Non-Archimedean Poincar\'e series and geodesics on the Bruhat-Tits tree

Abstract

Adapting a construction of Kurihara in the setting of Drinfeld modular forms, we define Poincar\'e series on the Drinfeld half-plane $\Omega$. 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 $\Gamma(\mathfrak{n})$ of $GL_2(\mathbb{F}_q[T])$, we construct explicit linearly independent families of Poincar\'e series by lifting certain $k$-forms from the components of the analytic reduction of $\Gamma(\mathfrak{n})\backslash\Omega$. We formulate conjectures on the vanishing orders at cusps, and we prove the first of them for an explicit family of Poincar\'e series by computing the corresponding expansions at the cusps; as an application, we obtain the Drinfeld modular forms $h$ and $\Delta$ as Poincar\'e 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 Poincar\'e series span the whole space of modular forms of given weight and type.