## 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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