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

We study two analogs, for modular forms over Fₐ (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 ₀ (n) with any prime ideal n in FₐT 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 Fₐ (T). Finally we present computational data on other kernel elements of these pairings.