Holomorphic Automorphic Forms and Cohomology.

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.

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.

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.

The authors investigate the correspondence between holomorphic automorphic forms on the upper half-plane with complex weight and parabolic cocycles. For integral weights at least 2 this correspondence is given by the Eichler integral. The authors use Knopp's generalization of this integral to real weights, and apply it to complex weights that are not an integer at least 2. They show that for these weights the generalized Eichler integral gives an injection into the first cohomology group with values in a module of holomorphic functions, and characterize the image. The authors impose no condition on the growth of the automorphic forms at the cusps. Their result concerns arbitrary cofinite discrete groups with cusps, and covers exponentially growing automorphic forms, like those studied by Borcherds, and like those in the theory of mock automorphic forms.

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