Related papers: Eisenstein series for the Weil representation

The vector valued theta series of a positive-definite even lattice is a modular form for the Weil representation of $\mathrm{SL}_2(\mathbb{Z})$. We show that the space of cusp forms for the Weil representation is generated by such…

It is known that average Siegel theta series lie in the space of Siegel Eisenstein series. Also, every lattice equipped with an even integral quadratic form lies in a maximal lattice. Here we consider average Siegel theta series of degree 2…

We develop the theory of Weyl group multiple Dirichlet series for root systems of type C. For an arbitrary root system of rank r and a positive integer n, these are Dirichlet series in r complex variables with analytic continuation and…

We describe connections between the Fourier coefficients of derivatives of Eisenstein series and invariants from the arithmetic geometry of the Shimura varieties $M$ associated to rational quadratic forms $(V,Q)$ of signature $(n,2)$. In…

We compute the Fourier coefficients of analogues of Kohnen and Zagier's modular forms $f_{k,D}$ of weight $2$ and negative discriminant. These functions can also be written as twisted traces of certain weight $2$ Poincar\'e series with…

This paper presents a very simple explicit description of Langlands Eisenstein series for ${\rm SL}(n,\mathbb Z)$. The functional equations of these Eisenstein series are heuristically derived from the functional equations of certain…

We consider modular functions (i.e., the Eisenstein series and Hecke-Maass forms) for the group PSL(2,Z). We fix a quadratic number field E. This gives rise to twisted (by a Hecke character of the field E) periods of a modular function…

We consider Fuchsian singularities of arbitrary genus and prove, in a conceptual manner, a formula for their Poincar\'e series. This uses Coxeter elements involving Eichler-Siegel transformations. We give geometrical interpretations for the…

This is the third part of a series of articles providing a foundation for the theory of Drinfeld modular forms of arbitrary rank. In the present article we construct and study some examples of Drinfeld modular forms. In particular we define…

We determine the unipotent orbits attached to degenerate Eisenstein series on general linear groups. This confirms a conjecture of David Ginzburg. This also shows that any unipotent orbit of general linear groups does occur as the unipotent…

We give all possible holomorphic Eisenstein series on $\Gamma_0(p)$, of rational weights greater than $2$, and with multiplier systems the same as certain rational-weight eta-quotients at all cusps. We prove they are modular forms and give…

New expressions are given for the Fourier expansions of non-holomorphic Eisenstein series with weight $k$. Among other applications, this leads to non-holomorphic analogs of formulas of Ramanujan, Grosswald and Berndt containing Eichler…

We study Whittaker coefficients for maximal parabolic Eisenstein series on metaplectic covers of split reductive groups. By the theory of Eisenstein series these coefficients have meromorphic continuation and functional equation. However…

We aim to provide a family of Drinfeld-Hecke eigenforms given in terms of a determinant of twisted Eisenstein series. Our main tool is the theory of vectorial Drinfeld modular forms, previously introduced by Pellarin [18] and extensively…

In this paper we give a classification of the asymptotic expansion of the $q$-expansion of reciprocals of Eisenstein series $E_k$ of weight $k$ for the modular group $\func{SL}_2(\mathbb{Z})$. For $k \geq 12$ even, this extends results of…

Let $k \geq 2$ and $N$ be positive integers and let $\chi$ be a Dirichlet character modulo $N$. Let $f(z)$ be a modular form in $M_k(\Gamma_0(N),\chi)$. Then we have a unique decomposition $f(z)=E_f(z)+S_f(z)$, where $E_f(z) \in…

Fourier coefficients of Eisenstein series figure prominently in the study of automorphic L-functions via the Langlands-Shahidi method, and in various other aspects of the theory of automorphic forms and representations. In this paper, we…

We investigate a Dirichlet series involving the Fourier-Jacobi coefficients of two cusp forms $F,G$ for orthogonal groups of signature $(2,n+2)$. In the case when $F$ is a Hecke eigenform and $G$ is a Maass lift of a Poincar\'e series, we…

In this paper we treat the Fourier coefficients of Siegel Eisenstein series of level $p$ with trivial or quadratic character, for an odd prime $p$. The Euler $p$-factor of the Fourier coefficient is called the ramified Siegel series. First…

Using general principles in the theory of vertex operator algebras and their twisted modules, we obtain a bosonic, twisted construction of a certain central extension of a Lie algebra of differential operators on the circle, for an…