Abstract
In this paper we prove a formula which relates the degree of a curve which is the image of a mapping \(z\longmapsto (f(z): g(z): h(z))\) constructed out of three linearly independent modular forms of the same integral or half-integral weight into \(\mathbb P^2\) and the degree of that map. Based on the formula, we present a test for birationality of the map. As an example, we compute the formula for the total degree i.e., the degree considered as a polynomial of two (independent) variables of the classical modular polynomial (or the degree of the canonical model of \(X_0(N)\)). We give an interesting example to the test for birationality which leads us to make a question on existence of specific explicit and simple model of \(X_0(N)\). We prove our claim when \(N=p\) is a prime.
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Notes
Otherwise, if \(X_0, X_1, X_2\) denote independent variables, then the resultant of \(P(X_0, X_1, X_2)\) and \(\frac{\partial }{\partial X_2}P(X_0, X_1, X_2)\) which is a homogeneous polynomial in \(X_0, X_1\) must be zero. But then \(P(X_0, X_1, X_2)\) and \(\frac{\partial }{\partial X_2}P(X_0, X_1, X_2)\) would have a common irreducible factor. This factor is obviously \(P(X_0, X_1, X_2)\). This a contradiction since this polynomial has the degree in \(X_2\) strictly greater than than its derivative with respect to \(X_2\).
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Communicated by A. Constantin.
This work has been supported in part by Croatian Science Foundation under the project 9364.
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Muić, G. On degrees and birationality of the maps \(X_0(N)\rightarrow \mathbb P^2\) constructed via modular forms. Monatsh Math 180, 607–629 (2016). https://doi.org/10.1007/s00605-016-0908-y
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DOI: https://doi.org/10.1007/s00605-016-0908-y