Holomorphic Automorphic Forms and Cohomology
We investigate the correspondence between holomorphic automorphic forms on the upper half-plane with complex weight and parabolic cocycles. For integral weights at least For real weights that are not an integer at least A tool in establishing these results is the relation to cohomology groups with v...
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| Hlavní autoři: | , , |
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| Médium: | E-kniha Kniha |
| Jazyk: | angličtina |
| Vydáno: |
Providence, Rhode Island
American Mathematical Society
2018
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| Vydání: | 1 |
| Edice: | Memoirs of the American Mathematical Society |
| Témata: | |
| ISBN: | 1470428555, 9781470428556 |
| ISSN: | 0065-9266, 1947-6221 |
| On-line přístup: | Získat plný text |
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- Introduction -- Cohomology with Values in Holomorphic Functions -- Definitions and notations -- Modules and cocycles -- The image of automorphic forms in cohomology -- One-sided averages -- Harmonic Functions -- Harmonic functions and cohomology -- Boundary germs -- Polar harmonic functions -- \redefinepart -- Cohomology with values in Analytic Boundary Germs -- \oldpart -- Highest weight spaces of analytic boundary germs -- Tesselation and cohomology -- Boundary germ cohomology and automorphic forms -- Automorphic forms of integral weights at least <inline-formula content-type="math/mathml"> 2 2 </inline-formula> and analytic boundary germ cohomology -- \redefinepart -- Miscellaneous -- \oldpart -- Isomorphisms between parabolic cohomology groups -- Cocycles and singularities -- Quantum automorphic forms -- Remarks on the literature -- Universal covering group and representations -- Indices
- Part III . Cohomology with values in Analytic Boundary Germs -- Chapter 8. Highest weight spaces of analytic boundary germs -- 8.1. Definition of highest weight space -- 8.2. General properties of highest weight spaces of analytic boundary germs -- 8.3. Splitting of harmonic boundary germs, Green's form -- 8.4. Periodic harmonic functions and boundary germs -- 8.5. Completion of the proof of Proposition 8.4 -- 8.6. Related work -- Chapter 9. Tesselation and cohomology -- 9.1. Tesselations of the upper half-plane -- 9.2. Resolutions based on a tesselation -- 9.3. Cocycles attached to automorphic forms -- 9.4. Derivatives of -functions -- 9.5. Related work -- Chapter 10. Boundary germ cohomology and automorphic forms -- 10.1. Spaces of global representatives for highest weight spaces -- 10.2. From parabolic cocycles to automorphic forms -- 10.3. Injectivity -- 10.4. From analytic boundary germ cohomology to automorphic forms -- 10.5. Completion of the proof of Theorem A for general weights -- 10.6. Related work -- Chapter 11. Automorphic forms of integral weights at least 2 and analytic boundary germ cohomology -- 11.1. Image of automorphic forms in mixed parabolic cohomology -- 11.2. Image of mixed parabolic cohomology classes in automorphic forms -- 11.3. Exact sequences for mixed parabolic cohomology groups -- 11.4. Automorphic forms and analytic boundary germ cohomology -- 11.5. Comparison with classical results -- 11.6. Related work -- Part IV . Miscellaneous -- Chapter 12. Isomorphisms between parabolic cohomology groups -- 12.1. Invariants under hyperbolic and parabolic elements -- 12.2. Modules of singularities -- 12.3. Mixed parabolic cohomology and parabolic cohomology -- 12.4. Related work -- Chapter 13. Cocycles and singularities -- 13.1. Cohomology with singularities in hyperbolic fixed points
- Cover -- Title page -- Introduction -- Part I . Cohomology with Values in Holomorphic Functions -- Chapter 1. Definitions and notations -- 1.1. Operators on functions on the upper and lower half-plane -- 1.2. Discrete group -- 1.3. Automorphic forms -- 1.4. Cohomology and mixed parabolic cohomology -- 1.5. Modules -- 1.6. Semi-analytic vectors -- 1.7. Isomorphic cohomology groups -- 1.8. Harmonic lifts of holomorphic automorphic forms -- Chapter 2. Modules and cocycles -- 2.1. The map from automorphic forms to cohomology -- 2.2. Cusp forms -- 2.3. The theorem of Knopp and Mawi -- 2.4. Modular group and powers of the Dedekind eta-function -- 2.5. Related work -- Chapter 3. The image of automorphic forms in cohomology -- 3.1. Mixed parabolic cohomology groups -- 3.2. The parabolic equation for an Eichler integral -- 3.3. Asymptotic behavior at infinity -- 3.4. Construction of solutions -- 3.5. Image of automorphic forms in the analytic cohomology -- 3.6. Proof of Theorem B -- 3.7. Related work -- Chapter 4. One-sided averages -- 4.1. One-sided averages with absolute convergence -- 4.2. Analytic continuation of one-sided averages -- 4.3. Parabolic cohomology groups -- 4.4. Related work -- Part II . Harmonic Functions -- Chapter 5. Harmonic functions and cohomology -- 5.1. The sheaf of harmonic functions -- 5.2. Harmonic lifts of automorphic forms -- Chapter 6. Boundary germs -- 6.1. Three sheaves on the real projective line -- 6.2. Relation between the sheaves of harmonic boundary and analytic boundary germs -- 6.3. Kernel function for the map from automorphic forms to boundary germ cohomology -- 6.4. Local study of the sheaf of analytic boundary germs -- 6.5. Related work -- Chapter 7. Polar harmonic functions -- 7.1. Polar expansion -- 7.2. Polar expansion of the kernel function -- 7.3. Related work
- 13.2. Mixed parabolic cohomology and condition at cusps -- 13.3. Recapitulation of the proof of Theorem E -- 13.4. Related work -- Chapter 14. Quantum automorphic forms -- 14.1. Quantum modular forms -- 14.2. Quantum automorphic forms -- 14.3. Quantum automorphic forms, cohomology, and automorphic forms -- 14.4. Related work -- Chapter 15. Remarks on the literature -- Appendix A. Universal covering group and representations -- A.1. Universal covering group -- A.2. Principal series -- A.3. Related work -- Bibliography -- Indices -- Index -- List of notations -- Back Cover