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Proceedings of ASARC Workshop Daecheon & Muju 2009
(2009 / vol.11 / no.2)
Function fields of certain arithmetic curves and application
Ja Kyung Koo, Dong Hwa Shin
Pages.   



Based on the classical theory of modular curves we describe the
function fields of the arithmetic curves
$X_1^dag(N)=langleoverline{Gamma}_1(N),~overline{Phi}_N
angleackslashmathfrak{H}^*$
where
$Phi_N=left(egin{smallmatrix}0&-1N&phantom{-}0end{smallmatrix}
ight)$
is the Fricke involution for $Ngeq2$. For the purpose we use the
modular invariant $j$ and finite products of Siegel functions. And,
we further construct primitive generators of the function fields of
these curves $X_1^dag(N)$ with genus zero in a systematic way by
means of Siegel functions only unlike Choi-Koo's method
(cite{C-K}). As an application we find the ray class invariants
over any imaginary quadratic fields other than
$mathbb{Q}(sqrt{-1})$ and $mathbb{Q}(sqrt{-3})$ by utilizing the
singular values of $j$ and Siegel functions which are different from
the Ramachandra's invariants (cite{Ramachandra}).



1. Introduction
2. Preliminaries
3. Function fields of $X_1^dag(N)$
4. Primitive generators of $mathcal{K}ig(X_1^dag(N)ig)$ of genus zero
5. Application to class fields



class fields, modular curves, modular units, Riemann surfaces, Siegel functions



11F03, 11G16, 11G30, 11R37, 14H55