Choosing tau for elliptic curves over the rational numbers with prescribed ramification data

2011-01-02T18:12:24Z

Let $r>2$ and let $b_1,b_2,\ldots,b_r$ be in $\mathbf{P}^1(\mathbf{Q})$. Let $B$ be the divisor $$B:= \sum [b_i].$$ We consider this data to be fixed. For $d>1$, we define $\textrm{Ell}(b_1,b_2,\ldots,b_r,d)$ as the set of (isomorphism classes of) elliptic curves $E$ over $\mathbf{Q}$ that admit a finite morphism $f:E\longrightarrow \mathbf{P}^1_\mathbf{Q}$ of degree $d$ which is etale outside $\{b_1,b_2,\ldots,b_r\} \subset \mathbf{P}^1(\mathbf{Q})$ .

Question 1. Let $E$ be in $\textrm{Ell}(b_1,b_2,\ldots,b_r,d)$ and choose a finite morphism $f:E\longrightarrow \mathbf{P}^1_\mathbf{Q}$ of degree $d$ which is etale outside $\{b_1,b_2,\ldots,b_r\} \subset \mathbf{P}^1(\mathbf{Q})$ . Let $X$ be the analytification of $E_\mathbf{C}$. There exists a $\tau$ in the complex upper half plane such that $X = \mathbf{C}/\mathbf{Z}+\tau\mathbf{Z}$. Can we choose $\tau$ (or $q=e^{2\pi i \tau}$ ) using the data $(b_1,b_2,\ldots,b_r,d,f)$?

Question 2. It follows from Faltings's theorem that the set $\textrm{Ell}(b_1,b_2,\ldots,b_r,d)$ is finite. Is there a more elementary proof of this?

EDIT: Let me describe how the elliptic curve is given (in the set-up I have in mind).

Let $U$ be an open subscheme of $\mathbf{P}^1_\mathbf{Z}$ with complement $D$. We suppose that the closed subscheme $D$ is a horizontal divisor on $\mathbf{P}^1_\mathbf{Z}$ such that the base change $D_\mathbf{Q}$ equals $B$ defined above. Let $V\longrightarrow U$ be a finite etale morphism, with $V$ connected. Let $g:Y\longrightarrow \mathbf{P}^1_\mathbf{Q}$ be the normalization of $\mathbf{P}^1_\mathbf{Q}$ in the function field of $V$. We make the following extra assumptions:

1. $Y$ has a $\mathbf{Q}$-rational point.

2. The genus of $Y$ equals 1.

So the morphism $f$ arises like this.

I'm actually more interested in the set-up described above without assumptions 1 and 2. I just figured it would be an easy case to start with because it could/should be handled more directly.

Answer by S. Carnahan for Choosing tau for elliptic curves over the rational numbers with prescribed ramification data

2011-01-03T14:49:08Z

I am having difficulty making sense of question 1. If you already know $f$, then you have $E$, which gives you an $SL_2(\mathbb{Z})$-orbit of values of $\tau$. This seems to be the best you can do.

For question 2, I think you can bound the number of degree $d$ field extensions of $\mathbb{C}(z)$ whose discriminant divides a certain polynomial (e.g., $\prod_{i=1}^r (z-b_i)^d$ ). This puts a bound on the number of curves of any genus with a degree $d$ map to the line ramified at the chosen points, and hence on the genus one curves.

Choosing tau for elliptic curves over the rational numbers with prescribed ramification data - MathOverflow

Choosing tau for elliptic curves over the rational numbers with prescribed ramification data

Answer by S. Carnahan for Choosing tau for elliptic curves over the rational numbers with prescribed ramification data