Let $\overline{pp}\left(n\right)$ denote the number of overpartition pairs of n. Bringmann and Lovejoy (2008) proved that for n ≥ 0, $\overline{pp}(3n+2)\equiv 0\left(mod3\right)$. They also proved that there are infinitely many Ramanujan-type congruences modulo every power of odd primes for $\overline{pp}\left(n\right)$. Recently, Chen and Lin (2012) established some Ramanujan-type identities and explicit congruences for $\overline{pp}\left(n\right)$. Furthermore, they also constructed infinite families of congruences for $\overline{pp}\left(n\right)$ modulo 3 and 5, and two congruence relations modulo 9. In this paper, we prove several new infinite...

Introduction. The recent discovery of an analogue of the Riemann-Siegel integral formula for Dirichlet series associated with cusp forms [2] naturally raises the question whether similar formulas might exist for other types of zeta functions. The proof of these formulas depends on the functional equation for the underlying Dirichlet series. In both cases, for ζ(s) and for the cusp form zeta functions, only a simple gamma factor is involved. The next simplest case arises when two such factors occur...

Utilizing the theory of the Poisson transform, we develop some new concrete models for the Hecke theory in a space $M_{\lambda }\left(N\right)$ of Maass forms with eigenvalue $1/4-{\lambda }^2$ on a congruence subgroup ${\Gamma }_1\left(N\right)$. We introduce the field $F_{\lambda }=\mathbb{Q}(\lambda ,\sqrt{n},n^{\lambda /2}\mid n\in \mathbb{N})$ so that $F_{\lambda }$ consists entirely of algebraic numbers if $\lambda =0$.The main result of the paper is the following. For a packet $\Phi =({\nu }_p\mid p\nmid N)$ of Hecke eigenvalues occurring in $M_{\lambda }\left(N\right)$ we then have that either every ${\nu }_p$ is algebraic over $F_{\lambda }$, or else $\Phi$ will – for some $m\in \mathbb{N}$ – occur in the first cohomology of a certain space $W_{\lambda ,m}$ which is a...

Which invariants of a Galois $p$-extension of local number fields $L/K$ (residue field of char $p$, and Galois group $G$) determine the structure of the ideals in $L$ as modules over the group ring ${\mathbb{Z}}_p\left[G\right]$, ${\mathbb{Z}}_p$ the $p$-adic integers? We consider this question within the context of elementary abelian extensions, though we also briefly consider cyclic extensions. For elementary abelian groups $G$, we propose and study a new group (within the group ring ${𝔽}_q\left[G\right]$ where ${𝔽}_q$ is the residue field) and its resulting ramification filtrations....