Related papers: On Constant Terms of Eisenstein Series

We prove a compatibility between parabolic restriction of Whittaker sheaves and restriction of representations under the geometric Casselman-Shalika equivalence. To do this, we establish various Hecke structures on geometric Eisenstein…

We study equivariant primitives of Eisenstein series for principal congruence subgroups and show that they are precisely the corresponding non-holomorphic Eisenstein series. We present closed formulas that naturally generalise existing…

The values of the so-called {\em Dedekind--Rademacher cocycle} at certain real quadratic arguments are shown to be global $p$-units in the narrow Hilbert class field of the associated real quadratic field, as predicted by conjectures of…

We show that every elliptic modular form of integral weight greater than $1$ can be expressed as linear combinations of products of at most two cusp expansions of Eisenstein series. This removes the obstruction of nonvanishing central…

Let F be a genus two Siegel newform and g a classical newform, both of squarefree levels and of equal weight l. We prove a pullback formula for certain Eisenstein series -- thus generalizing a construction of Shimura -- and use this to…

In a recent paper V. Vasyunin presented a proof of the reverse H\"older inequality with sharp constants for the weights satisfying the usual Muckenhoupt condition. In this paper we present the inverse, that is, we use the Bellman function…

We improve the constant $\frac{\pi}{2}$ in $L^1$-Poincar\'e inequality on Hamming cube. For Gaussian space the sharp constant in $L^1$ inequality is known, and it is $\sqrt{\frac{\pi}{2}}$. For Hamming cube the sharp constant is not known,…

By the theory of Eisenstein series, generating functions of various divisor functions arise as modular forms. It is natural to ask whether further divisor functions arise systematically in the theory of mock modular forms. We establish,…

In this paper we are concerned with the Bohnenblust--Hille type inequalities for certain polynomials of bounded degree but of very large number of variables. As the polynomials will be defined on groups, one can think about the problem as…

Let $F$ be a totally real field of degree $n$ and $p$ an odd prime. We prove the $p$-part of the integral Gross--Stark conjecture for the Brumer--Stark $p$-units living in CM abelian extensions of $F$. In previous work, the first author…

For a class of generalized holomorphic Eisenstein series, we establish complete asymptotic expansions (Theorems~1~and~2), which together with the explicit expression of the latter remainder (Theorem~3), naturally transfer to several new…

Let $p$ be an odd prime. We study the structure of the cyclotomic Greenberg-Selmer group attached to a general irreducible Artin motive over $\mathbb{Q}$ endowed with an ordinary $p$-stabilization. Under the Leopoldt and the weak $p$-adic…

We consider a sequence of composite bivariate Bernstein operators and the cubature formula associated with them. The upper bounds for the remainder term of the cubature formula are described in terms of moduli of continuity of order two.…

Using techniques from harmonic analysis, we derive several sharp stability estimates for the second order Heisenberg Uncertainty Principle. We also present the explicit lower and upper bounds for the sharp stability constants and compute…

As a natural generalization of the Euler-Mascheroni constant $\gamma$, Y. Ihara introduced the Euler-Kronecker constant $\gamma_K$ attached to any number field $K$. In this paper, we prove that a certain bound on $\gamma_K$ in a tower of…

For every point on the Jacobian of the modular curve $X_0(l)$ we define and study certain twisted holomorphic Eisenstein series. These are particular cases of a more general notion of twisted modular forms which correspond to sections on…

This paper is concerned with new results for the circular Eisenstein series $\varepsilon_r(z)$ as well as with a novel approach to Hilbert-Eisenstein series $\mathfrak h_r(z)$, introduced by Michael Hauss in 1995. The latter turn out to be…

We show that certain $p$-adic Eisenstein series for quaternionic modular groups of degree 2 become "real" modular forms of level $p$ in almost all cases. To prove this, we introduce a $U(p)$ type operator. We also show that there exists a…

In this paper, we prove the entirety of loop group Eisenstein series induced from cusp forms on the underlying finite dimensional group, by demonstrating their absolute convergence on the full complex plane. This is quite in contrast to the…

We show that indefinite theta series on cones converge and provide an explicit modular completion. Our completion rests on a convolution of the Gaussian with a piecewise constant function supported on the cone. Our main innovation is to…