scholarly journals Eisenstein Series and Modular Differential Equations

2012 ◽  
Vol 55 (2) ◽  
pp. 400-409 ◽  
Author(s):  
Abdellah Sebbar ◽  
Ahmed Sebbar
Keyword(s):  
Differential Equations ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Modular Functions ◽  
Tools And Techniques

AbstractThe purpose of this paper is to solve various differential equations having Eisenstein series as coefficients using various tools and techniques. The solutions are given in terms of modular forms, modular functions, and equivariant forms.

scholarly journals Holonomic relations for modular functions and forms: First guess, then prove

2020 ◽  
pp. 1-47
Author(s):  
Peter Paule ◽  
Cristian-Silviu Radu
Keyword(s):  
Differential Equations ◽  
Modular Forms ◽  
Major Theme ◽  
Modular Functions ◽  
Puiseux Series ◽  
Partition Congruences ◽  
Holonomic Functions

One major theme of this paper concerns the expansion of modular forms and functions in terms of fractional (Puiseux) series. This theme is connected with another major theme, holonomic functions and sequences. With particular attention to algorithmic aspects, we study various connections between these two worlds. Applications concern partition congruences, Fricke–Klein relations, irrationality proofs a laBeukers, or approximations to pi studied by Ramanujan and the Borweins. As a major ingredient to a “first guess, then prove” strategy, a new algorithm for proving differential equations for modular forms is introduced.

scholarly journals A formula for generating weakly modular forms with weight 12

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 1001-1004
Author(s):  
Aykut Aygunes
Keyword(s):  
Differential Equations ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Short Paper ◽  
Derivative Formula ◽  
Special Solutions

In this short paper, generally, we define a family of functions fk depends on the Eisenstein series with weight 2k, for k ( N. More detail, by considering the function fk, we define a derivative formula for generating weakly modular forms with weight 12. As a result for this, we claim that this formula gives an advantage to find the special solutions of some differential equations.

scholarly journals Differential equations satisfied by modular forms of level 5

2019 ◽  
Vol 15 (02) ◽  
pp. 213-250
Author(s):  
Kazuhide Matsuda
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Eisenstein Series ◽  
Modular Forms

This paper describes the derivation of the level 5 versions of Ramanujan’s system of ordinary differential equations satisfied by the Eisenstein series, [Formula: see text], and [Formula: see text]

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 A dominated convergence theorem for Eisenstein series

2021 ◽  
Author(s):  
Johann Franke
Keyword(s):  
Fourier Series ◽  
Dirichlet Series ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Convergence Theorem ◽  
Rational Functions ◽  
New Approach ◽  
Series Representations ◽  
Dominated Convergence ◽  
Generalized Dirichlet

AbstractBased on the new approach to modular forms presented in [6] that uses rational functions, we prove a dominated convergence theorem for certain modular forms in the Eisenstein space. It states that certain rearrangements of the Fourier series will converge very fast near the cusp $$\tau = 0$$ τ = 0 . As an application, we consider L-functions associated to products of Eisenstein series and present natural generalized Dirichlet series representations that converge in an expanded half plane.

scholarly journals Constructions of vector-valued modular forms of rank four and level one

2020 ◽  
Vol 16 (05) ◽  
pp. 1111-1152
Author(s):  
Cameron Franc ◽  
Geoffrey Mason
Keyword(s):  
Irreducible Representation ◽  
Differential Equations ◽  
Modular Forms ◽  
Modular Group ◽  
Tensor Products ◽  
Two Dimensional ◽  
Minimal Weight ◽  
Graded Module ◽  
Holomorphic Modular Forms ◽  
Vector Valued

This paper studies modular forms of rank four and level one. There are two possibilities for the isomorphism type of the space of modular forms that can arise from an irreducible representation of the modular group of rank four, and we describe when each case occurs for general choices of exponents for the [Formula: see text]-matrix. In the remaining sections we describe how to write down the corresponding differential equations satisfied by minimal weight forms, and how to use these minimal weight forms to describe the entire graded module of holomorphic modular forms. Unfortunately, the differential equations that arise can only be solved recursively in general. We conclude the paper by studying the cases of tensor products of two-dimensional representations, symmetric cubes of two-dimensional representations, and inductions of two-dimensional representations of the subgroup of the modular group of index two. In these cases, the differential equations satisfied by minimal weight forms can be solved exactly.

THE MINIMAL MODULAR FORM ON QUATERNIONIC

2020 ◽  
pp. 1-34
Author(s):  
Aaron Pollack
Keyword(s):  
Modular Form ◽  
Eisenstein Series ◽  
Explicit Form ◽  
Modular Forms ◽  
Fourier Expansion ◽  
Reductive Group ◽  
Automorphic Function ◽  
Unipotent Radical ◽  
Exceptional Groups ◽  
Dynkin Type

Suppose that $G$ is a simple reductive group over $\mathbf{Q}$ , with an exceptional Dynkin type and with $G(\mathbf{R})$ quaternionic (in the sense of Gross–Wallach). In a previous paper, we gave an explicit form of the Fourier expansion of modular forms on $G$ along the unipotent radical of the Heisenberg parabolic. In this paper, we give the Fourier expansion of the minimal modular form $\unicode[STIX]{x1D703}_{Gan}$ on quaternionic $E_{8}$ and some applications. The $Sym^{8}(V_{2})$ -valued automorphic function $\unicode[STIX]{x1D703}_{Gan}$ is a weight 4, level one modular form on $E_{8}$ , which has been studied by Gan. The applications we give are the construction of special modular forms on quaternionic $E_{7},E_{6}$ and $G_{2}$ . We also discuss a family of degenerate Heisenberg Eisenstein series on the groups $G$ , which may be thought of as an analogue to the quaternionic exceptional groups of the holomorphic Siegel Eisenstein series on the groups $\operatorname{GSp}_{2n}$ .

scholarly journals All modular forms of weight 2 can be expressed by Eisenstein series

2020 ◽  
Vol 6 (3) ◽  
Author(s):  
Martin Raum ◽  
Jiacheng Xia
Keyword(s):  
Modular Form ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Linear Combinations ◽  
Integral Weight ◽  
Elliptic Modular Form

Abstract We show that every elliptic modular form of integral weight greater than 1 can be expressed as linear combinations of products of at most two cusp expansions of Eisenstein series. This removes the obstruction of nonvanishing central $$\mathrm{L}$$ L -values present in all previous work. For weights greater than 2, we refine our result further, showing that linear combinations of products of exactly two cusp expansions of Eisenstein series suffice.

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