Expand all Collapse all | Results 1 - 9 of 9 |
1. 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 |
2. 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 |
3. CMB 2009 (vol 52 pp. 481)
Some Infinite Products of Ramanujan Type In his ``lost'' notebook, Ramanujan stated two results, which are equivalent to the identities
\[
\prod_{n=1}^{\infty} \frac{(1-q^n)^5}{(1-q^{5n})}
=1-5\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{d} d \Big) q^n
\]
and
\[
q\prod_{n=1}^{\infty} \frac{(1-q^{5n})^5}{(1-q^{n})}
=\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{n/d} d \Big) q^n.
\]
We give several more identities of this type.
Keywords:Power series expansions of certain infinite products Categories:11E25, 11F11, 11F27, 30B10 |
4. CMB 2006 (vol 49 pp. 526)
The Values of Modular Functions and Modular Forms Let $\Gamma_0$ be a Fuchsian group of the first kind of genus zero
and $\Gamma$ be a subgroup of $\Gamma_0$
of finite index of genus zero. We find universal recursive
relations giving the $q_{r}$-series coefficients of
$j_0$ by using those of the $q_{h_{s}}$-series of $j$, where $j$ is
the canonical Hauptmodul for $\Gamma$ and $j_0$ is a Hauptmodul
for $\Gamma_0$ without zeros on the complex upper half plane
$\mathfrak{H}$ (here $q_{\ell} := e^{2 \pi i z / \ell}$). We find universal recursive formulas for
$q$-series coefficients of any modular form on
$\Gamma_0^{+}(p)$ in terms of those of the canonical Hauptmodul $j_p^{+}$.
Categories:10D12, 11F11 |
5. CMB 2006 (vol 49 pp. 428)
Vector-Valued Modular Forms of Weight Two Associated With Jacobi-Like Forms We construct vector-valued modular forms of weight 2 associated to
Jacobi-like forms with respect to a symmetric tensor representation of
$\G$ by using the method of Kuga and Shimura as well as the
correspondence between Jacobi-like forms and sequences of modular forms.
As an application, we obtain vector-valued modular forms determined by
theta functions and by pseudodifferential operators.
Categories:11F11, 11F50 |
6. CMB 2006 (vol 49 pp. 296)
On the Modularity of Three Calabi--Yau Threefolds With Bad Reduction at 11 This paper investigates the modularity of three
non-rigid Calabi--Yau threefolds with bad reduction at 11. They are
constructed as fibre products of rational elliptic surfaces,
involving the modular elliptic surface of level 5. Their middle
$\ell$-adic cohomology groups are shown to split into
two-dimensional pieces, all but one of which can be interpreted in
terms of elliptic curves. The remaining pieces are associated to
newforms of weight 4 and level 22 or 55, respectively. For this
purpose, we develop a method by Serre to compare the corresponding
two-dimensional 2-adic Galois representations with uneven trace.
Eventually this method is also applied to a self fibre product of
the Hesse-pencil, relating it to a newform of weight 4 and level
27.
Categories:14J32, 11F11, 11F23, 20C12 |
7. CMB 2005 (vol 48 pp. 535)
On the Error Term in Duke's Estimate for the Average Special Value of $L$-Functions Let $\FF$ be an orthonormal basis for weight $2$
cusp forms of level $N$. We show that various weighted averages of
special values $L(f \tensor \chi, 1)$ over $f \in \FF$ are equal to $4
\pi c + O(N^{-1 + \epsilon})$, where $c$ is an explicit nonzero constant. A previous result of Duke gives an error
term of $O(N^{-1/2}\log N)$.
Categories:11F67, 11F11 |
8. CMB 2002 (vol 45 pp. 257)
Modular Forms Associated to Theta Functions We use the theory of Jacobi-like forms to construct modular forms for a
congruence subgroup of $\SL(2,\mathbb{R})$ which can be expressed as linear
combinations of products of certain theta functions.
Categories:11F11, 11F27, 33D10 |
9. CMB 1999 (vol 42 pp. 129)
Hecke Operations and the Adams $E_2$-Term Based on Elliptic Cohomology Hecke operators are used to investigate part of the $\E_2$-term of
the Adams spectral sequence based on elliptic homology. The main
result is a derivation of $\Ext^1$ which combines use of classical
Hecke operators and $p$-adic Hecke operators due to Serre.
Keywords:Adams spectral sequence, elliptic cohomology, Hecke operators Categories:55N20, 55N22, 55T15, 11F11, 11F25 |