ON THE THETA SERIES ATTACHED TO $D_{m}^{+}$–LATTICES

2006 ◽  
Vol 02 (01) ◽  
pp. 1-5 ◽  
Author(s):  
WINFRIED KOHNEN ◽  
RICCARDO SALVATI MANNI
Keyword(s):  
Positive Integer ◽  
Linear Combination ◽  
Eisenstein Series ◽  
Theta Series

We show that the theta series attached to the [Formula: see text]-lattice for any positive integer divisible by 8 can be explicitly expressed as a finite rational linear combination of products of two Eisenstein series.

EVALUATION OF CONVOLUTION SUMS AND FOR k = a · b = 21, 33, AND 35

2021 ◽  
pp. 1-20
Author(s):  
K. PUSHPA ◽  
K. R. VASUKI
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Divisor Function ◽  
Convolution Sums

Abstract The article focuses on the evaluation of convolution sums $${W_k}(n): = \mathop \sum \nolimits_{_{m < {n \over k}}} \sigma (m)\sigma (n - km)$$ involving the sum of divisor function $$\sigma (n)$$ for k =21, 33, and 35. In this article, our aim is to obtain certain Eisenstein series of level 21 and use them to evaluate the convolution sums for level 21. We also make use of the existing Eisenstein series identities for level 33 and 35 in evaluating the convolution sums for level 33 and 35. Most of the convolution sums were evaluated using the theory of modular forms, whereas we have devised a technique which is free from the theory of modular forms. As an application, we determine a formula for the number of representations of a positive integer n by the octonary quadratic form $$(x_1^2 + {x_1}{x_2} + ax_2^2 + x_3^2 + {x_3}{x_4} + ax_4^2) + b(x_5^2 + {x_5}{x_6} + ax_6^2 + x_7^2 + {x_7}{x_8} + ax_8^2)$$ , for (a, b)=(1, 7), (1, 11), (2, 3), and (2, 5).

scholarly journals -ADIC EISENSTEIN SERIES AND -FUNCTIONS OF CERTAIN CUSP FORMS ON DEFINITE UNITARY GROUPS

2014 ◽  
Vol 15 (3) ◽  
pp. 471-510 ◽  
Author(s):  
Ellen Eischen ◽  
Xin Wan
Keyword(s):  
Positive Integer ◽  
Eisenstein Series ◽  
Constant Term ◽  
Unitary Groups ◽  
Cusp Forms ◽  
Imaginary Field

We construct$p$-adic families of Klingen–Eisenstein series and$L$-functions for cusp forms (not necessarily ordinary) unramified at an odd prime$p$on definite unitary groups of signature$(r,0)$(for any positive integer$r$) for a quadratic imaginary field${\mathcal{K}}$split at$p$. When$r=2$, we show that the constant term of the Klingen–Eisenstein family is divisible by a certain$p$-adic$L$-function.

scholarly journals ACTION OF HECKE OPERATORS ON SIEGEL THETA SERIES I

2006 ◽  
Vol 02 (02) ◽  
pp. 169-186 ◽  
Author(s):  
LYNNE H. WALLING
Keyword(s):  
Quadratic Form ◽  
Linear Combination ◽  
Commutation Relation ◽  
Theta Series ◽  
Positive Definite ◽  
Hecke Operators ◽  
Elementary Functions ◽  
Positive Definite Quadratic Form

We apply the Hecke operators T(p) and [Formula: see text] to a degree n theta series attached to a rank 2k ℤ-lattice L, n ≤ k, equipped with a positive definite quadratic form in the case that L/pL is hyperbolic. We show that the image of the theta series under these Hecke operators can be realized as a sum of theta series attached to certain closely related lattices, thereby generalizing the Eichler Commutation Relation (similar to some work of Freitag and of Yoshida). We then show that the average theta series (averaging over isometry classes in a given genus) is an eigenform for these operators. We show the eigenvalue for T(p) is ∊(k - n, n), and the eigenvalue for T′j(p2) (a specific linear combination of T0(p2),…,Tj(p2)) is pj(k-n)+j(j-1)/2β(n,j)∊(k-j,j) where β(*,*), ∊(*,*) are elementary functions (defined below).

On the number of representations of a positive integer as a sum of two binary quadratic forms

2014 ◽  
Vol 10 (06) ◽  
pp. 1395-1420 ◽  
Author(s):  
Şaban Alaca ◽  
Lerna Pehlivan ◽  
Kenneth S. Williams
Keyword(s):  
Positive Integer ◽  
Linear Combination ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Definite Integral ◽  
Binary Quadratic Forms ◽  
Finite Linear Combination ◽  
Positive Integers ◽  
Finite Linear

Let ℕ denote the set of positive integers and ℤ the set of all integers. Let ℕ0 = ℕ ∪{0}. Let a1x2 + b1xy + c1y2 and a2z2 + b2zt + c2t2 be two positive-definite, integral, binary quadratic forms. The number of representations of n ∈ ℕ0 as a sum of these two binary quadratic forms is [Formula: see text] When (b1, b2) ≠ (0, 0) we prove under certain conditions on a1, b1, c1, a2, b2 and c2 that N(a1, b1, c1, a2, b2, c2; n) can be expressed as a finite linear combination of quantities of the type N(a, 0, b, c, 0, d; n) with a, b, c and d positive integers. Thus, when the quantities N(a, 0, b, c, 0, d; n) are known, we can determine N(a1, b1, c1, a2, b2, c2; n). This determination is carried out explicitly for a number of quaternary quadratic forms a1x2 + b1xy + c1y2 + a2z2 + b2zt + c2t2. For example, in Theorem 1.2 we show for n ∈ ℕ that [Formula: see text] where N is the largest odd integer dividing n and [Formula: see text]

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.

Trace of the Generating Series

2012 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-Wu Zhang ◽  
Wei Zhang
Keyword(s):  
Eisenstein Series ◽  
Abelian Varieties ◽  
Discrete Series ◽  
Theta Series ◽  
Shimura Curves ◽  
Trace Identity ◽  
Shimura Curve ◽  
Schwartz Functions ◽  
Generating Series ◽  
New Space

This chapter proves the theorem that asserts the modularity of the generating series and the theorem dealing with abelian varieties parametrized by Shimura curves. Before presenting the proofs, the chapter considers the new space of Schwartz functions and constructs theta series and Eisenstein series from such functions. It proceeds by discussing discrete series at infinite places, modularity of the generating series, degree of the generating series, and the trace identity. It also presents the pull-back formula for the compact and non-compact cases. In particular, it describes CM cycles on the Shimura curve, pull-back as cycles, degree of the pull-back, and some coset identities.

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 On integers which are representable as sums of large squares

2015 ◽  
Vol 11 (08) ◽  
pp. 2505-2511 ◽  
Author(s):  
Alessio Moscariello
Keyword(s):  
Positive Integer ◽  
Linear Combination ◽  
Integer Coefficients ◽  
Representation Of Integers

We prove that the greatest positive integer that is not expressible as a linear combination with integer coefficients of elements of the set {n2, (n + 1)2, …} is asymptotically O(n2), verifying thus a conjecture of Dutch and Rickett. Furthermore, we ask a question on the representation of integers as sum of four large squares.

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