\(L^ \infty\) norms of eigenfunctions of arithmetic surfaces
Let \(H= H^2\) denote the Poincaré upper half plane with Laplace operator \(\Delta= y^2 ({{\partial^2} \over {\partial x^2}}+ {{\partial^2} \over {\partial y^2}})\). Consider a discrete subgroup \(\Gamma\) of \(\text{SL}(2,\mathbb R)\) coming from a quaternion division algebra over \(\mathbb Q\). See \textit{M. Eichler} [Lectures on modular correspondences, Tata Inst. (1957); \url{http://www.math.tifr.res.in/~publ/ln/tifr09.pdf}], \textit{S. Katok} [Fuchsian groups. Chicago: The University of Chicago Press (1992; Zbl 0753.30001)], or \textit{M.-F. Vignéras}, Arithmétique des algèbres de quaternions. Berlin etc.: Springer-Verlag (1980; Zbl 0422.12008)]. Then \(X= \Gamma\setminus H\) is a compact hyperbolic surface. Consider an orthonormal basis \(\varphi_j\) of simultaneous eigenfunctions of \(\Delta\) and the Hecke operators \(T_n\) on \(L^2 (X)\). Let \(\Delta \varphi_j=- \lambda_j \varphi_j\), \(T_n \varphi_j= \lambda_j (n) \varphi_j\). The main theorem of the paper being reviewed says that for all \(j\) and \(\varepsilon>0\), we have \[ |\varphi_j |_\infty \ll_\varepsilon \lambda_j^{(5/ 24)+ \varepsilon} \qquad \text{and} \qquad |\varphi_j |_\infty\gg c\sqrt {\log \log \lambda_j} \] for infinitely many \(j\), where \(c\) is a positive constant. The upper bound beats the more general result of \textit{A. Seeger} and \textit{C. Sogge} [Indiana Univ. Math. J. 38, 669--682 (1989; Zbl 0703.35133)]\ in which \(5/24\) is replaced by \(1/4\). That the \(\varphi_j\)'s are not uniformly bounded is consistent with the numerical experiments of \textit{D. Hejhal} and \textit{B. Rackner} [Exp. Math. 1, 275--305 (1992; Zbl 0813.11035)]. See also \textit{P. Sarnak} [The Schur lectures (1992), Isr. Math. Conf. Proc. 8, 183--236 (1995; Zbl 0831.58045)]. This is also consistent with the basic conjecture (an analogue of the Ramanujan conjecture) that \[ |\varphi_j |_\infty \ll_\varepsilon \lambda_j^\varepsilon. \] Sarnack [loc. cit.]\ points out that upper bounds for \(|\varphi_j |_\infty\) lead to upper bounds for \(L\)-functions on their critical lines. So one is led to a bound for the Riemann zeta function on \(\text{Re} (s)= 1/2\) which beats the convexity bound but is not as good as what is known by other techniques. The proofs start -- as does the Selberg trace formula -- with the spectral expansion of a carefully chosen automorphic kernel. The spectral expansion says \[ \sum_{\gamma\in \Gamma} k(\gamma z,w)= \sum_{j=0}^\infty h(r_j) \varphi_j (z) \overline {\varphi_j (w)}. \] And one plugs in \[ h(r)= {{4\pi^2 \cosh {{\pi r} \over 2} \cosh {{\pi T} \over 2}} \over {\cosh \pi r+ \cosh \pi T}}. \] Use is also made of an inequality for Maass cusp forms for congruence subgroups of \(\text{SL}(2, \mathbb Z)\) in \textit{H. Iwaniec} [Acta Arith. 56, 65--82 (1990; Zbl 0702.11034)]\ and the Jacquet-Langlands correspondence between quaternion groups and congruence subgroups [see \textit{D. Hejhal}, Lect. Notes Math. 1135, 127--196 (1985; Zbl 0558.10019)]\ for a classical approach). An appendix sketches the modifications needed to prove the result in the case \(\text{SL}(2, \mathbb Z)\) when \(X\) is not compact. Then there is also a continuous spectrum of the Laplacian on \(L^2 (X)\) and one needs the Eisenstein series as well as Maass cusp forms \(\varphi_j\).
spectral expansion of an automorphic kernel
scientific article; zbMATH DE number 750875