Theta-lifts of Maass Waveforms
نویسنده
چکیده
Let O be an arbitrary order in an indeenite quaternion division algebra over Q. If O 1 is the group of elements in O with norm equal to 1, and H the complex upper half-plane, then X O := O 1 nH is a compact Riemann surface. Furthermore, let ? 0 (d) SL 2 (Z) be the Hecke congruence group of level d. Then X d := ? 0 (d)nH is a non-compact Riemann surface with nite volume. Let be the hyperbolic Laplace-operator on H. In certain situations, it is known that it is possible to relate the spectral resolutions of automorphic Laplacians in the compact case to the non-compact case. In this paper, we give an explicit construction of such correspondence in the case of Maaa waveforms. The construction uses Siegel theta functions and generalises the one in 8]. Furthermore, we prove that the theta-lifts commute with Hecke operators. Finally, we investigate to what extent the lifted forms are newforms or not.
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