2024/08/21 by Cécile Armana, Armana, Cécile
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2408.11473
openalex created_date 2024/08/21 · openalex publication_date 2024/08/21 · openalex updated_date 2026/08/01
We study two analogs, for modular forms over \mathbbFq(T), of the pairing between Hecke algebra and cusp forms given by the first coefficient in the expansion. For Drinfeld modular forms, the ℂ∞-pairing is provided by the first coefficient of their t-expansion at infinity. For ℤ-valued harmonic cochains, the ℤ-pairing is given by their Fourier coefficient with respect to the trivial ideal. We prove that, contrarily to classical cusp forms, both pairings in weight 2 are not perfect in a quite general setting, namely for the congruence subgroup Γ0(\mathfrakn) with any prime ideal \mathfrakn in \mathbbFq[T] of degree ≥ 5. We show it by exhibiting a common element of the Hecke algebra in the kernels of both pairings and proving that it is non-zero using computations with modular symbols over \mathbbFq(T). Finally we present computational data on other kernel elements of these pairings.