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.

This paper explores the computation of the Brauer-Manin obstruction on Del Pezzo surfaces of degree 2, with examples coming from the class of 'semi-diagonal' Del Pezzo surfaces of degree 2. It is conjectured that the failure of the Hasse principle for a broad class of varieties, including Del Pezzo surfaces, can always be explained by a non-trivial Brauer-Manin obstruction. We provide computational evidence in support of this conjecture for semi-diagonal Del Pezzo surfaces of degree 2. In addition, we determine the complete list of the possibilities for the finite abelian group H 1 (k, Pic X), where X is a Del Pezzo surface of any degree, thus completing a computation which had been previously carried out in various special cases only.

We analyze the Brauer-Manin obstruction to rational points on the K3 surfaces over Q given by double covers of P 2 ramified over a diagonal sextic. After finding an explicit set of generators for the geometric Picard group of such a surface, we find two types of infinite families of counterexamples to the Hasse principle explained by the algebraic Brauer-Manin obstruction. The first type of obstruction comes from a quaternion algebra, and the second type comes from a 3-torsion element of the Brauer group, which gives an affirmative answer to a question asked by Ieronymou and Skorobogatov.

Abstract. We give a correspondence which associates, to each Del Pezzo surface X of degree 6 over a field k of characteristic 0, a collection of data consisting of a Severi-Brauer variety/k and a set of points defined over some extension of k.The main results in this paper, and specifically Theorem 5.1, give a way to describe Del Pezzo surfaces of degree 6 over a field k of characteristic 0, via a correspondence with objects (Severi-Brauer varieties) which can be understood in a completely explicit way if k is sufficiently nice (e.g. k a number field). PreliminariesIn this paper, we will deal with varieties V over a field k of characteristic 0. If L/k is a field extension, then we write V L for the base extension V × Spec k Spec L, and V for V k .A Del Pezzo surface over a number field k is a smooth rational surface X whose anticanonical sheaf ω −1 X is ample. To each Del Pezzo surface X is associated a number d = (ω X , ω X ) (where (, ) denotes intersection number), called the degree of X.The results we need about Del Pezzo surfaces are summarized in the following proposition. We refer the interested reader to [Man74] for proofs and more details. The assumption that r ≤ 6 was made only to simplify the statements of (d) and (e); we will not be concerned with Del Pezzo surfaces of degree 1 or 2 in this paper.

Abstract. This paper explores a topic taken up recently by Logan and van Luijk, finding nontrivial 2-torsion elements of the Tate-Shafarevich group of the Jacobian of a genus-2 curve by exhibiting Brauer-Manin obstructions to rational points on certain quotients of principal homogeneous spaces of the Jacobian, whose desingularizations are explicit K3 surfaces. The main difference between the methods used in this paper and those of Logan and van Luijk is that the obstructions are obtained here from explicitly constructed quaternion algebras, rather than elliptic fibrations.