Author | Bruggeman, Roelof W. author |
---|---|

Title | Families of Automorphic Forms [electronic resource] / by Roelof W. Bruggeman |

Imprint | Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 1994 |

Connect to | http://dx.doi.org/10.1007/978-3-0346-0336-2 |

Descript | X, 318 p. online resource |

SUMMARY

Automorphic forms on the upper half plane have been studied for a long time. Most attention has gone to the holomorphic automorphic forms, with numerous applications to number theory. Maass, [34], started a systematic study of real analytic automorphic forms. He extended Hecke's relation between automorphic forms and Dirichlet series to real analytic automorphic forms. The names Selberg and Roelcke are connected to the spectral theory of real analytic automorphic forms, see, e. g. , [50], [51]. This culminates in the trace formula of Selberg, see, e. g. , Hejhal, [21]. Automorphicformsarefunctionsontheupperhalfplanewithaspecialtra- formation behavior under a discontinuous group of non-euclidean motions in the upper half plane. One may ask how automorphic forms change if one perturbs this group of motions. This question is discussed by, e. g. , Hejhal, [22], and Phillips and Sarnak, [46]. Hejhal also discusses the e?ect of variation of the multiplier s- tem (a function on the discontinuous group that occurs in the description of the transformation behavior of automorphic forms). In [5]-[7] I considered variation of automorphic forms for the full modular group under perturbation of the m- tiplier system. A method based on ideas of Colin de Verdì ere, [11], [12], gave the meromorphic continuation of Eisenstein and Poincarยด e series as functions of the eigenvalue and the multiplier system jointly. The present study arose from a plan to extend these results to much more general groups (discrete co?nite subgroups of SL (R))

CONTENT

Modular introduction -- Modular introduction -- General theory -- Automorphic forms on the universal covering group -- Discrete subgroups -- Automorphic forms -- Poincarรฉ series -- Selfadjoint extension of the Casimir operator -- Families of automorphic forms -- Transformation and truncation -- Pseudo Casimir operator -- Meromorphic continuation of Poincarรฉ series -- Poincarรฉ families along vertical lines -- Singularities of Poincarรฉ families -- Examples -- Automorphic forms for the modular group -- Automorphic forms for the theta group -- Automorphic forms for the commutator subgroup

Mathematics
Functions of real variables
Mathematics
Real Functions