Narrow Results

A compact Riemann surface $X$ of genus $g$ is called pseudo-real if it admits an anticonformal automorphism but no anticonformal involution. In this paper, we study pseudo-real $(q,n)$-gonal Riemann surfaces of genera greater or equal to two; these surfaces have anticonformal automorphisms of prime order $n$ such that the quotient spaces have genus $q$.

We show that any non-standard generating pair of a hyperbolic triangle group is represented by a special almost orbifold covering with a good marking.

Let Γ ⊂ SL₂(ℝ) be a Fuchsian group of the first kind. In this paper, we study the non-vanishing of the spanning set for the space of cuspidal modular forms of weight m ≥ 3 constructed in [5, Corollary 2.6.11]. Our approach is based on the generalization of the non-vanishing criterion for L¹-Poincaré series defined for locally compact groups and proved in [6, Theorem 4.1]. We obtain very sharp bounds for the non-vanishing of the spaces of cusp forms for general Γ having at least one cusp. We obtain explicit results for congruence subgroups Γ(N), Γ₀(N), and Γ₁(N) (N ≥ 1).

Let Γ be a Fuchsian group of the first kind which has a (regular) cusp at ∞ such as Γ₀(N). In this paper we use Kohnen's kernel function for Γ to analytically continue L(s, f) to the region Re(s) > m/2, where f is a cuspidal modular form of weight m ≥ 5 attached to Γ. We study non-vanishing of L(s, f) using our integral non-vanishing criterion.

Let $\mathrm{\Gamma }$ be a geometrically finite Fuchsian group acting on the upper half-plane $ℍ$. Let ${ℰ}$ denote the set of elliptic fixed points of $\mathrm{\Gamma }$ in $ℍ$. We give a lower bound on the minimal hyperbolic distance between points in ${ℰ}$. Our bound depends on a universal constant and the length of the smallest closed geodesic on $\mathrm{\Gamma }\backslash ℍ$.