Let $K$ be a non-archimedean local field. This paper gives an explicit isomorphism between the dual of the special representation of $G{L}_{n+1}\left(K\right)$ and the space of harmonic cochains defined on the Bruhat-Tits building of $G{L}_{n+1}\left(K\right)$, in the sense of E. de Shalit [11]. We deduce, applying the results of a paper of P. Schneider and U. Stuhler [9], that there exists a $G{L}_{n+1}\left(K\right)$-equivariant isomorphism between the cohomology group of the Drinfeld symmetric space and the space of harmonic cochains.

Let 𝓐₂(n) = Γ₂(n)∖𝔖₂ be the quotient of Siegel's space of degree 2 by the principal congruence subgroup of level n in Sp(4,ℤ). This is the moduli space of principally polarized abelian surfaces with a level n structure. Let 𝓐₂(n)* denote the Igusa compactification of this space, and ∂𝓐₂(n)* = 𝓐₂(n)* - 𝓐₂(n) its "boundary". This is a divisor with normal crossings. The main result of this paper is the determination of H(∂𝓐₂(n)*) as a module over the finite group Γ₂(1)/Γ₂(n). As an application...

Hecke groups $H\left({\lambda }_q\right)$ are the discrete subgroups of $\mathrm{P}SL(2,ℝ)$ generated by $S\left(z\right)=-{(z+{\lambda }_q)}^{-1}$ and $T\left(z\right)=-\frac{1}{z}$. The commutator subgroup of $H$(${\lambda }_q)$, denoted by $H^{\text{'}}\left({\lambda }_q\right)$, is studied in [2]. It was shown that $H^{\text{'}}\left({\lambda }_q\right)$ is a free group of rank $q-1$. Here the extended Hecke groups $\overline{H}\left({\lambda }_q\right)$, obtained by adjoining $R_1\left(z\right)=1/\overline{z}$ to the generators of $H\left({\lambda }_q\right)$, are considered. The commutator subgroup of $\overline{H}\left({\lambda }_q\right)$ is shown to be a free product of two finite cyclic groups. Also it is interesting to note that while in the $H\left({\lambda }_q\right)$ case, the index of $H^{\text{'}}\left({\lambda }_q\right)$ is changed by $q$, in the case of $\overline{H}\left({\lambda }_q\right)$, this number is either 4 for...

We present some completely normal elements in the maximal real subfields of cyclotomic fields over the field of rational numbers, relying on the criterion for normal element developed in [Jung H.Y., Koo J.K., Shin D.H., Normal bases of ray class fields over imaginary quadratic fields, Math. Z., 2012, 271(1–2), 109–116]. And, we further find completely normal elements in certain abelian extensions of modular function fields in terms of Siegel functions.

We propose an improved algorithm for computing mod ℓ Galois representations associated to a cusp form f of level one. The proposed method allows us to explicitly compute the case with ℓ = 29 and f of weight k = 16, and the cases with ℓ = 31 and f of weight k = 12,20,22. All the results are rigorously proved to be correct. As an example, we will compute the values modulo 31 of Ramanujan's tau function at some huge primes up to a sign. Also we will give an improved uper bound on...

We exhibit an algorithm to compute a Dirichlet domain for a Fuchsian group $\Gamma$ with cofinite area. As a consequence, we compute the invariants of $\Gamma$, including an explicit finite presentation for $\Gamma$.