On Near Prime-Order Elliptic Curves with Small Embedding Degrees

Article published in the proceeding of the conference CAI 2015 http://www.ims.uni-stuttgart.de/events/CAI2015In this paper, we generalize the method of Scott and Barreto in order to construct a family of pairing-friendly elliptic curve. We present an explicit algorithm to obtain generalized MNT families curves with any cofactors. We also analyze the complex multiplication equations of these curves and transform them into generalized Pell equation. As an example, we describe a way to generate Edwards curves with embedding degree 6

On near prime-order elliptic curves with small embedding degrees (Full version)

In this paper, we extend the method of Scott and Barreto and present an explicit and simple algorithm to generate families of generalized MNT elliptic curves. Our algorithm allows us to obtain all families of generalized MNT curves with any given cofactor. Then, we analyze the complex multiplication equations of these families of curves and transform them into generalized Pell equation. As an example, we describe a way to generate Edwards curves with embedding degree 6, that is, elliptic curves having cofactor h = 4

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

Relations among modular points on elliptic curves

Given a correspondence between a modular curve and an elliptic curve A we study the group of relations among the CM points of A. In particular we prove that the intersection of any finite rank subgroup of A with the set of CM points of A is finite. We also prove a local version of this global result with an effective bound valid also for certain infinite rank subgroups. We deduce the local result from a ``reciprocity'' theorem for CL (canonical lift) points on A. Furthermore we prove similar global and local results for intersections between subgroups of A and isogeny classes in A. Finally we prove Shimura curve analogues and, in some cases, higher-dimensional versions of these results.Comment: 48 page

Three lectures on Cox rings

Notes of an introductory course given at the conference "Torsors: Theory and Applications" in Edinburgh, January 2011.Comment: Minor corrections, 37 page

Modular embeddings of Teichmueller curves

Fuchsian groups with a modular embedding have the richest arithmetic properties among non-arithmetic Fuchsian groups. But they are very rare, all known examples being related either to triangle groups or to Teichmueller curves. In Part I of this paper we study the arithmetic properties of the modular embedding and develop from scratch a theory of twisted modular forms for Fuchsian groups with a modular embedding, proving dimension formulas, coefficient growth estimates and differential equations. In Part II we provide a modular proof for an Apery-like integrality statement for solutions of Picard-Fuchs equations. We illustrate the theory on a worked example, giving explicit Fourier expansions of twisted modular forms and the equation of a Teichmueller curve in a Hilbert modular surface. In Part III we show that genus two Teichmueller curves are cut out in Hilbert modular surfaces by a product of theta derivatives. We rederive most of the known properties of those Teichmueller curves from this viewpoint, without using the theory of flat surfaces. As a consequence we give the modular embeddings for all genus two Teichmueller curves and prove that the Fourier developments of their twisted modular forms are algebraic up to one transcendental scaling constant. Moreover, we prove that Bainbridge's compactification of Hilbert modular surfaces is toroidal. The strategy to compactify can be expressed using continued fractions and resembles Hirzebruch's in form, but every detail is different.Comment: revision including the referee's comments, to appear in Compositio Mat