Abstract
We use explicit results on modular forms (Muić, Ramanujan J 27:188–208, 2012) via uniformization theory to obtain embeddings of modular curves and more generally of compact Riemann surfaces attached to Fuchsian groups of the first kind in certain projective spaces. We obtain families of embeddings which vary smoothly with respect to a parameter in the upper-half plane. We study local expression for the divisors attached to the maps in the family.
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Notes
Lemma 2.1 (v) and the assumption \(S_m(\Gamma )\ne 0\) imply that \(t_m\ge g\). Hence, we can take \(k=k_{\xi , m}\).
If not, then by the definition of an elliptic point (see the beginning of Sect. 2), we find that the order of \(\Gamma _\xi \) is equal to \(2\). But in \(SL_2({\mathbb {R}})\) elements of order \(\le 2\) are \(\pm 1\). Then, we would get \(e_\xi =1\) which is a contradiction with the definition of an elliptic point.
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Communicated by A. Constantin.
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Muić, G. On embeddings of modular curves in projective spaces. Monatsh Math 173, 239–256 (2014). https://doi.org/10.1007/s00605-013-0593-z
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DOI: https://doi.org/10.1007/s00605-013-0593-z