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arXiv 2203.04853math.NT
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O（1，5）上CAP型的显式提升构造

An explicit lifting construction of CAP forms on O(1,5)

Hiro-aki Narita, Ameya Pitale, Siddhesh Wagh

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摘要：

我们在签名（1+，5-）的正交群O（1，5）上构造了非调和尖点形式.给定$\mathbb{Q}$上的一个确定的四元数代数B，它的正交群附着在秩为6的不定二次空间上，其各向异性部分由B的约化范数定义.我们的构造可以看作是[22]对任何定四元数代数的推广，我们注意到[22]采用了B的判别式为2的情况。与[22]不同的是，构造的方法是考虑从Maass尖点形式到O（1，5）的θ提升，遵循Borcherds的公式。我们的尖点形式产生的尖点表示进行了详细的研究。我们确定了尖点表示的所有局部分量，并证明了我们的尖点形式是CAP形式。

英文摘要：

We explicitly construct non-tempered cusp forms on the orthogonal group O(1,5) of signature (1+,5-). Given a definite quaternion algebra B over $\mathbb{Q}$, the orthogonal group is attached to the indefinite quadratic space of rank 6 with the anisotropic part defined by the reduced norm of B. Our construction can be viewed as a generalization of [22] to the case of any definite quaternion algebras, for which we note that [22] takes up the case where the discriminant of B is two. Unlike [22] the method of the construction is to consider the theta lifting from Maass cusp forms to O(1,5), following the formulation by Borcherds. The cuspidal representations generated by our cusp forms are studied in detail. We determine all local components of the cuspidal representations and show that our cusp forms are CAP forms.

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