Choosing tau for elliptic curves over the rational numbers with prescribed ramification data - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T07:31:53Z http://mathoverflow.net/feeds/question/50940 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/50940/choosing-tau-for-elliptic-curves-over-the-rational-numbers-with-prescribed-ramifi Choosing tau for elliptic curves over the rational numbers with prescribed ramification data Ariyan Javanpeykar 2011-01-02T18:12:24Z 2011-01-04T11:07:44Z <p>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 <code>$\textrm{Ell}(b_1,b_2,\ldots,b_r,d)$</code> 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 <code>$\{b_1,b_2,\ldots,b_r\} \subset \mathbf{P}^1(\mathbf{Q})$</code>. </p> <p><strong>Question 1.</strong> Let $E$ be in <code>$\textrm{Ell}(b_1,b_2,\ldots,b_r,d)$</code> and choose a finite morphism $f:E\longrightarrow \mathbf{P}^1_\mathbf{Q}$ of degree $d$ which is etale outside <code>$\{b_1,b_2,\ldots,b_r\} \subset \mathbf{P}^1(\mathbf{Q})$</code>. 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 <code>$q=e^{2\pi i \tau}$</code>) using the data $(b_1,b_2,\ldots,b_r,d,f)$? </p> <p><strong>Question 2.</strong> It follows from Faltings's theorem that the set <code>$\textrm{Ell}(b_1,b_2,\ldots,b_r,d)$</code> is finite. Is there a more elementary proof of this? </p> <p>EDIT: Let me describe how the elliptic curve is given (in the set-up I have in mind).</p> <p>Let $U$ be an open subscheme of <code>$\mathbf{P}^1_\mathbf{Z}$</code> with complement $D$. We suppose that the closed subscheme $D$ is a horizontal divisor on <code>$\mathbf{P}^1_\mathbf{Z}$</code> such that the base change <code>$D_\mathbf{Q}$</code> equals $B$ defined above. Let <code>$V\longrightarrow U$</code> be a finite etale morphism, with $V$ connected. Let <code>$g:Y\longrightarrow \mathbf{P}^1_\mathbf{Q}$</code> be the normalization of <code>$\mathbf{P}^1_\mathbf{Q}$</code> in the function field of $V$. We make the following extra assumptions:</p> <p><strong>1.</strong> $Y$ has a $\mathbf{Q}$-rational point.</p> <p><strong>2.</strong> The genus of $Y$ equals 1.</p> <p>So the morphism $f$ arises like this. </p> <p>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.</p> http://mathoverflow.net/questions/50940/choosing-tau-for-elliptic-curves-over-the-rational-numbers-with-prescribed-ramifi/51022#51022 Answer by S. Carnahan for Choosing tau for elliptic curves over the rational numbers with prescribed ramification data S. Carnahan 2011-01-03T14:49:08Z 2011-01-03T14:49:08Z <p>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.</p> <p>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., <code>$\prod_{i=1}^r (z-b_i)^d$</code>). 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.</p>