On degrees and birationality of the maps $$X_0(N)\rightarrow \mathbb P^2$$ X 0 ( N ) → P 2 constructed via modular forms
| Autor: | Goran Muić |
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| Rok vydání: | 2016 |
| Předmět: |
Discrete mathematics
Polynomial Degree (graph theory) General Mathematics Image (category theory) 010102 general mathematics Modular form 010103 numerical & computational mathematics 01 natural sciences Prime (order theory) Combinatorics Simple (abstract algebra) Canonical model Linear independence 0101 mathematics Mathematics |
| Zdroj: | Monatshefte für Mathematik. 180:607-629 |
| ISSN: | 1436-5081 0026-9255 |
| DOI: | 10.1007/s00605-016-0908-y |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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