scholarly journals Burgess-like subconvexity for

2019 ◽  
Vol 155 (8) ◽  
pp. 1457-1499 ◽  
Author(s):  
Han Wu
Keyword(s):  
Eisenstein Series ◽  
Number Field ◽  
Main Tool ◽  
Previous Method ◽  
Triple Product ◽  
Extended Theory ◽  
Product Formulas ◽  
Cuspidal Representations

We generalize our previous method on the subconvexity problem for $\text{GL}_{2}\times \text{GL}_{1}$ with cuspidal representations to Eisenstein series, and deduce a Burgess-like subconvex bound for Hecke characters, that is, the bound $|L(1/2,\unicode[STIX]{x1D712})|\ll _{\mathbf{F},\unicode[STIX]{x1D716}}\mathbf{C}(\unicode[STIX]{x1D712})^{1/4-(1-2\unicode[STIX]{x1D703})/16+\unicode[STIX]{x1D716}}$ for varying Hecke characters $\unicode[STIX]{x1D712}$ over a number field $\mathbf{F}$ with analytic conductor $\mathbf{C}(\unicode[STIX]{x1D712})$ . As a main tool, we apply the extended theory of regularized integrals due to Zagier developed in a previous paper to obtain the relevant triple product formulas of Eisenstein series.

scholarly journals On a Certain Function Analogous to log|η(z)|

1970 ◽  
Vol 40 ◽  
pp. 193-211 ◽  
Author(s):  
Tetsuya Asai
Keyword(s):  
Eisenstein Series ◽  
Number Field ◽  
Algebraic Number ◽  
Rational Number ◽  
Modular Group ◽  
Classical Case ◽  
Algebraic Number Field ◽  
View Point ◽  
Limit Formula

The purpose of this paper is to give the limit formula of the Kronecker’s type for a non-holomorphic Eisenstein series with respect to a Hubert modular group in the case of an arbitrary algebraic number field. Actually, we shall generalize the following result which is well-known as the first Kronecker’s limit formula. From our view-point, this classical case is corresponding to the case of the rational number field Q.

Local behavior of Iwasawa’s invariants

2017 ◽  
Vol 13 (04) ◽  
pp. 1013-1036 ◽  
Author(s):  
Sören Kleine
Keyword(s):  
Number Field ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Local Behavior ◽  
Locally Maximal

Let [Formula: see text] be a number field, let [Formula: see text] denote a fixed rational prime. We study the local behavior of Iwasawa’s invariants as functions on the set [Formula: see text] of all [Formula: see text]-extensions of [Formula: see text]. With respect to a certain topology on [Formula: see text] that takes care of ramification, we prove that for each [Formula: see text] the [Formula: see text]-invariant of [Formula: see text] is locally maximal among the [Formula: see text]-invariants, and we give sufficient conditions for the [Formula: see text]-invariant to be locally maximal (e.g., a vanishing [Formula: see text]-invariant). This concerns a question raised by R. Greenberg in 1973. Our main result also provides information about [Formula: see text]- (and even [Formula: see text]-) invariants in the case of a nonvanishing [Formula: see text]-invariant. The main tool used in the proof is a new result based on the stabilization of certain ranks, which considerably generalizes a theorem of T. Fukuda.

scholarly journals Residues of Eisenstein series and the automorphic cohomology of reductive groups

2013 ◽  
Vol 149 (7) ◽  
pp. 1061-1090 ◽  
Author(s):  
Harald Grobner
Keyword(s):  
Algebraic Group ◽  
Eisenstein Series ◽  
Number Field ◽  
Automorphic Forms ◽  
Reductive Groups ◽  
Algebraic Representation ◽  
Reductive Algebraic Group ◽  
Automorphic Representations ◽  
Residual Spectrum ◽  
Eisenstein Cohomology

AbstractLet $G$ be a connected, reductive algebraic group over a number field $F$ and let $E$ be an algebraic representation of ${G}_{\infty } $. In this paper we describe the Eisenstein cohomology ${ H}_{\mathrm{Eis} }^{q} (G, E)$ of $G$ below a certain degree ${q}_{ \mathsf{res} } $ in terms of Franke’s filtration of the space of automorphic forms. This entails a description of the map ${H}^{q} ({\mathfrak{m}}_{G} , K, \Pi \otimes E)\rightarrow { H}_{\mathrm{Eis} }^{q} (G, E)$, $q\lt {q}_{ \mathsf{res} } $, for all automorphic representations $\Pi $ of $G( \mathbb{A} )$ appearing in the residual spectrum. Moreover, we show that below an easily computable degree ${q}_{ \mathsf{max} } $, the space of Eisenstein cohomology ${ H}_{\mathrm{Eis} }^{q} (G, E)$ is isomorphic to the cohomology of the space of square-integrable, residual automorphic forms. We discuss some more consequences of our result and apply it, in order to derive a result on the residual Eisenstein cohomology of inner forms of ${\mathrm{GL} }_{n} $ and the split classical groups of type ${B}_{n} $, ${C}_{n} $, ${D}_{n} $.

Poles of Siegel Eisenstein Series on U(n, n)

1999 ◽  
Vol 51 (1) ◽  
pp. 164-175 ◽  
Author(s):  
Victor Tan
Keyword(s):  
Eisenstein Series ◽  
Number Field ◽  
Complex Plane ◽  
Unitary Group

AbstractLet U(n, n) be the rank n quasi-split unitary group over a number field. We show that the normalized Siegel Eisenstein series of U(n, n) has at most simple poles at the integers or half integers in certain strip of the complex plane.

A construction of residues of Eisenstein series and related square-integrable classes in the cohomology of arithmetic groups of low k-rank

2019 ◽  
Vol 31 (5) ◽  
pp. 1225-1263
Author(s):  
Neven Grbac ◽  
Joachim Schwermer
Keyword(s):  
Eisenstein Series ◽  
Number Field ◽  
Reductive Algebraic Group ◽  
Cohomology Of Arithmetic Groups ◽  
Geometric Conditions ◽  
Global Properties ◽  
Eisenstein Cohomology ◽  
Cuspidal Automorphic Representations ◽  
Totally Real ◽  
Low K

AbstractThe cohomology of an arithmetic congruence subgroup of a connected reductive algebraic group defined over a number field is captured in the automorphic cohomology of that group. The residual Eisenstein cohomology is by definition the part of the automorphic cohomology represented by square-integrable residues of Eisenstein series. The existence of residual Eisenstein cohomology classes depends on a subtle combination of geometric conditions (coming from cohomological reasons) and arithmetic conditions in terms of analytic properties of automorphic L-functions (coming from the study of poles of Eisenstein series). Hence, there are almost no unconditional results in the literature regarding the very existence of non-trivial residual Eisenstein cohomology classes. In this paper, we show the existence of certain non-trivial residual cohomology classes in the case of the split symplectic, and odd and even special orthogonal groups of rank two, as well as the exceptional group of type {\mathrm{G}_{2}}, defined over a totally real number field. The construction of cuspidal automorphic representations of {\mathrm{GL}_{2}} with prescribed local and global properties is decisive in this context.

Some Results on L-Indistinguishability for SL(r)

1983 ◽  
Vol 35 (6) ◽  
pp. 1075-1109 ◽  
Author(s):  
Freydoon Shahidi
Keyword(s):  
Positive Integer ◽  
Weyl Group ◽  
Recent Work ◽  
Eisenstein Series ◽  
Number Field ◽  
Parabolic Subgroup ◽  
Cusp Form ◽  
Constant Multiple ◽  
Levi Decomposition ◽  
Image Position

Fix a positive integer r. Let AF be the ring of adeles of a number field F. For a parabolic subgroup P of SLr, we fix a Levi decomposition P = MN, and we letLet be the Weyl group of . It follows from a recent work of James Arthur [1,2] (also cf. [3]) that, among the terms appearing in the trace formula for SLr(AF), coming from the Eisenstein series, are those which are a constant multiple (depending only on M and w) of1where σ is a cusp form on M(AF) satisfying wσ ≅ σ,and in the notation of [2, 3]).

A converse theorem for metaplectic Eisenstein series on the double and triple covers of SL2(ℚ(−D))

2016 ◽  
Vol 12 (06) ◽  
pp. 1625-1639
Author(s):  
Vladislav Petkov
Keyword(s):  
Eisenstein Series ◽  
Number Field ◽  
Quadratic Number ◽  
Converse Theorem ◽  
Quadratic Number Field ◽  
Converse Theorems ◽  
Double Dirichlet Series ◽  
Imaginary Quadratic Number ◽  
Metaplectic Cover ◽  
Integer Weight

In this work, we prove a converse theorem for metaplectic Eisenstein series on the [Formula: see text]th metaplectic cover of the group [Formula: see text], where [Formula: see text] is an imaginary quadratic number field containing the [Formula: see text]th roots of unity. This is an analog to previous converse theorems relating certain double Dirichlet series to the Mellin transforms of Eisenstein series of half-integer weight. We also propose a way to generalize this result to any number field.

CONSTRUCTION OF EISENSTEIN SERIES FOR Γ0(p)

2009 ◽  
Vol 05 (05) ◽  
pp. 765-778 ◽  
Author(s):  
SHAUN COOPER
Keyword(s):  
Eisenstein Series ◽  
Congruence Subgroup ◽  
Gauss Sums ◽  
Triple Product ◽  
Simple Construction ◽  
Dedekind Eta Function ◽  
Modular Transformation ◽  
Product Identity ◽  
Jacobi Triple Product Identity ◽  
Extra Difficulty

A simple construction of Eisenstein series for the congruence subgroup Γ0(p) is given. The construction makes use of the Jacobi triple product identity and Gauss sums, but does not use the modular transformation for the Dedekind eta-function. All positive integral weights are handled in the same way, and the conditionally convergent cases of weights 1 and 2 present no extra difficulty.

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