Advanced Topics in the Arithmetic of Elliptic Curves

Cyclic Coverings, Calabi-Yau Manifolds and Complex Multiplication

“…Elliptic curves with CM has been well studied by number theorists. In [44] Appendix C, §3 there is a list of 13 isomorphy classes of elliptic curves with complex multiplication containing all classes represented by the preceding 4 examples. Two examples of the list, which have the j invariants 54000 and 16581375, are given by the equations y 2 = x 3 − 15x + 22, y 2 = x 3 − 595x + 5586.…”

Some consequences of Masser’s counting theorem on elliptic curves

“…The theory of complex multiplication (see [33], II, Proposition 1.4 and Corollary 1.5 for the case of maximal orders, [18], 8, §1 for the general case) yields an isogeny of degree m defined over k:…”

“…Up to isomorphism, which does not change the height, there is a finite number of possible curves E ( [33], II, Proposition 2.1, or [18], 10, §2). Therefore, we can choose the constant to depend only on E in the above inequality.…”

Prime Divisors of Sequences Associated to Elliptic Curves

“…We assume that the coefficients are chosen to lie in ‫ޚ‬ and equation (1) defines a minimal model. For background, definitions and all properties of elliptic curves used in this paper, consult [19], [22] and [21]. Let ‫ދ‬ denote an algebraic number field of degree d = ‫ދ[‬ : ‫]ޑ‬ over ‫.ޑ‬ Throughout the paper, E(‫)ދ‬ denotes the group of ‫-ދ‬rational points of E and O denotes the point at infinity, the identity for the group of ‫-ދ‬rational points.…”