Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.23940 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915520145522688 |
|---|---|
| author | Chen, Shih-Yu |
| author_facet | Chen, Shih-Yu |
| contents | For a globally generic cuspidal automorphic representation $\mathitΠ$ of a quasi-split reductive group $G$ over $\mathbb Q$, E. Lapid and Z. Mao proposed a conjecture on the decomposition of the global Whittaker functionals on $\mathitΠ$ into products of an adjoint $L$-value of $\mathitΠ$ and the local Whittaker functionals. In this paper, we consider the algebraic aspect of the Lapid-Mao conjecture. More precisely, when $\mathitΠ$ is $C$-algebraic, we show that the algebraicity of the adjoint $L$-value can be expressed in terms of the Petersson norm of Whittaker-rational cusp forms in $\mathitΠ$, subject to the validity of the Lapid-Mao conjecture. For unitary similitude groups, we also establish an unconditional and more refined algebraicity result. Additionally, we give an explicit formula for the case $G={\rm U}(2,1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_23940 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algebraicity of adjoint $L$-functions for quasi-split groups Chen, Shih-Yu Number Theory For a globally generic cuspidal automorphic representation $\mathitΠ$ of a quasi-split reductive group $G$ over $\mathbb Q$, E. Lapid and Z. Mao proposed a conjecture on the decomposition of the global Whittaker functionals on $\mathitΠ$ into products of an adjoint $L$-value of $\mathitΠ$ and the local Whittaker functionals. In this paper, we consider the algebraic aspect of the Lapid-Mao conjecture. More precisely, when $\mathitΠ$ is $C$-algebraic, we show that the algebraicity of the adjoint $L$-value can be expressed in terms of the Petersson norm of Whittaker-rational cusp forms in $\mathitΠ$, subject to the validity of the Lapid-Mao conjecture. For unitary similitude groups, we also establish an unconditional and more refined algebraicity result. Additionally, we give an explicit formula for the case $G={\rm U}(2,1)$. |
| title | Algebraicity of adjoint $L$-functions for quasi-split groups |
| topic | Number Theory |
| url | https://arxiv.org/abs/2509.23940 |