arXiv subjects

Fumitake Hyodo

Publications and source records attributed to Fumitake Hyodo.

3 recordsLinked to original sources

arXiv2022

A formal power series over a noncommutative Hecke ring and the rationality of the Hecke series for $GSp_4$

The present paper studies Hecke rings derived by the automorphism groups of certain algebras $L_p$ over the ring of $p$-adic integers. Our previous work considered the case where $L_p$ is the Heisenberg Lie algebra (of dimension 3) over the ring of $p$-adic integers. Although this Hecke ring is noncommutative, we showed that a formal power series with coefficients in this Hecke ring satisfies an identity similar to the rationality of the Hecke series for $GL_2$ due to E.~Hecke. In the present paper, we establish an analogous result in the case of the Heisenberg Lie algebra of dimension 5 over the ring of $p$-adic integers. In this case, our identity is similar to the rationality of the Hecke series for $GSp_4$, due to G.~Shimura.

math.NT↗

arXiv2021

Global properties of a Hecke ring associated with the Heisenberg Lie algebra

This study concerns (not necessarily commutative) Hecke rings associated with certain algebras and describes a formal Dirichlet series with coefficients in the Hecke rings, which can be used to generalize Shimura's series. Considering the case of the Heisenberg Lie algebra, an analog of the identity for Shimura's series derived employing the rationality theorem, presented by Hecke and Tamagawa, is established. Moreover, this analog recovers the explicit formula for the pro-isomorphic zeta function of the Heisenberg Lie algebra shown by Grunewald, Segal and Smith.

math.NT↗

arXiv2021

A note on a Hecke ring associated with the Heisenberg Lie algebra

This paper focuses on the theory of the Hecke rings associated with the general linear groups originally studied by Hecke and Shimura et al., and moreover generalizes its notions to Hecke rings associated with the automorphism groups of certain algebras. Then, in the case of the Heisenberg Lie algebra, we show an analog of the classical theory.

math.NT↗