Narrow Results

In this paper, we give a description of the generators of the prime level congruence subgroups of braid groups. Also, we give a new presentation of the symplectic group over a finite field, and we calculate the symmetric quotients of the prime level congruence subgroups of braid groups. Finally, we find a finite generating set for the level-3 congruence subgroup of the braid group on three strands.

After Zagier proved that the traces of singular moduli are Fourier coefficients of a weakly holomorphic modular form, various arithmetic properties of the traces of singular values of modular functions mostly on the full modular group have been found. The purpose of this paper is to generalize the results for modular functions on congruence subgroups with arbitrary level.

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.

It is known that there is a one-to-one correspondence between equivalence classes of primitive indefinite binary quadratic forms and primitive hyperbolic conjugacy classes of the modular group. Due to such a correspondence, Sarnak obtained the asymptotic formula for the class number sum in order of the fundamental unit by using the prime geodesic theorem for the modular group. In the present paper, we propose asymptotic formulas of the class number sums over discriminants on arithmetic progressions. Since there are relations between the arithmetic properties of the discriminants and the conjugacy classes in the finite groups given by the modular group and its congruence subgroups, we can get the desired asymptotic formulas by arranging the Tchebotarev-type prime geodesic theorem. While such asymptotic formulas were already given by Raulf, the approaches are quite different, the expressions of the leading terms of our asymptotic formulas are simpler and the estimates of the remainder terms are sharper.

We determine the normalizer in SL₂(ℝ) of several families of congruence subgroups of SL₂(ℤ). In addition, we show how these tools can be used to evaluate the groups of automorphisms and the discriminant kernels of a large family of isotropic lattices of signature (2,1).

Deligne proved in [Extensions centrales non résiduellement finies de groupes arithmetiques, C. R. Acad. Sci. Paris 287 (1978) 203–208] (see also 7.1 in [R. Hill, Fractional weights and non-congruence subgroups, in Automorphic Forms and Representations of Algebraic Groups Over Local Fields, eds. H. Saito and T. Takahashi, Surikenkoukyuroku Series, Vol. 1338 (2003), pp. 71–80]) that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus $g>1$ must be integral or half integral. Actually he proved that for a system $v(M)$ of complex numbers of absolute value 1

can be an automorphy factor only if $2r$ is integral. We give a different proof for this. It uses Mennicke’s result [Zur Theorie der Siegelschen Modulgruppe, Math. Ann. 159 (1965) 115–129] that subgroups of finite index of the Siegel modular group are congruence subgroups and some techniques from [Solution of the congruence subgroup problem for ${\mathrm{\text{SL}}}_n(n\ge 3)$ and ${\mathrm{\text{Sp}}}_{2n}(n\ge 2)$, Publ. Math. Inst. Hautes Études Sci. 33 (1967) 59–137] of Bass–Milnor–Serre.