By Roelof W. Bruggeman
This publication provides a scientific therapy of actual analytic automorphic varieties at the higher part airplane for basic confinite discrete subgroups. those automorphic kinds are allowed to have exponential development on the cusps and singularities at different issues to boot. it really is proven that the Poincaré sequence and Eisenstein sequence happen in households of automorphic kinds of this common kind. those households are meromorphic within the spectral parameter and the multiplier method together. the overall a part of the publication closes with a research of the singularities of those families.
The paintings is aimed essentially at mathematicians engaged on genuine analytic automorphic types. even though, the publication also will inspire readers on the graduate point (already versed within the topic and in spectral concept of automorphic types) to delve into the sector extra deeply. An introductory bankruptcy explicates major principles, and 3 concluding chapters are replete with examples that make clear the final thought and effects constructed therefrom.
"It is made abundantly transparent that this perspective, of households of automorphic capabilities looking on various eigenvalue and multiplier platforms, is either deep and fruitful." - MathSciNet
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Additional info for Families of automorphic forms
The Fourier term of order (P, n) is F˜P,n f (gP k(η)a(tu )k(ψ)) = einη FP,n f (u)eilψ , for 0 < u < σ, η, ψ ∈ R. Again ω F˜P,n f = F˜P,n ωf , and FP,n ωf = lP,n (l)FP,n f , with (n − l)2 nl lP,n (l) = −(u2 + u)∂u2 − (2u + 1)∂u + − . 4 Diﬀerential equation. The domain of FP,n f is a subset of (0, ∞), depending on the domain of f . It always contains an interval (σ, ∞) if P ∈ X ∞ , and an interval (0, σ) if P ∈ PY . If f satisﬁes ωf = λf on U , all FP,n f are solutions of the second order linear diﬀerential equation lP,n (l)F = λF .
5. 4 we consider the action of these diﬀerential operators on the Fourier terms. The ideas we follow in this chapter may be found in Chapter IV of , except that Maass allows no singularities in H, and less growth at the cusps. 5 we have ﬁxed a ﬁnite set P ⊂ X = Γ\H containing at least the cusps. We have called the elements of P exceptional points. We consider in this section the Fourier expansion of an automorphic form at an exceptional point P ∈ P, and we deﬁne the condition ‘regular at P ’.
One may also consider the action of Γ ˜ . H∗ = H ∪ cuspidal points for Γ ˜ Each Γ-orbit of cuspidal points corresponds to at least two boundary segments of ˜ a given fundamental domain, hence the number of Γ-orbits in the cuspidal points ˜ is ﬁnite. We call such a Γ-orbit a cusp. 1 Connected neighborhoods of Γ 2 2 2 as represented in the standard fundamental domain. 7, describes how to put a topology on H∗ such that Γ\H ∗ ˜ is a Hausdorﬀ topological space. 1 in . ) ∼ ˜ ˜ G/ ˜ K ˜ and X = Γ\H ˜ ∗.
Families of automorphic forms by Roelof W. Bruggeman