Non-perfect pairings between Hecke algebra and modular forms over function fields

Abstract

We study two analogs, for modular forms over ${\mathbb{F}}_q(T)$, of the pairing between Hecke algebra and cusp forms given by the first coefficient in the expansion. For Drinfeld modular forms, the ${\mathbb{C}}_{\infty }$-pairing is provided by the first coefficient of their $t$-expansion at infinity. For $\mathbb{Z}$-valued harmonic cochains, the $\mathbb{Z}$-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 $\hat{I}{“}_0(\mathfrak{n})$ with any prime ideal $\mathfrak{n}$ in ${\mathbb{F}}_q[T]$ of degree $\ge 5$. 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 ${\mathbb{F}}_q(T)$. Finally we present computational data on other kernel elements of these pairings.