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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.23940 |
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Table of 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 $G={\rm U}(2,1)$.