Modular curve parametrizing two cyclic subgroups of an elliptic curve - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T05:32:01Z http://mathoverflow.net/feeds/question/95128 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95128/modular-curve-parametrizing-two-cyclic-subgroups-of-an-elliptic-curve Modular curve parametrizing two cyclic subgroups of an elliptic curve OldMacdonaldHadaForm 2012-04-25T08:31:16Z 2012-04-25T11:04:30Z <p>The aim of this question is to better understand the following moduli space/modular curve, for which I propose (temporarily) the name $Y_0(M,N)$. We define $Y_0(M,N)$ as the moduli space parametrizing an elliptic curve $E$, together with two cyclic subgroups $G$ and $H$, of order respectively $M$ and $N$, of the group of degree $0$ line bundles modulo linear equivalence on $E$.</p> <p>(note that the subgroup generated by $G$ and $H$ need not be of rank $2$, although it will be so generically)</p> <p>When $N=1$, $Y_0(M,N)$ is just the well known modular curve $Y_0(M)$.</p> <p>Questions:</p> <p>1) Are there references for this moduli space/modular curve? Is this well studied/well known?</p> <p>2) I think of this as the fiber product $Y_0(M) \times_{\mathcal{M}_{1,1}} Y_0(N)$ (the fiber product over the moduli space of elliptic curves). Is this correct?</p> <p>3) What can we say about the curve $Y_0(M,N)$? Is it irreducible? Are there some kind of formulas for its genus as there are for $Y_0(M)$?</p> http://mathoverflow.net/questions/95128/modular-curve-parametrizing-two-cyclic-subgroups-of-an-elliptic-curve/95148#95148 Answer by S. Carnahan for Modular curve parametrizing two cyclic subgroups of an elliptic curve S. Carnahan 2012-04-25T11:04:30Z 2012-04-25T11:04:30Z <p>$Y_0(M,N)$ can be reinterpreted as the moduli space of diagrams $E_1 \to E \leftarrow E_2$ of elliptic curves, where the arrows are cyclic isogenies of degree $M$ and $N$. From this viewpoint, it is naturally a fiber product of $Y_0(N)$ with $Y_0(M)$, as you suggested. It is in general a disjoint union of modular curves, because the kernels of dual isogenies may have isomorphism types and restrictions of the Weil pairing that are not unique.</p> <p>In good characteristic, you may enumerate components by finding all abelian groups generated by a cyclic group of order $M$ and a cyclic group of order $N$, and finding all equivalence classes of $\mu_{(M,N)}$-valued symplectic forms on these groups. In characteristic $p$, you have to work out cases, depending on how many powers of $p$ divide $M$ and $N$. For example, if $M=N=p$, then $Y_0(p)$ is two copies of the affine line, glued at supersingular points, and the fiber product is four copies of the affine line, also glued at those points.</p> <p>Most of the information you want can be extracted from Katz and Mazur's <em>Arithmetic Moduli of Elliptic Curves</em>, although I don't know where a table of genera of components can be found. You may have to explicitly enumerate elliptic points and cusp orders, then use Hurwitz.</p>