Related papers: Eisenstein series identities based on partial frac…
We carry out "Hecke summation" for the classical Eisenstein series $E_k$ in an adelic setting. The connection between classical and adelic functions is made by explicit calculations of local and global intertwining operators and Whittaker…
In this paper, a transformation formula under modular substitutions is derived for a large class of generalized Eisenstein series. Appearing in the transformation formulae are generalizations of Dedekind sums involving the periodic…
Borisov and Gunnells have proved that certain linear combinations of products of Eisenstein series are Eisenstein series themselves, in analogy with the Manin relations for modular symbols. We devise a new method for determining and proving…
In this paper we consider weakly holomorphic modular forms (i.e. those meromorphic modular forms for which poles only possibly occur at the cusps) of weight $2-k\in 2\Z$ for the full modular group $\SL_2(\Z)$. The space has a distinguished…
We study the double shuffle relations satisfied by the double zeta values of level 2, and introduce the double Eisenstein series of level 2 which satisfy the double shuffle relations. We connect the double Eisenstein series to modular forms…
In this paper we deal with Drinfeld modular forms, defined and taking values in complete fields of positive characteristic. Our aim is to study a sequence of families of Drinfeld modular forms depending on a parameter t that produces, for…
We introduce the formal double Eisenstein space $\mathcal{E}_k$, which is a generalization of the formal double zeta space $\mathcal{D}_k$ of Gangl-Kaneko-Zagier, and prove analogues of the sum formula and parity result for formal double…
In this paper we study iterated Eisenstein {\tau}-integrals and multiple Eisenstein L-series, they are functions on the complex upper half plane and form two Q-algebras. They reduce to iterated Eisenstein integrals and multiple Hecke…
We study the $\mthbb{Q}$-vector space generated by the double zeta values with character of conductor $4$. For this purpose, we define associated double Eisenstein series and investigate their relation with modular forms of level $4$.
Using results from the theory of modular forms, we reprove and extend a result of Romik about lacunary recurrence relations for Eisenstein series.
We say a power series $a_0+a_1q+a_2q^2+\cdots$ is \emph{multiplicative} if $n\mapsto a_n/a_1$ for positive integers $n$ is a multiplicative function. Given the Eisenstein series $E_{2k}(q)$, we consider formal multiplicative power series…
The weight two Eisenstein series may be considered as the first example of a Katz $p$-adic modular form. Classically, its values are defined for the primes of ordinary reduction. We offer a modified definition which applies uniformly to all…
In this paper we consider certain classes of generalized double Eisenstein series by simple differential calculations of trigonometric functions. In particular, we give four new transformation formula for some double Eisenstein series. We…
Consider $k\ge 2$ distinct, linearly independent, homogeneous linear recurrences of order $k$ satisfying the same recurrence relation. We prove that the recurrences are related to a decomposable form of degree $k$, and there is a very broad…
We give a formula for the Eisenstein cohomology of local systems on the partial compactification of the moduli of principally polarized abelian varieties given by rank 1 degenerations. For genus 2 we give a formula for the full Eisenstein…
We consider logarithmic vector- and matrix-valued modular forms of integral weight $k$ associated with a $p$-dimensional representation $\rho: SL_2(\mathbb{Z}) \to GL_p(\mathbb{C})$ of the modular group, subject only to the condition that…
We give simple explicit formulas for deformation quantization of Poisson-Lie groups and of similar Poisson manifolds which can be represented as moduli spaces of flat connections on surfaces. The star products depend on a choice of…
This article is a research exposition based on the author's talk at the International Colloquium on Automorphic Representations and L-Functions, 2012, held at TIFR, Mumbai. We consider some special cases of the following question: when is a…
MacMahon's partition functions and their extensions provide equations that identify prime numbers as solutions. These results depend on the theory of (mixed weight) quasimodular forms on $SL_2(\mathbb{Z})$. Two of the authors, along with…
Analytic continuation and functional equation of a Dirichlet series constructed from two (not necessarily cuspidal) holomorphic modular forms is discussed, where either weights of the modular forms or characters are not necessarily equal to…