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

TL;DR

This paper investigates the failure of perfect pairings between Hecke algebras and modular forms over function fields, revealing new phenomena distinct from classical cases through theoretical proofs and computational evidence.

Contribution

It demonstrates that, unlike classical cusp forms, the pairings are not perfect in a broad setting for certain congruence subgroups, providing explicit kernel elements and computational data.

Findings

Pairings are not perfect for weight 2 forms over function fields.

Existence of non-zero kernel elements in the Hecke algebra.

Computational evidence supports the theoretical results.

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 $\Gamma_0(\mathfrak{n})$ with any prime ideal $\mathfrak{n}$ in $\mathbb{F}_{q}[T]$ of degree $\geq 5$. 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.