra action 181

11.7. Characterization of the weight k discrete series of GL2(R) 186

Groundwork

12. Cusp forms as elements of

LQ(UJ)

195

12.1. From Dirichlet characters to Hecke characters 195

12.2. From cusp forms to functions on G(A) 197

12.3. Comparison of classical and adelic Fourier coefficients 198

12.4. Characterizing the image of Sk(iV,u/) in LQ(U) 201

13. Construction of the test function / 206

13.1. The non-archimedean component of / 206

13.2. Spectral properties of R(f) 213

14. Explicit computations for /

k

and /* 221

The Trace Formula

15. Introduction to the trace formula for R(f) 227

16. Terms that contribute to K(x, y) 230

17. Truncation of the kernel 231

18. Bounds for £

7

l/fcrSfe)! 240

19. Integrability of kj(g) 248

20. The hyperbolic terms as weighted orbital integrals 259

21. Simplifying the unipotent term 270

22. The trace formula 276

Computation of the Trace

23. The identity term 279

24. The hyperbolic terms 280

24.1. A lemma about orbital integrals 280

24.2. The archimedean orbital integral and weighted orbital integral 281

24.3. Simplification of the hyperbolic term 283

24.4. Calculation of the local orbital integrals 283

24.5. The global hyperbolic result 286

25. The unipotent term 288

25.1. Explicit evaluation of the zeta integral at oo 288

25.2. Computation of the non-archimedean local zeta functions 291