Related papers: Higher order group cohomology and the Eichler-Shim…
We construct bases for the spaces of higher order modular forms of all orders and weights. We also provide a cohomological interpretation of these forms.
Higher order cohomology of arithmetic groups is expressed in terms of (g,K)-cohomology. Generalizing results of Borel, it is shown that the latter can be computed using functions of (uniform) moderate growth. A higher order versions of…
The Eichler-Shimura isomorphism realizes the automorphic representation generated by an automorphic newform in certain cohomology of an arithmetic group. In this short note, we give a cohomological interpretation of the Eichler-Shimura…
When studying deformations of an $A$-module $M$, Laudal and Yau showed that one can consider 1-cocycles in the Hochschild cohomology of $A$ with coefficients in the bi-module $End_k(M).$ With this in mind, the use of higher order Hochschild…
In this note we give a generalization for the higher order Hochschild cohomology and show that the secondary Hochschild cohomology is a particular case of this new construction.
In the literature, the standard approach to finding bases of spaces of modular forms is via modular symbols and the homology of modular curves. By using the Eichler-Shimura isomorphism, a work by Wang shows how one can use a cohomological…
We know that coalgebra measurings behave like generalized maps between algebras. In this note, we show that coalgebra measurings between commutative algebras induce morphisms between higher order Hochschild homology groups of algebras. By…
Hochschild (co)homology and Pirashvili's higher order Hochschild (co)homology are useful tools for a variety of applications including deformations of algebras. When working with higher order Hochschild (co)homology, we can consider the…
We introduce and analyze a new geometric structure on topological surfaces generalizing the complex structure. To define this so called higher complex structure we use the punctual Hilbert scheme of the plane. The moduli space of higher…
Bayer and Blanco-Chac\'on have recently defined quadratic modular symbols for the Shimura curves X(D,N) attached to Eichler orders of level N of an indefinite quaternion rational algebra of discriminant D. In this paper, we give a…
We compute the dimensions of the Hochschild cohomology of symmetric groups over prime fields in low degrees. This involves us in studying some partition identities and generating functions of the dimensions in any fixed degree of the…
We formulate a detailed conjectural Eichler-Shimura type formula for the cohomology of local systems on a Picard modular surface associated to the group of unitary similitudes $\mathrm{GU}(2,1,\mathbb{Q}(\sqrt{-3}))$. The formula is based…
We define families of invariants for elements of the mapping class group of S, a compact orientable surface. Fix any characteristic subgroup H of pi_1(S) and restrict to J(H), any subgroup of mapping classes that induce the identity modulo…
The notion of a higher bundle gerbe is introduced to give a geometric realization of the higher degree integral cohomology of certain manifolds. We consider examples using the infinite dimensional spaces arising in gauge theories.
Eichler and Zagier developed a theory of Jacobi forms to understand and extend Maass' work on the Saito-Kurokawa conjecture. Later Skoruppa introduced skew-holomorphic Jacobi forms, which play an important role in understanding liftings of…
Let V be a representation space of a finite group G. We determine the group structure of the first homology of the equivariant diffeomorphism group of V. Then we can apply it to the calculation of the first homology of the corresponding…
For a group $G$, we construct a quasi morphism from its left orderings and the map from the space of left orderings to the second bounded cohomology. We show that these maps reflect various properties of the group orderings.
We introduce Hochschild (co-)homology of morphisms of schemes or analytic spaces and study its fundamental properties. In analogy with the cotangent complex we introduce the so called (derived) Hochschild complex of a morphism; the…
Explicit bases for the spaces of holomorphic cusp forms of all even positive weights and all orders are constructed. The dimensions of these spaces are computed.
We construct the cohomology groups with compact support of stacks of shtukas with $\mathbb Z_{\ell}$-coefficients. We construct the cuspidal cohomology groups and prove that they are $\mathbb Z_{\ell}$-modules of finite type. We prove that…