scholarly journals Siegel theta series for indefinite quadratic forms

2021 ◽  
Vol 7 (3) ◽  
Author(s):  
Christina Roehrig
Keyword(s):  
Differential Equation ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Theta Series ◽  
Second Order ◽  
Transformation Behavior ◽  
Modular Transformation ◽  
Indefinite Quadratic

AbstractThe modular transformation behavior of theta series for indefinite quadratic forms is well understood in the case of elliptic modular forms due to Vignéras, who deduced that solving a differential equation of second order serves as a criterion for modularity. In this paper, we will give a generalization of this result to Siegel theta series.

scholarly journals Shimura correspondence for level p2 and the central values of L-series, II

2014 ◽  
Vol 10 (07) ◽  
pp. 1595-1635
Author(s):  
Ariel Pacetti ◽  
Gonzalo Tornaría
Keyword(s):  
Linear Combination ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Theta Series ◽  
Geometric Interpretation ◽  
Integral Weight ◽  
Free Level ◽  
Hecke Eigenform ◽  
Special Points ◽  
Shimura Correspondence

Given a Hecke eigenform f of weight 2 and square-free level N, by the work of Kohnen, there is a unique weight 3/2 modular form of level 4N mapping to f under the Shimura correspondence. Furthermore, by the work of Waldspurger the Fourier coefficients of such a form are related to the quadratic twists of the form f. Gross gave a construction of the half integral weight form when N is prime, and such construction was later generalized to square-free levels. However, in the non-square free case, the situation is more complicated since the natural construction is vacuous. The problem being that there are too many special points so that there is cancellation while trying to encode the information as a linear combination of theta series. In this paper, we concentrate in the case of level p2, for p > 2 a prime number, and show how the set of special points can be split into subsets (indexed by bilateral ideals for an order of reduced discriminant p2) which gives two weight 3/2 modular forms mapping to f under the Shimura correspondence. Moreover, the splitting has a geometric interpretation which allows to prove that the forms are indeed a linear combination of theta series associated to ternary quadratic forms. Once such interpretation is given, we extend the method of Gross–Zagier to the case where the level and the discriminant are not prime to each other to prove a Gross-type formula in this situation.

scholarly journals Theta-functions and Hilbert modular forms

1978 ◽  
Vol 69 ◽  
pp. 97-106 ◽  
Author(s):  
Stephen S. Kudla
Keyword(s):  
Theta Function ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Integral Kernel ◽  
Theta Functions ◽  
Cusp Forms ◽  
Hilbert Modular Forms ◽  
Hilbert Modular ◽  
History Of ◽  
Indefinite Quadratic

The purpose of this note is to show how the theta-functions attached to certain indefinite quadratic forms of signature (2, 2) can be used to produce a map from certain spaces of cusp forms of Nebentype to Hilbert modular forms. The possibility of making such a construction was suggested by Niwa [4], and the techniques are the same as his and Shintani’s [6]. The construction of Hilbert modular forms from cusp forms of one variable has been discussed by many people, and I will not attempt to give a history of the subject here. However, the map produced by the theta-function is essentially the same as that of Doi and Naganuma [2], and Zagier [7]. In particular, the integral kernel Ω(τ, z1, z2) of Zagier is essentially the ‘holomorphic part’ of the theta-function.

Binary theta series and modular forms with complex multiplication

2014 ◽  
Vol 10 (04) ◽  
pp. 1025-1042 ◽  
Author(s):  
Ernst Kani
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Complex Multiplication ◽  
Theta Series ◽  
Character Formula ◽  
Binary Quadratic Forms

The main purpose of this paper is to give an intrinsic interpretation of the space Θ(D) generated by the binary theta series ϑf attached to the positive binary quadratic forms f whose discriminant has the form D(f) = D/t2, for some integer t. It turns out that [Formula: see text], the space of modular forms of weight 1 and of level |D| which have complex multiplication (CM) by their Nebentypus character [Formula: see text]. As an application, we obtain a structure theorem of the space [Formula: see text]. The proof of this theorem rests on the results of [The space of binary theta series, Ann. Sci. Math. Québec36 (2012) 501–534] together with a characterization of the newforms f which have CM by their Nebentypus character in terms of properties of the associated Deligne–Serre Galois representationρf.

Eigenvalue Problem for the Second Order Differential Equation with Nonlocal

2006 ◽  
Vol 11 (1) ◽  
pp. 13-32 ◽  
Author(s):  
B. Bandyrskii ◽  
I. Lazurchak ◽  
V. Makarov ◽  
M. Sapagovas
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Variable Coefficient ◽  
Order Differential Equation ◽  
Integral Condition ◽  
Second Order ◽  
Computational Results ◽  
Discrete Method ◽  
Complex Eigenvalues ◽  
Nonlocal Integral Condition

The paper deals with numerical methods for eigenvalue problem for the second order ordinary differential operator with variable coefficient subject to nonlocal integral condition. FD-method (functional-discrete method) is derived and analyzed for calculating of eigenvalues, particulary complex eigenvalues. The convergence of FD-method is proved. Finally numerical procedures are suggested and computational results are schown.

scholarly journals Oscillatory behavior of a second order nonlinear advanced differential equation with mixed neutral terms

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hongwei Shi ◽  
Yuzhen Bai
Keyword(s):  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Oscillation Criteria ◽  
Order Nonlinear Differential Equation

AbstractIn this paper, we present several new oscillation criteria for a second order nonlinear differential equation with mixed neutral terms of the form $$ \bigl(r(t) \bigl(z'(t)\bigr)^{\alpha }\bigr)'+q(t)x^{\beta } \bigl(\sigma (t)\bigr)=0,\quad t\geq t_{0}, $$(r(t)(z′(t))α)′+q(t)xβ(σ(t))=0,t≥t0, where $z(t)=x(t)+p_{1}(t)x(\tau (t))+p_{2}(t)x(\lambda (t))$z(t)=x(t)+p1(t)x(τ(t))+p2(t)x(λ(t)) and α, β are ratios of two positive odd integers. Our results improve and complement some well-known results which were published recently in the literature. Two examples are given to illustrate the efficiency of our results.

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