Let O be an order in the imaginary quadratic field K. For positive integers M | N , we determine the least degree of an O-CM point on the modular curve X(M, N ) /K(ζ M ) and also on the modular curve X(M, N ) /Q(ζ M ) : that is, we treat both the case in which the complex multiplication is rationally defined and the case in which we do not assume that the complex multiplication is rationally defined. To prove these results we establish several new theorems on rational cyclic isogenies of CM elliptic curves. In particular, we extend a result of Kwon [Kw99] that determines the set of positive integers N for which there is an O-CM elliptic curve E admitting a cyclic, Q(j(E))-rational N -isogeny.

We give the complete list of possible torsion subgroups of elliptic curves with complex multiplication over number fields of degree 1-13. Additionally we describe the algorithm used to compute these torsion subgroups and its implementation.

Spodoptera frugiperda (J.E. Smith) (Lepidoptera: Noctuidae), the fall armyworm is the most economically important maize pest in the western hemisphere. This research focused on the genetic variability of the maize host strain because there is a lack of information in this area of S. frugiperda research. Amplified fragment length polymorphism (AFLP) was used to assess the genetic variability of S. frugiperda over a large geographic area. Twenty populations were collected from the maize, one population was collected from princess tree, one population was collected from lemon tree, and one population was collected from bermudagrass. The 23 populations were from Mexico, the continental United States, Puerto Rico, Brazil, and Argentina. The objective of this research was to evaluate whether the majority of genetic variability was within populations or between populations. The AFLP results showed that the majority of the genetic variability is within populations and not between populations, indicating minor gene flow and suggesting that S. frugiperda in the Western Hemisphere are an interbreeding population.

We study torsion subgroups of elliptic curves with complex multiplication (CM) defined over number fields which admit a real embedding. We give a complete classification of the groups which arise up to isomorphism as the torsion subgroup of a CM elliptic curve defined over a number field of odd degree: there are infinitely many. However, if we fix an odd integer d and consider number fields of degree dp as p ranges over all prime numbers, all but finitely many torsion subgroups that appear for CM elliptic curves actually occur in a degree dividing d. This implies an absolute bound on the size of torsion subgroups of CM elliptic curves defined over number fields of degree dp. In the case where d = 1, there are six "Olson groups" which arise as torsion subgroups of CM elliptic curves over Q, and there are precisely 17 "non-Olson" CM elliptic curves defined over a number field of (variable) prime degree. ContentsIt follows from Theorem 2.1a) below that for every Olson group G and every d ≥ 2, there are infinitely many degree d number fields F for which there is a CM elliptic curve E /F with E(F )[tors] ∼ = G. Similarly, whenever d 1 | d 2 , the list of torsion subgroups of CM elliptic curves in degree d 2 will contain the corresponding list in degree d 1 . It is more penetrating to ask which new groups arise in degree d: for d ∈ Z + , let T CM (d) be the set of isomorphism classes of torsion subgroups of CM elliptic curves defined over number fields of degree d, and for d ≥ 2 we putFrom [CCRS14, §4] we compile the following table.

Let E/K be an elliptic curve defined over a number field, and let p be a prime number such that E(K) has full p-torsion. We show that the order of the p-part of the Shafarevich-Tate group of E/L is unbounded as L varies over degree p extensions of K. The proof uses O'Neil's period-index obstruction. We deduce the result from the fact that, under the same hypotheses, there exist infinitely many elements of the Weil-Châtelet group of E/K of period p and index p 2 .