Shimura curve computations via K3 surfaces of Neron-Severi rank at least 19

It is known that K3 surfaces S whose Picard number rho (= rank of the Neron-Severi group of S) is at least 19 are parametrized by modular curves X, and these modular curves X include various Shimura modular curves associated with congruence subgroups of quaternion algebras over Q. In a family of such K3 surfaces, a surface has rho=20 if and only if it corresponds to a CM point on X. We use this to compute equations for Shimura curves, natural maps between them, and CM coordinates well beyond what could be done by working with the curves directly as we did in ``Shimura Curve Computations'' (1998) = Comment: 16 pages (1 figure drawn with the LaTeX picture environment); To appear in the proceedings of ANTS-VIII, Banff, May 200

Points of Low Height on Elliptic Curves and Surfaces, I: Elliptic surfaces over P^1 with small d

For each of n=1,2,3 we find the minimal height h^(P) of a nontorsion point P of an elliptic curve E over C(T) of discriminant degree d=12n (equivalently, of arithmetic genus n), and exhibit all (E,P) attaining this minimum. The minimal h^(P) was known to equal 1/30 for n=1 (Oguiso-Shioda) and 11/420 for n=2 (Nishiyama), but the formulas for the general (E,P) were not known, nor was the fact that these are also the minima for an elliptic curve of discriminant degree 12n over a function field of any genus. For n=3 both the minimal height (23/840) and the explicit curves are new. These (E,P) also have the property that that mP is an integral point (a point of naive height zero) for each m=1,2,...,M, where M=6,8,9 for n=1,2,3; this, too, is maximal in each of the three cases.Comment: 15 pages; some lines in the TeX source are commented out with "%" to meet the 15-page limit for ANTS proceeding

On some congruence properties of elliptic curves

In this paper, as a result of a theorem of Serre on congruence properties, a complete solution is given for an open question (see the text) presented recently by Kim, Koo and Park. Some further questions and results on similar types of congruence properties of elliptic curves are also presented and discussed.Comment: 11 pages, The title is changed. Thanks to a result of J.-P. Serre from his letter on June 15, 2009 to the author, a complete solution for an open question of Kim, Koo and Park is obtained in this fifth revised version. Some related questions and results are also presented and discusse

A closed form expression for the Drinfeld modular polynomial Φ<em>_T (X, Y )</em>

In this paper we give a closed-form expression for the Drinfeld modular polynomial for arbitrary q and prove a conjecture of Schweizer. A new identity involving the Catalan numbers plays a central role

Shimura Curve Computations

We give some methods for computing equations for certain Shimura curves, natural maps between them, and special points on them. We then illustrate these methods by working out several examples in varying degrees of detail. For instance, we compute coordinates for all the rational CM points on the curves X*(1) associated with the quaternion algebras over Q ramified at {2, 3}, {2, 5}, {2, 7}, and {3, 5}. We conclude with a list of open questions that may point the way to further computational investigation of these curves

Computing CM points on Shimura curves arising from compact arithmetic triangle groups

Let Γ ⊂ P SL2(R) be a cocompact arithmetic triangle group, i.e. a Fuchsian triangle group that arises from the unit group of a quaternion algebra over a totally real number field. The group Γ acts on the upper half-plane H; the quotient XC = Γ \H is a Shimura curve, and there is a map j: XC → P 1 C. We algorithmically apply the Shimura reciprocity law to compute CM points j(zD) ∈ P 1 C and their Galois conjugates so as to recognize them as purported algebraic numbers. We conclude by giving some examples of how this method works in practice

Class invariants by the CRT method

We adapt the CRT approach for computing Hilbert class polynomials to handle a wide range of class invariants. For suitable discriminants $D$, this improves its performance by a large constant factor, more than 200 in the most favourable circumstances. This has enabled record-breaking constructions of elliptic curves via the CM method, including examples with $|D|>10^{15}$