arXiv subjects
Publications and source records attributed to Iva Kodrnja.
In this paper we find the number of homogeneous polynomials of degree d such that they vanish on cuspidal modular forms of even weight $m\geq 2$ that form a basis for $S_m(Γ_0(N))$. We use these cuspidal forms to embedd $X_0(N)$ to projective space and we find the Hilbert polynomial of the graded ideal of the projective curve that is the image of this embedding.
In this paper we discuss applications of our earlier work in studying certain Galois groups and splitting fields of rational functions in $\mathbb Q\left(X_0(N)\right)$ using Hilbert's irreducibility theorem and modular forms. We also consider computational aspect of the problem using MAGMA and SAGE.
This paper is a continuation of our previous works where we study maps from $X_0(N)$, $N \ge 1$, into $\mathbb P^2$ constructed via modular forms of the same weight and criteria that such a map is birational (see [12]). In the present paper our approach is based on the theory of primitive elements in finite separable field extensions. We prove that in most of the cases the constructed maps are birational, and we consider those such that the resulting equation of the image in $\mathbb P^2$ is simplest possible.
We find plane models for all $X_0(N)$, $N\geq 2$. We observe a map from the modular curve $X_0(N)$ to the projective plane constructed using modular forms of weight $12$ for the group $Γ_0(N)$; the Ramanujan function $Δ$, $Δ(N\cdot)$ and the third power of Eisestein series of weight $4$, $E_4^3$, and prove that this map is birational equivalence for every $N\geq 2$. The equation of the model is the minimal polynomial of $Δ(N\cdot)/Δ$ over $\mathbb{C}(j)$.
In this paper we find projective plane models of $X_0(N)$ by constructing maps from $X_0(N)$ to the projective plane using modular forms. We use eta-quotients of weight 12. We find those eta-quotients of weight 12 which have maximal order of zero at the cusp $\infty$.