Explicit modular towers

We give a general recipe for explicitly constructing asymptotically optimal towers of modular curves such as {X0(l)}n>1. We illustrate the method by giving equations for eight towers with various geometric features. We conclude by observing that such towers are all of a specific recursive form and speculate that perhaps every tower of this form which attains the Drinfeld-Vlăduţ bound is modular. Introduction. Explicit equations for modular curves have attracted interest at least since the classical work of Fricke and Klein. Recent renewed interest in such equations has been stimulated on the one hand by the availability of software for symbolic computation and on the other hand by specific applications. In [E] we considered the use of modular curves to count rational points on elliptic curves over large finite fields, and illustrated some other applications of equations for the curves X0(N) with small N (say N < 10). Another kind of application is to coding theory: good Goppa codes [G] require curves of large genus g over a fixed finite field k = Fq whose number of rational points grows as a positive multiple of g. Drinfeld and Vlăduţ showed that as g → ∞ no multiple greater than (q − 1)g is possible. Ihara [I] and, independently, Tsfasman, Vlăduţ, and Zink [TVZ] showed that this upper bound is attained by the supersingular points on appropriate modular curves when q is a square. For this application modular curves — elliptic, Shimura, or Drinfeld — are needed whose level is too high to apply the methods of [E] directly, and in general one does not expect to have any pleasant model for a curve of high genus. However, if the curve is of smooth level then it tops a tower of O(log q) covers of low degree, and one may hope to obtain equations for the curve by writing those covers explicitly. In this paper we show how to do this recursively for towers such as {X0(l)}n>1. It turns out that only information about the first few levels of the tower is needed, and that this information can be obtained for modular elliptic curves using the methods of [E], and for some Shimura curves using only the ramification structure. We then illustrate the method by giving explicit formulas for eight asymptotically optimal towers: six of elliptic modular curves, namely X0(l) for l = 2, 3, 4, 5, 6, and X0(3 · 2); and two of Shimura modular curves. Over any finite field whose characteristic does not divide the level of these modular curves, the towers are tamely ramified, making it easy to calculate the genus of every curve in the tower. [This contrasts with the wildly ramified tower of [GS1], whose genus computation required some ingenuity; we show elsewhere that that tower too is modular, of Drinfeld type.] For each finite field k over which one of our towers is asymptotically optimal, the optimality can then be shown by elementary means, independent of the tower’s modular provenance, by exhibiting the coordinates of the rational (supersingular) points. These formulas may also have other uses, e.g. in finding explicit modular parametrizations of elliptic curves with smooth conductor, or in connection with generalizations of the arithmetic-geometric mean (which corresponds to the X0(2) tower) as in [S1,S2]; we hope to pursue these connections in future papers. We conclude this paper with a speculation concerning the modularity of “any” asymptotically optimal tower.