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Estimates of automorphic functions. (English) Zbl 1081.11037
Consider $$\text{SL}(2,\mathbb{R})$$ acting on the upper half plane $$\mathbb{H}$$ by fractional linear transformations and let $$\Gamma$$ be a discrete cocompact subgroup. We set $$Y=\Gamma\setminus\mathbb{H}$$. The Laplace operator acts on functions on $$Y$$ with discrete eigenvalues $$0< \mu_1\leq\mu_2,\dots$$. Let $$\phi_i$$ denote the normalized eigenfunction with eigenvalue $$\mu_i$$. Fix two eigenfunctions $$\phi$$ and $$\phi'$$ and consider $c_i= \int_Y \phi\phi'\phi_i \,dy.$ Define $$\lambda_i$$ such that $$\mu_i={1\over 4}(1- \lambda^2_i)$$, and define $$b_i= |c_i|^2\exp({\pi\over 2} |\lambda_i|)$$. The main theorem of this paper states that there exists a constant $$A$$ such that for any $$T> 0$$, $\sum_{T\leq |\lambda_i|\leq 2T} b_i\leq A.$ The authors claim that this theorem is a stronger version of the main theorem in [$$(*)$$ J. Bernstein and A. Reznikov, Analytic continuation of representations and estimaes of automorphic forms, Ann. Math. (2) 150, No. 1, 329–352 (1999; Zbl 0934.11023)] where they prove that $$\sum_{|\lambda_i|\leq T} b_i\leq C(\ln T)^3$$. An important step of the proof is to interpret $$c_i$$ as the value of the $$\text{PGL}(2,\mathbb{R})$$-invariant linear functional on the tensor product three principal series representations of $$\text{PGL}(2,\mathbb{R})$$, evaluated at the tensor product of the spherical vectors. Such linear functionals are known to be one-dimensional.
Estimates of the $$c_i$$’s have important number-theoretic implications. This is explained in the introduction of this paper as well as in [$$(*)$$].

MSC:
 11F70 Representation-theoretic methods; automorphic representations over local and global fields 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods 11F12 Automorphic forms, one variable
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