Related papers: Bianchi period polynomials: Hecke action and congr…

We reveal a correspondence between the homological torsion of the Bianchi groups and new geometric invariants, which are effectively computable thanks to their action on hyperbolic space. We use it to explicitly compute their integral group…

We prove multiplicity one for vector valued holomorphic Siegel modular forms of weights greater or equal to 3 and the full Siegel modular group and give a trace formula for the action of the Hecke operators T(p) in the regular cases.

Consider a connected reductive algebraic group $ G $ and a symmetric subgroup $ K $. Let $ \mathfrak{X} = K/B_K \times G/P $ be a double flag variety of finite type, where $ B_K $ is a Borel subgroup of $ K $, and $ P $ a parabolic subgroup…

We prove that Ramanujan-type congruences for integral weight modular forms away from the level and the congruence prime are equivalent to specific congruences for Hecke eigenvalues. In particular, we show that Ramanujan-type congruences are…

In this article we report on a surprising relation between the transfer operators for the congruence subgroups $\Gamma_{0}(n)$ and the Hecke operators on the space of period functions for the modular group $\PSL (2,\mathbb{Z})$. For this we…

We discuss several arithmetic aspects of Bianchi groups, especially from a computational point of view. In particular, we consider computing the homology of Bianchi groups together with the Hecke action, connections with automorphic forms,…

In this paper we define Banach spaces of overconvergent half-integral weight $p$-adic modular forms and Banach modules of families of overconvergent half-integral weight $p$-adic modular forms over admissible open subsets of weight space.…

We construct a basis for the space of half-integral weight Siegel Eisenstein series of level 4N where N is odd and square-free. Then we restrict our attention to those Eisenstein series generated from elements of $\Gamma_0(4)$, commenting…

We show that Ramanujan-type congruences are preserved by the action of the shallow Hecke algebra and provide several structure results for them. We discover a dichotomy between congruences originating in Hecke eigenvalues and congruences on…

Let $S_{k}(\Gamma_0(N),\chi)$ denote the space of holomorphic cuspforms with Dirichlet character $\chi$ and modular subgroup $\Gamma_0(N)$. We will characterize the space of newforms $S_{k}^{new}(\Gamma_0(N),\chi)$ as the intersection of…

For a bounded Hankel matrix $\Gamma$, we describe the structure of the Schmidt subspaces of $\Gamma$, namely the eigenspaces of $\Gamma^* \Gamma$ corresponding to non zero eigenvalues. We prove that these subspaces are in correspondence…

We consider actions of the current Lie algebras $\mathfrak{gl}_{n}[t]$ and $\mathfrak{gl}_{k}[t]$ on the space of polynomials in $kn$ anticommuting variables. The actions depend on parameters $\bar{z}=(z_{1}\dots z_{k})$ and…

This is a report on recent work, with Wen-Ching Winnie Li and Ling Long. In that work explicit formulas are given, involving hypergeometric character sums, for the traces of Hecke operators $T_p$ acting spaces of cusp forms $S_k(\Gamma)$ of…

We compute the intertwining relation between the Hecke operators and the Siegel lowering operators on Siegel modular forms of arbitrary level $N$ and character $\chi$ by using formulas for the action of the Hecke operators on Fourier…

This is a further investigation of our approach to group actions in homological algebra in the settings of homology of {\Gamma}-simplicial groups, particularly of {\Gamma}-equivariant homology and cohomology of {\Gamma}-groups. This…

We generalize Merel's work on universal Fourier expansions to Bianchi modular forms over Euclidean imaginary quadratic fields, under the assumption of the nondegeneracy of a pairing between Bianchi modular forms and Bianchi modular symbols.…

We study the biholomorphic action of the Heisenberg group $\mathbb{H}_n$ on the Siegel domain $D_{n+1}$ ($n \geq 1$). Such $\mathbb{H}_n$-action allows us to obtain decompositions of both $D_{n+1}$ and the weighted Bergman spaces…

We show how there is a natural action on the cohomology groups attached to certain subgroups of GL_n(F) of the Hecke operators defined as elements in an adelic double coset algebra. Our main result is, that if a system of eigenvalues for…

We give formulae for the Chen-Ruan orbifold cohomology for the orbifolds given by a Bianchi group acting on complex hyperbolic 3-space. The Bianchi groups are the arithmetic groups PSL\_2(A), where A is the ring of integers in an imaginary…

We show that a surjective homomorphism $\varphi \colon \Gamma \to K$ of (discrete) groups induces an isomorphism $H^\bullet_b(K; V) \to H^\bullet_b(\Gamma; \varphi^{-1} V)$ in bounded cohomology for all dual normed $K$-modules $V$ if and…