scholarly journals EISENSTEIN–KRONECKER SERIES VIA THE POINCARÉ BUNDLE

2019 ◽  
Vol 7 ◽  
Author(s):  
JOHANNES SPRANG
Keyword(s):  
Elliptic Curve ◽  
Eisenstein Series ◽  
Theta Functions ◽  
Algebraic Construction ◽  
Real Analytic ◽  
Classical Construction

A classical construction of Katz gives a purely algebraic construction of Eisenstein–Kronecker series using the Gauß–Manin connection on the universal elliptic curve. This approach gives a systematic way to study algebraic and $p$ -adic properties of real-analytic Eisenstein series. In the first part of this paper we provide an alternative algebraic construction of Eisenstein–Kronecker series via the Poincaré bundle. Building on this, we give in the second part a new conceptional construction of Katz’ two-variable $p$ -adic Eisenstein measure through $p$ -adic theta functions of the Poincaré bundle.

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 The Saito–Kurokawa lift and Siegel’s theta series

2017 ◽  
Vol 13 (07) ◽  
pp. 1679-1693
Author(s):  
Roland Matthes
Keyword(s):  
Spectral Analysis ◽  
Eisenstein Series ◽  
Short Proof ◽  
Theta Series ◽  
Converse Theorem ◽  
Selberg Integral ◽  
Real Analytic

The aim of this paper is to give another short proof of the Saito–Kurokawa lift based on a converse theorem of Imai as was already done by Duke and Imamoglu. In contrast to their proof we avoid spectral analysis but use a real analytic Eisenstein series in a suitable Rankin–Selberg integral involving Siegel’s theta series.

Weil Representation and Waldspurger Formula

2012 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-Wu Zhang ◽  
Wei Zhang
Keyword(s):  
Integral Representation ◽  
Eisenstein Series ◽  
Local Field ◽  
Integral Formula ◽  
Theta Functions ◽  
Direct Consequence ◽  
Weil Representation ◽  
Computational Results ◽  
Weil Representations ◽  
Central Value

This chapter reviews some basic results on Weil representations, theta liftings and Eisenstein series. In particular, it introduces a proof of the Waldspurger formula. The theory of Weil representation is applied to an integral representation of the Rankin–Selberg L-function and to a proof of Waldspurger's central value formula. The chapter mostly follows Waldspurger's treatment with some modifications including Kudla's construction of incoherent Eisenstein series. It first describes the classical theory of Weil representation for an orthogonal space over a local field before discussing theta functions, the Siegel–Weil formula, and normalized local Shimizu lifting. The main result is an integral formula for the L-series using a kernel function. The Waldspurger formula is a direct consequence of the Siegel–Weil formula. After presenting the proof of Waldspurger formula, the chapter lists some computational results on three types of incoherent Eisenstein series.

scholarly journals Eisenstein Series Identities Involving the Borweins' Cubic Theta Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Ernest X. W. Xia ◽  
Olivia X. M. Yao
Keyword(s):  
Eisenstein Series ◽  
Theta Functions ◽  
Elliptic Functions ◽  
The Other ◽  
Fundamental Theory ◽  
Convolution Sums

Based on the theories of Ramanujan's elliptic functions and the (p,k)-parametrization of theta functions due to Alaca et al. (2006, 2007, 2006) we derive certain Eisenstein series identities involving the Borweins' cubic theta functions with the help of the computer. Some of these identities were proved by Liu based on the fundamental theory of elliptic functions and some of them may be new. One side of each identity involves Eisenstein series, the other products of the Borweins' cubic theta functions. As applications, we evaluate some convolution sums. These evaluations are different from the formulas given by Alaca et al.

scholarly journals ON A CERTAIN CONVOLUTION OF POLYLOGARITHMS

2011 ◽  
Vol 86 (3) ◽  
pp. 461-472 ◽  
Author(s):  
HIROFUMI TSUMURA
Keyword(s):  
Zeta Function ◽  
Eisenstein Series ◽  
Riemann Zeta Function ◽  
Double Series ◽  
The Riemann Zeta Function ◽  
Sum Of Products ◽  
Riemann Zeta ◽  
Real Analytic ◽  
Double Integrals

AbstractIn this paper, we consider certain double series analogous to Tornheim’s double series and real analytic Eisenstein series. By computing double integrals in two ways, we express the double series as a sum of products of polylogarithms. The technique generalises one given by Kanemitsu, Tanigawa and Yoshimoto. Evaluating the double series at particular points gives new evaluations for certain double series in terms of values of the Riemann zeta function and the dilogarithm which are analogues of formulas of Mordell and Goncharov.

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