$p$-adic hyperbolic planes and modular forms

Abstract:

For $K$ a number field and ${\mathbf {p}}$ a finite prime of $K$, we define a ${\mathbf {p}}$-adic hyperbolic plane and study its geometry under the action of $G{L_2}({K_{\mathbf {p}}})$. Seeking an operator with properties analogous to those of the non-Euclidean Laplacian of the classical hyperbolic plane, we investigate the fundamental invariant integral operator, the Hecke operator ${T_{\mathbf {p}}}$. Letting $S$ be a finite set of primes of a totally real $K$ (including all the infinite ones), a modular group $\Gamma (S)$ is defined. This group acts discontinuously on a product of classical and ${\mathbf {p}}$-adic hyperbolic planes. $S$-modular forms and their associated Dirichlet series are studied.