We introduce an interpolation between Euler integral and Laplace integral: Euler-Laplace integral. We claim that, when parameters $\delta$ of the integrand are non-resonant, the $\mathcal{D}$-module corresponding to Euler-Laplace integral is naturally isomorphic to GKZ hypergeometric system $M_{A}(\delta)$ where $A$ is a generalization of Cayley configuration. As a topological counterpart of this isomorphism, we establish an isomorphism between certain rapid decay homology group and holomorphic solutions of $M_{A}(\delta)$.

We establish the intersection theory of the rapid decay homology group and formulate the twisted period relation in this setting. We claim that there is a standard method of constructing a basis of the rapid decay homology group which can be related to GKZ hypergeometric series. This can be carried out with the aid of a convergent regular triangulation $T$. When $T$ is unimodular, we can obtain a closed formula of the homology intersection number. Finally, we obtain a Laurent series expansion formula of the cohomology intersection number in terms of the combinatorics of $T$.

Let $k$ be a number field and $p$ a prime number. It is conjectured by Greenberg that the Iwasawa $\lambda$- and $\mu$-invariants of the cyclotomic $\mathbf{Z}_{p}$-extension of $k$ always vanish if $k$ is totally real. In this article, we will discuss a weak version of Greenberg’s conjecture, and give results analogous to Greenberg’s and Ozaki’s results.