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1. CMB 2009 (vol 52 pp. 583)
Computing Polynomials of the Ramanujan $t_n$ Class Invariants We compute the minimal polynomials of the Ramanujan values $t_n$,
where $n\equiv 11 \mod 24$, using the Shimura reciprocity law.
These polynomials can be used for defining the Hilbert class field
of the imaginary quadratic field $\mathbb{Q}(\sqrt{-n})$ and have
much smaller coefficients than the Hilbert polynomials.
Categories:11R29, 33E05, 11R20 |
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 1999 (vol 42 pp. 427)
Ramanujan and the Modular $j$-Invariant A new infinite product $t_n$ was introduced by S.~Ramanujan on the
last page of his third notebook. In this paper, we prove
Ramanujan's assertions about $t_n$ by establishing new connections
between the modular $j$-invariant and Ramanujan's cubic theory of
elliptic functions to alternative bases. We also show that for
certain integers $n$, $t_n$ generates the Hilbert class field of
$\mathbb{Q} (\sqrt{-n})$. This shows that $t_n$ is a new class
invariant according to H.~Weber's definition of class invariants.
Keywords:modular functions, the Borweins' cubic theta-functions, Hilbert class fields Categories:33C05, 33E05, 11R20, 11R29 |