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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.16873 |
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| _version_ | 1866915629670334464 |
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| author | Mahendraker, Siddharth |
| author_facet | Mahendraker, Siddharth |
| contents | Let $E/F$ be a quadratic extension of number fields. We introduce truncated geometric and spectral RTF distributions associated to a Galois symmetric pair $G \subset \mathrm{Res}_{E/F} G_E$, subject to the constraint that $G$ and $\mathrm{Res}_{E/F} G_E$ have the same split rank, and formulate a precise coarse RTF identity. Specializing to $SL_{2, F} \subset \mathrm{Res}_{E/F} SL_{2, E}$, we show that the truncated geometric RTF distribution converges, and is given by a linear polynomial in the truncation parameter. We then compute the fine geometric expansion explicitly, including the contribution of the regularized relative unipotent orbital integrals. We propose a geometric viewpoint which guided the computation of these unipotent terms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_16873 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Relative Trace Formula for Galois Periods Mahendraker, Siddharth Number Theory Representation Theory Let $E/F$ be a quadratic extension of number fields. We introduce truncated geometric and spectral RTF distributions associated to a Galois symmetric pair $G \subset \mathrm{Res}_{E/F} G_E$, subject to the constraint that $G$ and $\mathrm{Res}_{E/F} G_E$ have the same split rank, and formulate a precise coarse RTF identity. Specializing to $SL_{2, F} \subset \mathrm{Res}_{E/F} SL_{2, E}$, we show that the truncated geometric RTF distribution converges, and is given by a linear polynomial in the truncation parameter. We then compute the fine geometric expansion explicitly, including the contribution of the regularized relative unipotent orbital integrals. We propose a geometric viewpoint which guided the computation of these unipotent terms. |
| title | The Relative Trace Formula for Galois Periods |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2511.16873 |