Abstract
In this chapter we consider the particular case of congruence covers \(Y (N) =\varGamma (N)\setminus \mathcal{H}\) or \(Y _{0}(N) =\varGamma _{0}(N)\setminus \mathcal{H}\) of the modular surface. In the first sections we pay particular attention to the case of the modular group SL(2, Z).
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Notes
- 1.
The coefficient \(\frac{1} {2}\) comes from the fact that the element − I ∈ SL(2, Z) acts trivially on \(\mathcal{H}\).
- 2.
When \(\mathop{\mathrm{Re}}\nolimits (s) > 1\), this is obvious since s(1 − s) doesn’t lie in R +.
- 3.
We shall see that this explains why the constant functions belong to the subspace \(\overline{\mathcal{E}(\varGamma \setminus \mathcal{H})} \subset L^{2}(\varGamma \setminus \mathcal{H})\) in the decomposition of Proposition 4.9 below.
- 4.
Note that according to Fubini’s theorem we can speak of the constant term (as a measurable function) f 0 of any integrable function f on D. The subspace \(\overline{\mathcal{C}(\varGamma \setminus \mathcal{H})}\) then consists of functions whose constant term is almost everywhere equal to 0.
- 5.
This is no problem at zero since ψ is of compact support.
- 6.
Since f is cuspidal, f Y = f.
- 7.
This is an abuse of notation: the Eisenstein series E(⋅ , 1∕2 + ir) is not square integrable.
- 8.
If the geodesic is one of the γ(p∕q, p′∕q′) defining the triangulation, its coding is simply \((\ldots, 0, 1, 0,\ldots )\).
- 9.
Recall that the continued fraction expansion of β is
$$\displaystyle{\beta = b_{0} + \frac{1} {b_{1} + \frac{1} {b_{2}+\ldots }} = [b_{0},b_{1},\ldots ],}$$where the (b i ) form a sequence of integers (a finite sequence, if β ∈ Q) with b 0 ∈ Z and b i > 0 for i > 0.
- 10.
We recover here the same alternative as at the end of the preceding subsection on the subject of solving the Pell equation by means of continued fractions.
- 11.
It is natural to include the factor of \(\sqrt{y}\) in order to obtain a sum of Whittaker functions.
- 12.
Indeed, one checks easily that the function \(L(s,\lambda ^{k})\) is non-zero by writing it as an Euler product in the region of absolute convergence.
- 13.
Starting from a genus 1 subgroup of SL(2, Z) one can indeed form cyclic covers of genus 1 and m cusps. As m gets large, it is possible to show (see [22]) that these covers have m − 1 small eigenvalues. On the other hand Huxley [60] (see also Otal [94] as well as Otal and Rosas [95]) proves that for a genus 1 surface the cuspidal spectrum is contained in \(]1/4, +\infty [\).
- 14.
In this context the term “hyperbolic” has a different meaning from the hyperbolic geometry that we consider in this book.
- 15.
In the case of the field \(\mathbf{Q}(\sqrt{2})\) such a character is necessarily trivial. We write it as \(\lambda ^{0}\).
- 16.
See § 6.3.1 for a rapid introduction to the theory of representations of finite groups.
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Bergeron, N. (2016). Maaß Forms. In: The Spectrum of Hyperbolic Surfaces. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-27666-3_4
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