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Search: All articles in the CMB digital archive with keyword modular form

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1. CMB Online first

Gu, Miao; Martin, Gregory George
Factorization tests and algorithms arising from counting modular forms and automorphic representations
A theorem of Gekeler compares the number of non-isomorphic automorphic representations associated with the space of cusp forms of weight $k$ on $\Gamma_0(N)$ to a simpler function of $k$ and $N$, showing that the two are equal whenever $N$ is squarefree. We prove the converse of this theorem (with one small exception), thus providing a characterization of squarefree integers. We also establish a similar characterization of prime numbers in terms of the number of Hecke newforms of weight $k$ on $\Gamma_0(N)$. It follows that a hypothetical fast algorithm for computing the number of such automorphic representations for even a single weight $k$ would yield a fast test for whether $N$ is squarefree. We also show how to obtain bounds on the possible square divisors of a number $N$ that has been found to not be squarefree via this test, and we show how to probabilistically obtain the complete factorization of the squarefull part of $N$ from the number of such automorphic representations for two different weights. If in addition we have the number of such Hecke newforms for even a single weight $k$, then we show how to probabilistically factor $N$ entirely. All of these computations could be performed quickly in practice, given the number(s) of automorphic representations and modular forms as input.

Keywords:modular form, automorphic representation, squarefree number, primality testing, factorization algorithm
Categories:11F70, 11N25, 11N60, 11Y05, 11Y16

2. CMB 2017 (vol 61 pp. 376)

Sebbar, Abdellah; Al-Shbeil, Isra
Elliptic Zeta Functions and Equivariant Functions
In this paper we establish a close connection between three notions attached to a modular subgroup. Namely the set of weight two meromorphic modular forms, the set of equivariant functions on the upper half-plane commuting with the action of the modular subgroup and the set of elliptic zeta functions generalizing the Weierstrass zeta functions. In particular, we show that the equivariant functions can be parameterized by modular objects as well as by elliptic objects.

Keywords:modular form, equivariant function, elliptic zeta function
Categories:11F12, 35Q15, 32L10

3. CMB 2017 (vol 60 pp. 329)

Le Fourn, Samuel
Nonvanishing of Central Values of $L$-functions of Newforms in $S_2 (\Gamma_0 (dp^2))$ Twisted by Quadratic Characters
We prove that for $d \in \{ 2,3,5,7,13 \}$ and $K$ a quadratic (or rational) field of discriminant $D$ and Dirichlet character $\chi$, if a prime $p$ is large enough compared to $D$, there is a newform $f \in S_2(\Gamma_0(dp^2))$ with sign $(+1)$ with respect to the Atkin-Lehner involution $w_{p^2}$ such that $L(f \otimes \chi,1) \neq 0$. This result is obtained through an estimate of a weighted sum of twists of $L$-functions which generalises a result of Ellenberg. It relies on the approximate functional equation for the $L$-functions $L(f \otimes \chi, \cdot)$ and a Petersson trace formula restricted to Atkin-Lehner eigenspaces. An application of this nonvanishing theorem will be given in terms of existence of rank zero quotients of some twisted jacobians, which generalises a result of Darmon and Merel.

Keywords:nonvanishing of $L$-functions of modular forms, Petersson trace formula, rank zero quotients of jacobians
Categories:14J15, 11F67

4. CMB 2014 (vol 57 pp. 485)

Franc, Cameron; Mason, Geoffrey
Fourier Coefficients of Vector-valued Modular Forms of Dimension $2$
We prove the following Theorem. Suppose that $F=(f_1, f_2)$ is a $2$-dimensional vector-valued modular form on $\operatorname{SL}_2(\mathbb{Z})$ whose component functions $f_1, f_2$ have rational Fourier coefficients with bounded denominators. Then $f_1$ and $f_2$ are classical modular forms on a congruence subgroup of the modular group.

Keywords:vector-valued modular form, modular group, bounded denominators
Categories:11F41, 11G99

5. CMB 2013 (vol 57 pp. 845)

Lei, Antonio
Factorisation of Two-variable $p$-adic $L$-functions
Let $f$ be a modular form which is non-ordinary at $p$. Loeffler has recently constructed four two-variable $p$-adic $L$-functions associated to $f$. In the case where $a_p=0$, he showed that, as in the one-variable case, Pollack's plus and minus splitting applies to these new objects. In this article, we show that such a splitting can be generalised to the case where $a_p\ne0$ using Sprung's logarithmic matrix.

Keywords:modular forms, p-adic L-functions, supersingular primes
Categories:11S40, 11S80

6. CMB 2012 (vol 56 pp. 520)

Elbasraoui, Abdelkrim; Sebbar, Abdellah
Equivariant Forms: Structure and Geometry
In this paper we study the notion of equivariant forms introduced in the authors' previous works. In particular, we completely classify all the equivariant forms for a subgroup of $\operatorname{SL}_2(\mathbb{Z})$ by means of the cross-ratio, the weight 2 modular forms, the quasimodular forms, as well as differential forms of a Riemann surface and sections of a canonical line bundle.

Keywords:equivariant forms, modular forms, Schwarz derivative, cross-ratio, differential forms

7. CMB 2011 (vol 55 pp. 400)

Sebbar, Abdellah; Sebbar, Ahmed
Eisenstein Series and Modular Differential Equations
The 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.

Keywords:differential equations, modular forms, Schwarz derivative, equivariant forms
Categories:11F11, 34M05

8. CMB 2005 (vol 48 pp. 180)

Cynk, SÅ‚awomir; Meyer, Christian
Geometry and Arithmetic of Certain Double Octic Calabi--Yau Manifolds
We study Calabi--Yau manifolds constructed as double coverings of $\mathbb{P}^3$ branched along an octic surface. We give a list of 87 examples corresponding to arrangements of eight planes defined over $\mathbb{Q}$. The Hodge numbers are computed for all examples. There are 10 rigid Calabi--Yau manifolds and 14 families with $h^{1,2}=1$. The modularity conjecture is verified for all the rigid examples.

Keywords:Calabi--Yau, double coverings, modular forms
Categories:14G10, 14J32

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