Siegel zeros of Eisenstein series

Let $G$ be a linear split algebraic group. The degenerate Eisenstein series associated to a maximal parabolic subgroup $E_{P}(f^{0},s,g)$ with the spherical section $f^{0}$ is studied in the first part of the thesis. In this part, we study the poles of $E_{P}(f^{0},s,g)$ in the region $\operatorname{Re} s >0$. We determine when the leading term in the Laurent expansion of $E_{P}(f^{0},s,g)$ around $s=s_0$ is square integrable. The second part is devoted to finding identitie...

We state and give complete proof of the results of Siegel about the zeros of the auxiliary function of Riemann $\mathop{\mathcal R}(s)$. We point out the importance of the determination of the limit to the left of the zeros of $\mathop{\mathcal R}(s)$ with positive imaginary part, obtaining the term $-\sqrt{T/2\pi}P(\sqrt{T/2\pi})$ that would explain the periodic behaviour observed with the statistical study of the zeros of $\mathop{\mathcal R}(s)$. We precise also the connec...

We give a general identity relating Eisenstein series on general linear groups. We do it by constructing an Eisenstein series, attached to a maximal parabolic subgroup and a pair of representations, one cuspidal and the other a character, and express it in terms of a degenerate Eisenstein series. In the local fields analogue, we prove the convergence in a half plane of the local integrals, and their meromorphic continuation. In addition, we find that the unramified calculatio...

Eisenstein series are ubiquitous in the theory of automorphic forms. The traditional proofs of the meromorphic continuation of Eisenstein series, due to Selberg and Langlands, start with cuspidal Eisenstein series as a special case, and deduce the general case from spectral theory. We present a "soft" proof which relies only on rudimentary Fredholm theory (needed only in the number field case). It is valid for Eisenstein series induced from an arbitrary automorphic form. ...

We locate all but $O(\sqrt{k\log{k}})$ zeroes of the half integral weight Eisenstein series $E_\infty(z,k)$ of $\Gamma_0(4)$ for $k$ sufficiently large. To do this, we relate $E_\infty(z,k)$ to $\Gamma_0(4)$'s other Eisenstein series, $E_0(z,k)$ and $E_\frac{1}{2}(z,k)$, which are easier to study in the regions of which zeroes occur. We will use trigonometric approximations of $E_0(z,k)$ and $E_\frac{1}{2}(z,k)$ in order to locate the zeroes.

Based on Garland's work, in this paper we construct the Eisenstein series on the adelic loop groups over a number field, induced from either a cusp form or a quasi-character which is assumed to be unramified. We compute the constant terms, prove their absolute and uniform convergence under the affine analog of Godement's criterion. For the case of quasi-characters the resulting formula is an affine Gindikin-Karpelevich formula. Then we prove the convergence of Eisenstein seri...

This manuscript has two goals: 1. To write an explicit description of the degenerate residual spectrum of the split, simple, simply-connected, exceptional groups of type $E_n$ (for $n=6,7,8$). 2. To set a practical guide for similar calculations and, in particular, to describe various methods of ``computational representation theory'' relevant to the study of residues of automorphic Eisenstein series. In Part I we supply background information and notations from the theory ...

We describe a plan how to prove an effective Siegel theorem (about the exceptional Dirichlet character). We give a brief outline in Section 0. We give a more detailed plan in Sections 1-5. The missing details (mostly routine elementary estimations) are in Part Two, which is very long. I am happy to send the pdf-file of Part Two to anybody who requests it by email.

The Fourier coefficients of the Siegel-Eisenstein series are p-adically continued for all primes p, as meromorphic functions, using the reciprocal of a product of L-functions. A construction of p-adic meromorphic families of such series is given in relation to the geometry of homogeneous spaces. Applications are given to p-adic L-functions, to Siegel's Mass Formula, to p-adic analytic families of automorphic representations. Based on author's talk for the Conference Automorph...

In this paper we consider the constant term $\phi_K(y,s)$ of the non-normalized Eisenstein series attached to $\PSL(2,\cO_K)$, where $K$ is either $\Q$ or an imaginary quadratic field of class number one. The main purpose of this paper is to show that for every $a\ge 1$ the zeros of the Dirichlet series $\phi_K(a,s)$ admit a spectral interpretation in terms of eigenvalues of a natural self-adjoint operator $\Delta_a$. This implies that, except for at most two real zeros, all ...