Non-Archimedean Poincaré series and geodesics on the Bruhat-Tits tree
Abstract
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(n) of GL2(𝔽q[T]), we construct explicit linearly independent families of Poincaré series by lifting certain k-forms from the components of the analytic reduction of Γ(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 𝔽q(T) that split at ∞, we show that the Poincaré series span the whole space of modular forms of given weight and type.
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BibTeX
@misc{bergerguesneau2026nonarchimedean,
title = {{Non-Archimedean Poincaré series and geodesics on the Bruhat-Tits tree}},
author = {Milan Berger-Guesneau and Mihran Papikian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.23964},
eprint = {2609.23964},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23964},
}
RIS
TY - UNPB
TI - Non-Archimedean Poincaré series and geodesics on the Bruhat-Tits tree
AU - Milan Berger-Guesneau
AU - Mihran Papikian
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2609.23964
UR - https://arxiv.org/abs/2609.23964
AB - 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(n) of GL_2(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(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 𝔽_q(T) that split at ∞, we show that the Poincaré series span the whole space of modular forms of given weight and type.
ER -