Spectral decomposition and Siegel-Veech transforms for strata: The case of marked tori
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| Format: | Preprint |
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2024
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| _version_ | 1866911834071629824 |
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| author | Athreya, Jayadev Lagacé, Jean Möller, Martin Raum, Martin |
| author_facet | Athreya, Jayadev Lagacé, Jean Möller, Martin Raum, Martin |
| contents | Generalizing the well-known construction of Eisenstein series on the modular curves, Siegel-Veech transforms provide a natural construction of square-integrable functions on strata of differentials on Riemannian surfaces. This space carries actions of the foliated Laplacian derived from the SL(2,R)-action as well as various differential operators related to relative period translations.
In the paper we give spectral decompositions for the stratum of tori with two marked points. This is a homogeneous space for a special affine group, which is not reductive and thus does not fall into well-studied cases of the Langlands program, but still allows to employ techniques from representation theory and global analysis. Even for this simple stratum exhibiting all Siegel-Veech transforms requires novel configurations of saddle connections. We also show that the contiunuous spectrum of the foliated Laplacian is much larger than the space of Siegel-Veech transforms, as opposed to the case of the modular curve. This defect can be remedied by using instead a compound Laplacian involving relative period translations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_06597 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral decomposition and Siegel-Veech transforms for strata: The case of marked tori Athreya, Jayadev Lagacé, Jean Möller, Martin Raum, Martin Number Theory Geometric Topology Spectral Theory Generalizing the well-known construction of Eisenstein series on the modular curves, Siegel-Veech transforms provide a natural construction of square-integrable functions on strata of differentials on Riemannian surfaces. This space carries actions of the foliated Laplacian derived from the SL(2,R)-action as well as various differential operators related to relative period translations. In the paper we give spectral decompositions for the stratum of tori with two marked points. This is a homogeneous space for a special affine group, which is not reductive and thus does not fall into well-studied cases of the Langlands program, but still allows to employ techniques from representation theory and global analysis. Even for this simple stratum exhibiting all Siegel-Veech transforms requires novel configurations of saddle connections. We also show that the contiunuous spectrum of the foliated Laplacian is much larger than the space of Siegel-Veech transforms, as opposed to the case of the modular curve. This defect can be remedied by using instead a compound Laplacian involving relative period translations. |
| title | Spectral decomposition and Siegel-Veech transforms for strata: The case of marked tori |
| topic | Number Theory Geometric Topology Spectral Theory |
| url | https://arxiv.org/abs/2404.06597 |