Timeline for Modular curve X(2)

6 events

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Oct 3, 2016 at 7:04	comment	added	Will Sawin		@AdelBETINA One needs to know that the two-torsion points are defined over $k$. There $y$ coordinates are zero, and by assumption their $x$ coordinates are in $k$. Now apply a linear transformation to $x$ to send one point to $0$ and another to $1$. However, this only produces an equation of the form $D y^2 = x(x-1)(x-\lambda)$. The $D$ is related to the fact that it is a coarse moduli space and not a fine one.
Oct 2, 2016 at 19:42	comment	added	Adel BETINA		I have another question : If $E$ is a curve of genus one over a field $k$ and $E(k)$ is not empty ($E$ is an elliptic curve), by Riemann-Roch we can find an equation of $E$ given by $P(X,Y)$ such that the maximal degree of $X$ is three and the maximal degree of $Y$ is two, how we can find an Equation of the type $y^2=x(x-1)(x-\lambda)$ ?
Oct 2, 2016 at 12:09	vote	accept	Adel BETINA
Oct 2, 2016 at 2:46	comment	added	S. Carnahan♦		@user1952009 Objects of $\mathfrak{M}(2)$ are families of elliptic curves $E \to B$ with level 2 structure. That is, such a family with level two structure is "the same as" a map $B \to \mathfrak{M}(2)$. Take $B = \mathbb{P}^1_{\mathbb{Q}}$, and compose with the coarse moduli map $\mathfrak{M}(2) \to X(2)$.
Oct 1, 2016 at 23:00	comment	added	reuns		Sorry I know modular forms but I get nothing of algebraic geometry. What is the map $\mathbb{P}^1 \to X(2)$ ?
Oct 1, 2016 at 21:41	history	answered	Will Sawin	CC BY-SA 3.0