Theta functions, fourth moments of eigenforms, and the sup-norm problem I
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2020
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866916676951343104 |
|---|---|
| author | Khayutin, Ilya Steiner, Raphael S. |
| author_facet | Khayutin, Ilya Steiner, Raphael S. |
| contents | We give sharp point-wise bounds in the weight-aspect on fourth moments of modular forms on arithmetic hyperbolic surfaces associated to Eichler orders. Therefore we strengthen a result of Xia and extend it to co-compact lattices. We realize this fourth moment by constructing a holomorphic theta kernel on $\mathbf{G} \times \mathbf{G} \times \mathbf{SL}_{2}$, for $\mathbf{G}$ an indefinite inner-form of $\mathbf{SL}_2$ over $\mathbb{Q}$, based on the Bergman kernel, and considering its $L^2$-norm in the Weil variable. The constructed theta kernel further gives rise to new elementary theta series for integral quadratic forms of signature $(2,2)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_07194 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Theta functions, fourth moments of eigenforms, and the sup-norm problem I Khayutin, Ilya Steiner, Raphael S. Number Theory 11F72 We give sharp point-wise bounds in the weight-aspect on fourth moments of modular forms on arithmetic hyperbolic surfaces associated to Eichler orders. Therefore we strengthen a result of Xia and extend it to co-compact lattices. We realize this fourth moment by constructing a holomorphic theta kernel on $\mathbf{G} \times \mathbf{G} \times \mathbf{SL}_{2}$, for $\mathbf{G}$ an indefinite inner-form of $\mathbf{SL}_2$ over $\mathbb{Q}$, based on the Bergman kernel, and considering its $L^2$-norm in the Weil variable. The constructed theta kernel further gives rise to new elementary theta series for integral quadratic forms of signature $(2,2)$. |
| title | Theta functions, fourth moments of eigenforms, and the sup-norm problem I |
| topic | Number Theory 11F72 |
| url | https://arxiv.org/abs/2009.07194 |