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1. CMB Online first
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 |
2. CMB Online first
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 |
3. CMB 2012 (vol 56 pp. 520)
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 Category:11F11 |
4. CMB 2011 (vol 55 pp. 400)
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 |
5. CMB 2005 (vol 48 pp. 180)
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 |