On singular modular forms for principal congruence subgroups

We show that spaces of vector–valued singular modular forms for principal congruence subgroups of the symplectic group Sp(n,ℤ) of integral weight are generated by suitable finite dimensional families of Siegel theta series. This is obtained as an application of some results concerning the action of trace operators on non–homogeneous theta series.     

Cohomology of certain congruence subgroups of the modular group

In this note we compute the integral cohomology groups of the subgroups of and the corresponding subgroups of its quotient, the classical modular group,

     

Basic subgroups in modular abelian group algebras

Suppose F is a perfect field of char F = p ≠ 0 and G is an arbitrary abelian multiplicative group with a p-basic subgroup B and p-component G _p . Let FG be the group algebra with normed group of all units V(FG) and its Sylow p-subgroup S(FG), and let I _p (FG; B) be the nilradical of the relative augmentation ideal I(FG; B) of FG with respect to B. The main results that motivate this article are that 1 + I _p (FG; B) is basic in S(FG), and B(1 + I _p (FG; B)) is p-basic in V(FG) provided G is p-mixed. These achievements extend in some way a result of N. Nachev (1996) in Houston J. Math. when G is p-primary. Thus the problem of obtaining a (p-)basic subgroup in FG is completely resolved provided that the field F is perfect. Moreover, it is shown that G _p (1 + I _p (FG; B))/G _p is basic in S(FG)/G _p , and G(1 + I _p (FG; B))/G is basic in V(FG)/G provided G is p-mixed. As consequences, S(FG) and S(FG)/G _p are both starred or divisible groups. All of the listed assertions enlarge in a new aspect affirmations established by us in Czechoslovak Math. J. (2002), Math. Bohemica (2004) and Math. Slovaca (2005) as well.     

On the some normal subgroups of the extended modular group

In this paper, we give the group structures and the signatures of some normal subgroups of the extended modular group Π containing the principal congruence subgroup Γ(12).     

Construction of free subgroups in the group of units of modular group algebras

Let KGbe the group algebra of a p¹ -group Gover a field Kof characteristic p > 0, and let U(KG)be its group of units. If KGcontains a nontrivial bicyclic unit and if Kis not algebraic over its prime field, then we prove that the free product Z_p? Z_p? Z_pcan be embedded in U(KG).     

Automorphic forms for elliptic function fields

Let F be the function field of an elliptic curve X over ${\mathbb{F}_q}$ . In this paper, we calculate explicit formulas for unramified Hecke operators acting on automorphic forms over F. We determine these formulas in the language of the graph of an Hecke operator, for which we use its interpretation in terms of ${\mathbb{P}^1}$ -bundles on X. This allows a purely geometric approach, which involves, amongst others, a classification of the ${\mathbb{P}^1}$ -bundles on X. We apply the computed formulas to calculate the dimension of the space of unramified cusp forms and the support of a cusp form. We show that a cuspidal Hecke eigenform does not vanish in the trivial ${\mathbb{P}^1}$ -bundle. Further, we determine the space of unramified F′-toroidal automorphic forms where F′ is the quadratic constant field extension of F. It does not contain non-trivial cusp forms. An investigation of zeros of certain Hecke L-series leads to the conclusion that the space of unramified toroidal automorphic forms is spanned by the Eisenstein series E( · , s) where s?+?1/2 is a zero of the zeta function of X—with one possible exception in the case that q is even and the class number h equals q?+?1.     

Zeta dimension formula for picard modular cusp forms of neat natural congruence subgroups

Let be the full Picard modular group of the imaginary quadratic number fieldK. For all natural congruence subgroups Γ_k (m), m ≥ 3, acting freely on the two-dimensional complex unit ball, we prove an explicit polynomial formula for the dimensions of spaces of cusp forms of weightn ≥ 2. The coefficients of these polynomials in the natural variablesm, n are expressed by higher third and first Bernoulli numbers of the Dirichlet character χk ^{of K} and by values of Euler factors of the Riemann Zeta function and such factors of theL-series of χk^{at 2 or 3}, respectively. The proof is based on detailed knowledge about classification of Picard modular surfaces. It combines algebraic geometric methods (Riemann-Roch, Vanishing- and Proportionality Theorem, curvature, structure of algebraic groups) with modern and classical number theoretic ones (representation densities, Tamagawa measure, strong approximation, functional equation forL-series).     

On the number of subgroups of given index in the modular group

A new recurrence for the number of subgroups of given index in the modular group is derived. The proof requires the derivation of a recurrence for a sequencea _nb_n from recurrences for thea _n andb _n. We show that this is always possible if thea _n andb _n satisfy polynomial recurrences. We also include a short proof of a result ofW. W. Stothers on the parity of the number of subgroups of given index in the modular group.     

Maximal divisible subgroups in modular group rings of p-mixed abelian groups

The isomorphism structure of the maximal divisible subgroup of the subgroup V _p (R(G); H) Id R(G) of the normalized unit group V R(G) in a commutative group ring R(G) is completely described only in terms of R, G and H whenever R is a commutative unital ring of prime characteristic p and G is a p-mixed abelian group. In particular, the maximal divisible subgroup of V R(G) is characterized. This extends a result due to Nachev (Commun. Algebra, 1995) as well as a result due to the author (Commun. Algebra, 2010).