非阿基米德Poincaré级数与Bruhat-Tits树上的测地线

中文摘要 AI 辅助

改编Kurihara在Drinfeld模形式领域的一个构造，我们在Drinfeld半平面$\Omega$上定义Poincaré级数。这些级数由与Bruhat-Tits树上的测地线自然相关的亚纯$1$-形式的乘积构成。我们在测地线的有限性条件下建立了收敛性，在若干情形下验证了这一条件，并给出了所得尖点形式非零的充分条件。对于$GL_2(\mathbb{F}_q[T])$的主同余子群$\Gamma(\mathfrak{n})$，我们通过从$\Gamma(\mathfrak{n})\backslash\Omega$的解析约化分量提升某些$k$-形式，构造了显式的线性无关的Poincaré级数族。我们提出了关于尖点处消失阶的猜想，并通过计算相应的尖点展开式，对一个显式的Poincaré级数族证明了其中的第一个猜想；作为应用，我们得到Drinfeld模形式$h$和$\Delta$（差一个符号）可表示为Poincaré级数。最后，对于与$\mathbb{F}_q(T)$上在$\infty$处分裂的四元数代数相伴的余紧群，我们证明了这些Poincaré级数张成给定权和类型的模形式的整个空间。

英文摘要

Adapting a construction of Kurihara in the setting of Drinfeld modular forms, we define Poincaré series on the Drinfeld half-plane $Ω$. These series are built from products of meromorphic $1$-forms that are naturally associated to geodesics on the Bruhat-Tits tree. We establish convergence under a finiteness condition on the geodesics, verify this condition in several cases, and give sufficient conditions for the resulting cusp forms to be nonzero. For the principal congruence subgroup $Γ(\mathfrak{n})$ of $GL_2(\mathbb{F}_q[T])$, we construct explicit linearly independent families of Poincaré series by lifting certain $k$-forms from the components of the analytic reduction of $Γ(\mathfrak{n})\backslashΩ$. We formulate conjectures on the vanishing orders at cusps, and we prove the first of them for an explicit family of Poincaré series by computing the corresponding expansions at the cusps; as an application, we obtain the Drinfeld modular forms $h$ and $Δ$ as Poincaré series (up to a sign). Finally, for cocompact groups attached to quaternion algebras over $\mathbb{F}_q(T)$ that split at $\infty$, we show that the Poincaré series span the whole space of modular forms of given weight and type.

发表机构

• Pennsylvania State University（宾夕法尼亚州立大学）

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