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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2303.16114 |
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| _version_ | 1866910833176477696 |
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| author | Loeffler, David Rivero, Óscar |
| author_facet | Loeffler, David Rivero, Óscar |
| contents | We prove algebraicity results for critical $L$-values attached to the group $\text{GSp}_4 \times \text{GL}_2$, and for Gan--Gross--Prasad periods which are conjecturally related to central $L$-values for $\text{GSp}_4 \times \text{GL}_2 \times \text{GL}_2$. Our result for $\text{GSp}_4 \times \text{GL}_2$ gives a new proof (by a very different method) of a recent result of Morimoto, and will be used in a sequel paper to construct a new $p$-adic $L$-function for $\text{GSp}_4 \times \text{GL}_2$. The results for Gross--Prasad periods appear to be new. A key aspect is the computation of certain archimedean zeta integrals, whose $p$-adic counterparts are also studied in this note. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_16114 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Algebraicity of $L$-values for $\text{GSp}_4 \times \text{GL}_2$ and $\text{GSp}_4 \times \text{GL}_2 \times \text{GL}_2$ Loeffler, David Rivero, Óscar Number Theory We prove algebraicity results for critical $L$-values attached to the group $\text{GSp}_4 \times \text{GL}_2$, and for Gan--Gross--Prasad periods which are conjecturally related to central $L$-values for $\text{GSp}_4 \times \text{GL}_2 \times \text{GL}_2$. Our result for $\text{GSp}_4 \times \text{GL}_2$ gives a new proof (by a very different method) of a recent result of Morimoto, and will be used in a sequel paper to construct a new $p$-adic $L$-function for $\text{GSp}_4 \times \text{GL}_2$. The results for Gross--Prasad periods appear to be new. A key aspect is the computation of certain archimedean zeta integrals, whose $p$-adic counterparts are also studied in this note. |
| title | Algebraicity of $L$-values for $\text{GSp}_4 \times \text{GL}_2$ and $\text{GSp}_4 \times \text{GL}_2 \times \text{GL}_2$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2303.16114 |