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1. CMB 2009 (vol 52 pp. 53)
Cusp Forms Like $\Delta$ Let $f$ be a square-free integer and denote by $\Gamma_0(f)^+$ the
normalizer of $\Gamma_0(f)$ in $\SL(2,\R)$. We find the analogues of
the cusp form $\Delta$ for the groups $\Gamma_0(f)^+$.
Categories:11F03, 11F22, 30F35 |
2. CMB 2007 (vol 50 pp. 334)
Determination of Hauptmoduls and Construction of Abelian Extensions of Quadratic Number Fields We obtain Hauptmoduls of genus zero congruence
subgroups of the type $\Gamma_0^+(p):=\linebreak\Gamma_0(p)+w_p$, where $p$ is
a prime and $w_p$ is the Atkin--Lehner involution. We then use the
Hauptmoduls, along with modular functions on $\Gamma_1(p)$
to construct families of cyclic extensions of quadratic number
fields. Further examples of cyclic extension of bi-quadratic and
tri-quadratic number fields are also given.
Categories:11F03, 11G16, 11R20 |
3. CMB 2007 (vol 50 pp. 196)
Plane Quartic Twists of $X(5,3)$ Given an odd surjective Galois representation $\varrho\from \G_\Q\to\PGL_2(\F_3)$ and a
positive integer~$N$, there exists a twisted modular curve $X(N,3)_\varrho$
defined over $\Q$ whose rational points classify the quadratic $\Q$-curves of degree $N$
realizing~$\varrho$. This paper gives a method to provide an explicit plane quartic model for
this curve in the genus-three case $N=5$.
Categories:11F03, 11F80, 14G05 |
4. CMB 2007 (vol 50 pp. 234)
A Remark on a Modular Analogue of the Sato--Tate Conjecture The original Sato--Tate Conjecture concerns the angle distribution
of the eigenvalues arising from non-CM elliptic curves. In this paper,
we formulate a modular analogue of the Sato--Tate Conjecture and prove
that the angles arising from non-CM holomorphic Hecke
eigenforms with non-trivial central characters are not distributed
with respect to the Sate--Tate measure
for non-CM elliptic curves. Furthermore, under a reasonable conjecture,
we prove that the expected distribution is uniform.
Keywords:$L$-functions, Elliptic curves, Sato--Tate Categories:11F03, 11F25 |
5. CMB 2002 (vol 45 pp. 36)
Modular Equations and Discrete, Genus-Zero Subgroups of $\SL(2,\mathbb{R})$ Containing $\Gamma(N)$ Let $G$ be a discrete subgroup of $\SL(2,\R)$ which contains
$\Gamma(N)$ for some $N$. If the genus of $X(G)$ is zero, then there
is a unique normalised generator of the field of $G$-automorphic
functions which is known as a normalised Hauptmodul. This paper gives
a characterisation of normalised Hauptmoduls as formal $q$ series
using modular polynomials.
Categories:11F03, 11F22, 30F35 |