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Sparse equidistribution problems, period bounds and subconvexity. (English) Zbl 1214.11051

Let \(\Gamma\subset SL_2({\mathbb R})\) be a cocompact lattice. The author proves that there exists \(\gamma_{\text{max}}\) depending on \(\Gamma\) such that for every \( x_0\in\Gamma\backslash \text{SL}_2({\mathbb R}) \) the set \[ \left\{x_0\begin{pmatrix} 1&j^{1+\gamma}\\ 0&1 \end{pmatrix}:j\in{\mathbb N} \right\} \] is equidistributed. If \(\lambda_1\) is the smallest nonzero eigenvalue of the Laplacian on \(\Gamma\backslash{\mathbb H}\), let \[ \alpha=\begin{cases} 0,&\lambda_1\geq \frac14\\ \sqrt{\frac14-\lambda_1},&\text{else}. \end{cases} \] Then we can take \(\gamma_{\text{max}}=\frac{(1-2\alpha)^2}{16(3-2\alpha)}\). The author presents other results on subconvexity of the triple product period in the level aspect over number fields and on bounding Fourier coefficients of automorphic forms.

MSC:

11F03 Modular and automorphic functions
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F30 Fourier coefficients of automorphic forms
58J51 Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity
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