2013/05/10 by Goran Muić, Muić, Goran
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Algebraic structures and combinatorial models
paper · pdf · doi:10.48550/arxiv.1305.2428
In this paper we compute 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 even weight ≥ 4 into \mathbb P2. We prove that in most cases this map is a birational equivalence and defined over \mathbb Z. We use this to construct models of X0(N), N≥ 2, using modular forms in M12(Γ0(N)) with integral q--expansion. The models have degree equal to ψ(N) (a classical Dedekind psi function). When genus is at least one, we show the existence of models constructed using cuspidal forms in S4(Γ0(N)) of degree ≤ ψ(N)/3 and in S6(Γ0(11)) of degree 4. As an example of a different kind, 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 X0(N)).