We give a sufficient criterion for generic local functional equations for submodule zeta functions associated to nilpotent algebras of endomorphisms defined over number fields. This allows us, in particular, to prove various conjectures on such functional equations for ideal zeta functions of nilpotent Lie lattices. Via the Mal'cev correspondence, these results have corollaries pertaining to zeta functions enumerating normal subgroups of finite index in finitely generated nilpotent groups, most notably finitely generated free nilpotent groups of any given class.
In @cite_14 , Woodward computed the ideal zeta functions of the full upper triangular @math -matrices over @math , as well as a number of combinatorially defined quotients of these algebras. He gives sufficient criteria for local functional equations, as well as some examples suggesting that these criteria might be necessary.
@cite_14
2068681643
We prove a two-part theorem on local ideal zeta functions of Lie rings of upper-triangular matrices. First, we prove that these local zeta functions display a strong uniformity. Secondly, we prove that these zeta functions satisfy a local functional equation. Some explicit examples of these zeta functions are also presented. Finally, we consider certain quotients of these Lie rings, showing that the strong uniformity continues to hold, and that under certain circumstances the functional equation does too.