The least degree of a CM point on a modular curve

Abstract

For a modular curve X=X0(N)$X = X_0(N)$, X1(N)$X_1(N)$ or X1(M,N)$X_1(M,N)$ defined over Q$\mathbb {Q}$, we denote by dCM(X)$d_{\operatorname{CM}}(X)$ the least degree of a CM point on X$X$. For each discriminant normalΔ<0$\Delta &lt; 0$, we determine the least degree of a point on X0(N)$X_0(N)$ with CM by the order of discriminant Δ$\Delta$. This places us in a position to study dCM(X)$d_{\operatorname{CM}}(X)$ as an ‘arithmetic function’ and we do so, obtaining various upper bounds, lower bounds and typical bounds. We deduce that all but finitely many curves in each of the families have sporadic CM points. Finally, we supplement these results with a computational study, for example, computing dCM(X0false(Nfalse))$d_{\operatorname{CM}}(X_0(N))$ and dCM(X1false(Nfalse))$d_{\operatorname{CM}}(X_1(N))$ exactly for N⩽106$N \leqslant 10^6$ and determining whether X0(N)$X_0(N)$ (respectively, X1(N)$X_1(N)$, X1(M,N)$X_1(M,N)$) has sporadic CM points for all but 106 values of N$N$ (respectively, 227 values of N$N$, 146 pairs false(M,Nfalse)$(M,N)$ with M⩾2$M \geqslant 2$).

Computing quadratic points on modular curves 𝑋₀(𝑁)

“…Next, the non-cuspidal rational points on X + 0 (116) correspond to CM points by [32, Theorem 0.1], and so P must be a CM point. From the data associated to the paper [13] (available at https://github.com/fsaia/least-cm-degree/blob/master/Least%20Degrees/X0), we obtain five possible pairs (j, L) of j-invariants j and quadratic fields L for the pair (j(P ), K). Each of these pairs in fact already arises as the j-invariant and field of definition of one of the points P 1 , .…”

Cyclic isogenies of elliptic curves over a fixed quadratic field

“…Hence, after computing the pullbacks of the rational points on X + 0 (N), we have all the quadratic points on X 0 (N). Using data provided to us by the authors of [9] and which can be obtained using [9, Theorem 3.7], we can conclude over which quadratic fields there exist CM points and how many there are. This allows us to conclude that the only non-CM points are the points on X 0 (125) defined over Q( √ 509).…”

Torsion for CM elliptic curves defined over number fields of degree 2p

“…( (4,4), (5,10), (6,6), (7,7), (7,49), (10,10), (49, 49)}. These leaves only (M, N ) = (3, 3), but Z/3Z × Z/3Z occurs already in degree 2 by [6].…”

“…For the collection of all CM elliptic curves, the classification of torsion subgroups is known for any d ≤ 13 or for any odd d > 13; see [6,22,5,4]. We note that CM elliptic curves produce many examples of sporadic points on modular curves (see, for example, [7]), so this provides further motivation for studying this class in particular.…”