The first author and Bump defined Schubert Eisenstein series by restricting the summation in a degenerate Eisenstein series to a particular Schubert variety. In the case of $mathrm{GL}_3$ over $mathbb{Q}$ they proved that these Schubert Eisenstein series have meromorphic continuations in all parameters and conjectured the same is true in general. We revisit their conjecture and relate it to the program of Braverman, Kazhdan, Lafforgue, Ng^o, and Sakellaridis aimed at establishing generalizations of the Poisson summation formula. We prove the Poisson summation formula for certain schemes closely related to Schubert varieties and use it to refine and establish the conjecture of the first author and Bump in many cases.
Generalizing a result of cite{Z1991, CPZ} about elliptic modular forms, we give a closed formula for the sum of all Hilbert Hecke eigenforms over a totally real number field with strict class number $1$, multiplied by their period polynomials, as a single product of the Kronecker series.
A generalized Riemann hypothesis states that all zeros of the completed Hecke $L$-function $L^*(f,s)$ of a normalized Hecke eigenform $f$ on the full modular group should lie on the vertical line $Re(s)=frac{k}{2}.$ It was shown by Kohnen that there exists a Hecke eigenform $f$ of weight $k$ such that $L^*(f,s) eq 0$ for sufficiently large $k$ and any point on the line segments $Im(s)=t_0, frac{k-1}{2} < Re(s) < frac{k}{2}-epsilon, frac{k }{2}+epsilon < Re(s) < frac{k+1}{2},$ for any given real number $t_0$ and a positive real number $epsilon.$ This paper concerns the non-vanishing of the product $L^*(f,s)L^*(f,w)$ $(s,win mathbb{C})$ on average.
We extend to positive real weights Haberlands formula giving a cohomological description of the Petersson scalar product of modular cusp forms of positive even weight. This relation is based on the cup product of an Eichler cocycle and a Knopp cocycle. We also consider the cup product of two Eichler cocycles attached to modular forms. In the classical context of integral weights at least $2$ this cup product is uninteresting. We show evidence that for real weights this cup product may very well be non-trivial. We approach the question whether the cup product is a non-trivial coinvariant by duality with a space of entire modular forms. Under suitable conditions on the weights this leads to an explicit triple integral involving three modular forms. We use this representation to study the cup product numerically.
This work is devoted to the algebraic and arithmetic properties of Rankin-Cohen brackets allowing to define and study them in several natural situations of number theory. It focuses on the property of these brackets to be formal deformations of the algebras on which they are defined, with related questions on restriction-extension methods. The general algebraic results developed here are applied to the study of formal deformations of the algebra of weak Jacobi forms and their relation with the Rankin-Cohen brackets on modular and quasimodular forms.
We investigate non-vanishing properties of $L(f,s)$ on the real line, when $f$ is a Hecke eigenform of half-integral weight $k+{1over 2}$ on $Gamma_0(4).$
This work gives a version of the Eichler-Shimura isomorphism with a non-abelian $H^1$ in group cohomology. Manin has given a map from vectors of cusp forms to a noncommutative cohomology set by means of iterated integrals. We show Manins map is injective but far from surjective. By extending Manins map we are able to construct a bijective map and remarkably this establishes the existence of a non-abelian version of the Eichler-Shimura map.
We investigate the correspondence between holomorphic automorphic forms on the upper half-plane with complex weight and parabolic cocycles. For integral weights at least 2 this correspondence is given by the Eichler integral. Knopp generalized this to real weights. We show that for weights that are not an integer at least 2 the generalized Eichler integral gives an injection into the first cohomology group with values in a module of holomorphic functions, and characterize the image. We impose no condition on the growth of the automorphic forms at the cusps. For real weights that are not an integer at least 2 we similarly characterize the space of cusp forms and the space of entire automorphic forms. We give a relation between the cohomology classes attached to holomorphic automorphic forms of real weight and the existence of harmonic lifts. A tool in establishing these results is the relation to cohomology groups with values in modules of analytic boundary germs, which are represented by harmonic functions on subsets of the upper half-plane. Even for positive integral weights cohomology with these coefficients can distinguish all holomorphic automorphic forms, unlike the classical Eichler theory.
We give an explicit upper bound for the first sign change of the Fourier coefficients of an arbitrary non-zero Siegel cusp form $F$ of even integral weight on the Siegel modular group of arbitrary genus $ ggeq 2 $.
We first survey the known results on functional equations for the double zeta-function of Euler type and its various generalizations. Then we prove two new functional equations for double series of Euler-Hurwitz-Barnes type with complex coefficients. The first one is of general nature, while the second one is valid when the coefficients are Fourier coefficients of a cusp form.
