Shimura lift of real analytic Poincaré series and Hilbert modular Eisenstein series

1998 ◽  
Vol 229 (3) ◽  
pp. 547-574 ◽  
Author(s):  
Roland Matthes
Keyword(s):  
Eisenstein Series ◽  
Poincaré Series ◽  
Poincare Series ◽  
Shimura Lift ◽  
Hilbert Modular ◽  
Real Analytic

Theta Series, Eisenstein Series and Poincaré Series over Function Fields

2004 ◽  
Vol 56 (2) ◽  
pp. 406-430 ◽  
Author(s):  
Ambrus Pál
Keyword(s):  
Eisenstein Series ◽  
Finite Fields ◽  
Function Fields ◽  
Theta Series ◽  
Poincaré Series ◽  
Poincare Series ◽  
Basic Properties

AbstractWe construct analogues of theta series, Eisenstein series and Poincaré series for function fields of one variable over finite fields, and prove their basic properties.

scholarly journals EXACT FORMULAS FOR COEFFICIENTS OF JACOBI FORMS

2011 ◽  
Vol 07 (03) ◽  
pp. 825-833 ◽  
Author(s):  
KATHRIN BRINGMANN ◽  
OLAV K. RICHTER
Keyword(s):  
Fourier Coefficients ◽  
Theta Functions ◽  
Poincaré Series ◽  
Jacobi Forms ◽  
Mock Theta Functions ◽  
Explicit Formulas ◽  
Poincare Series ◽  
Real Analytic

In previous work, we introduced harmonic Maass–Jacobi forms. The space of such forms includes the classical Jacobi forms and certain Maass–Jacobi–Poincaré series, as well as Zwegers' real-analytic Jacobi forms, which play an important role in the study of mock theta functions and related objects. Harmonic Maass–Jacobi forms decompose naturally into holomorphic and non-holomorphic parts. In this paper, we give exact formulas for the Fourier coefficients of the holomorphic parts of harmonic Maass–Jacobi forms and, in particular, we obtain explicit formulas for the Fourier coefficients of weak Jacobi forms.

EISENSTEIN AND POINCARÉ SERIES ON SL(3,R)

2009 ◽  
Vol 05 (08) ◽  
pp. 1447-1475 ◽  
Author(s):  
LEON EHRENPREIS
Keyword(s):  
Fourier Transform ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Eisenstein Series ◽  
Algebraic Variety ◽  
Wave Operator ◽  
Fourier Transforms ◽  
Poincaré Series ◽  
Partial Differential ◽  
Poincare Series

This work continues the ideas presented in the author's book, The Universality of the Radon Transform (Oxford, 2003), which deals with the group SL(2,R). The complication that arises for G = SL(3,R) comes from the fact that there are now two fundamental representations. This has the consequence that the wave operator, which plays a central role in our work on SL(2,R) is replaced by an overdetermined system of partial differential equations. The analog of the wave operator is defined using an MN invariant orbit of G acting on the direct sum of the symmetric squares of the fundamental representations. The relation of orbits, or, in general, of any algebraic variety, to a system of partial differential equations comes via the Fundamental Principle, which shows how Fourier transforms of functions or measures on an algebraic variety correspond to solutions of the system of partial differential equations defined by the equations of the variety. In particular, we can start with the sum T of the delta functions of the orbit of the group Γ = SL(2,Z) on the light cone. We then take its Fourier transform, using a suitable quadratic form. We then decompose the Fourier transform under the commuting group of G. In this way, we obtain a Γ invariant distribution which has a natural restriction to the orbit G/K, which is the symmetric space of G. This restriction is (essentially) the nonanalytic Eisenstein series. We can compute the periods of the Eisenstein series over various orbits of subgroups of G by means of the Euclidean Planchere formula. A more complicated form of these ideas is needed to define Poincaré series.

scholarly journals Automorphic pseudodifferential operators, Poincaré series and Eisenstein series

1999 ◽  
Vol 59 (1) ◽  
pp. 45-52
Author(s):  
Min Ho Lee
Keyword(s):  
Eisenstein Series ◽  
Pseudodifferential Operators ◽  
Poincaré Series ◽  
Cusp Forms ◽  
Explicit Formulas ◽  
Poincare Series

We construct Poincaré series and Eisenstein series for automorphic pseudodifferential operators, and show that the space of automorphic pseudodifferential operators associated to cusp forms is generated by Poincaré series. We also obtain explicit formulas for such Poincaré series and Eisenstein series.

scholarly journals Eisenstein series and an asymptotic for the K-Bessel function

2021 ◽  
Author(s):  
Jimmy Tseng
Keyword(s):  
Asymptotic Expansion ◽  
Bessel Function ◽  
Eisenstein Series ◽  
Uniform Estimate ◽  
Positive Real ◽  
Dominant Term ◽  
Complex Order ◽  
Fixed Parameter ◽  
Real Argument ◽  
Real Analytic

AbstractWe produce an estimate for the K-Bessel function $$K_{r + i t}(y)$$ K r + i t ( y ) with positive, real argument y and of large complex order $$r+it$$ r + i t where r is bounded and $$t = y \sin \theta $$ t = y sin θ for a fixed parameter $$0\le \theta \le \pi /2$$ 0 ≤ θ ≤ π / 2 or $$t= y \cosh \mu $$ t = y cosh μ for a fixed parameter $$\mu >0$$ μ > 0 . In particular, we compute the dominant term of the asymptotic expansion of $$K_{r + i t}(y)$$ K r + i t ( y ) as $$y \rightarrow \infty $$ y → ∞ . When t and y are close (or equal), we also give a uniform estimate. As an application of these estimates, we give bounds on the weight-zero (real-analytic) Eisenstein series $$E_0^{(j)}(z, r+it)$$ E 0 ( j ) ( z , r + i t ) for each inequivalent cusp $$\kappa _j$$ κ j when $$1/2 \le r \le 3/2$$ 1 / 2 ≤ r ≤ 3 / 2 .

scholarly journals Poincaré series, 3d gravity and averages of rational CFT

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Viraj Meruliya ◽  
Sunil Mukhi ◽  
Palash Singh
Keyword(s):  
Poincaré Series ◽  
Simons Theory ◽  
Partition Functions ◽  
Moody Algebra ◽  
Modular Invariant ◽  
Poincare Series ◽  
Single Genus ◽  
3D Gravity ◽  
Chern Simons ◽  
Chern Simons Theory

Abstract We investigate the Poincaré series approach to computing 3d gravity partition functions dual to Rational CFT. For a single genus-1 boundary, we show that for certain infinite sets of levels, the SU(2)k WZW models provide unitary examples for which the Poincaré series is a positive linear combination of two modular-invariant partition functions. This supports the interpretation that the bulk gravity theory (a topological Chern-Simons theory in this case) is dual to an average of distinct CFT’s sharing the same Kac-Moody algebra. We compute the weights of this average for all seed primaries and all relevant values of k. We then study other WZW models, notably SU(N)1 and SU(3)k, and find that each class presents rather different features. Finally we consider multiple genus-1 boundaries, where we find a class of seed functions for the Poincaré sum that reproduces both disconnected and connected contributions — the latter corresponding to analogues of 3-manifold “wormholes” — such that the expected average is correctly reproduced.

Two Applications of Change of Rings Theorems for Poincare-Series

1979 ◽  
Vol 73 (2) ◽  
pp. 163
Author(s):  
Jurgen Herzog ◽  
Manfred Steurich
Keyword(s):  
Poincaré Series ◽  
Poincare Series

scholarly journals Poincaré series for filtrations defined by discrete valuations with arbitrary center

2013 ◽  
Vol 377 ◽  
pp. 66-75 ◽  
Author(s):  
Antonio Campillo ◽  
Ann Lemahieu
Keyword(s):  
Poincaré Series ◽  
Poincare Series
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