this paper (section 2), and to Denis Simon for the reference [10].

This thesis is available for Library use on the understanding that it is copy-right material and that no quotation from the thesis may be published without proper acknowledgement. I certify that all the material in this thesis which is not my own work has been clearly identified and that no material is included for which a degree has previously been conferred upon me.

1.1. Motivation. The most important classes of smooth connected linear algebraic groups G over a field k are semisimple groups, tori, and unipotent groups. The first two classes are unified by the theory of reductive groups, and if k is perfect then an arbitrary G is canonically built up from all three classes in the sense that

Let k be a field of characteristic zero and k an algebraic closure of k. For a geometrically integral variety X over k, we write k(X) for the function field of X = X ×k k. If X has a smooth k-point, the natural embedding of multiplicative groups k ∗ ֒ → k(X) ∗ admits a Galois-equivariant retraction. In the first part of the paper, over local and then over global fields, equivalent conditions to the existence of such a retraction are given. They are expressed in terms of the Brauer group of X. In the second part of the paper, we restrict attention to varieties which are homogeneous spaces of connected but otherwise arbitrary algebraic groups, with connected geometric stabilizers. For k local or global, for such a variety X, in many situations but not all, the existence of a Galois-equivariant retraction to k ∗ ֒ → k(X) ∗ ensures the existence of a k-rational point on X. For homogeneous spaces of linear algebraic groups, the technique also handles the case where k is the function field of a complex surface.

Abstract. We study the question of whether algebraic curves of a given genus g defined over a field K must have points rational over the maximal abelian extension K ab of K. We give: (i) an explicit family of diagonal plane cubic curves without Q ab-points, (ii) for every number field K, a genus one curve C /Q with no K ab-points, and (iii) for every g ≥ 4 an algebraic curve C /Q of genus g with no Q ab-points. In an appendix, we discuss varieties over Q((t)), obtaining in particular a curve of genus 3 without (Q((t)) ab-points. Convention: All varieties over a field K are assumed to be nonsingular, projective and (as is especially important for what follows) geometrically irreducible. 1.

This survey paper contains two parts. The first one is a written version of a lecture given at the “Random Matrix Theory and L-functions ” workshop organized at the Newton Institute in July 2004. This was meant as a very concrete and down to earth introduction to elliptic curves with some description of how random matrices become a tool for the (conjectural) understanding of the rank of Mordell-Weil groups by means of the Birch and Swinnerton-Dyer Conjecture; the reader already acquainted with the basics of the theory of elliptic curves can certainly skip it. The second part was originally the write-up of a lecture given for a workshop on the Birch and Swinnerton-Dyer Conjecture itself, in November 2003 at Princeton University, dealing with what is known and expected about the variation of the rank in families of elliptic curves. Thus it is also a natural continuation of the first part. In comparison with the original text and in accordance with the focus of the first part, more details about the input and confirmations of Random Matrix Theory have been added. Acknowledgments. I would like to thank the organizers of both workshops for

Abstract. We give an elementary, self-contained exposition concerning counterexamples to the Hasse Principle. Our account, which uses only techniques from standard undergraduate courses in number theory and algebra, focusses

In this thesis we consider sequences of non-negative integers which arise from counting the periodic points of a map T: X → X, where X is a non-empty set. Some of the main results obtained are concerned with the counting of the periodic points of an endomorphism of a group, in particular when the group is locally nilpotent, for which class of groups a local-global property is established. The ideas developed are applied to some classical sequences, including the Bernoulli and Euler numbers, which are shown to have certain ‘dynamical’ properties. We also consider the Lehmer-Pierce construction for sequences of integers, looking at possible generalizations and their associated measures.

Fix a squarefree integer N and let f be a newform of weight 2 for Γ0(N); we assume that f does not have complex multiplication. It was shown in [14] and [15] that for a set of primes l of density 1 the naive deformation theory of the mod l Galois representation associated to f is unobstructed (in the sense that the universal deformation ring is a power series ring over the Witt vectors). In [31] these methods were modified to obtain results on the deformation problems studied by Taylor-Wiles. In this paper we extend the results of Flach and Mazur to the case of newforms f of weight κ ≥ 2 for Γ1(N). We now state our results more precisely. Fix l> max{5, κ+1}, let f be as above and let H be the associated l-adic representation: H is a free module of rank 2 over a certain Hecke algebra A, which itself is a finite, flat, local, Gorenstein Zl-algebra. Let T be the Tate twist End 0 AH(1) of the module of trace zero endomorphisms of H. Using techniques of Flach we construct a collection of cohomology classes {cp} in H1 (Q, T) with tightly controlled ramification. With some mild additional hypotheses, applying the methods of Kolyvagin to these classes yields a certain annihilator η ∈ A of the Selmer group H1 f (Q, T ∗ ) of the Cartier dual of T. This Selmer group is dual to the differentials ΩR⊗RA, where R is the universal minimally ramified deformation ring of the residual representation of H. In the case that η is a unit this then implies that both R and A are isomorphic to the ring of Witt vectors over the residue field of A. In the general case, following Mazur we show that our construction yields a derivation from A to the Selmer group H1 f (Q, T/ηT); it follows by a formal argument that the natural surjection R ↠ A induces an isomorphism ΩR ⊗R A ∼ = ΩA. Although not the strongest possible result, this does provide a great deal of information on the structure of the ring R. (It is possible that any such map R ↠ A must be an isomorphism, although as far as I know this question remains open.) We also show that the isomorphism ΩR⊗RA ∼ = ΩA is characterized by the fact that

Elliptic curves. An elliptic curve is a curve of genus one with a distinguished rational point. It can be described by a homogeneous equation of the form E: Y 2 Z = 4X 3 + aXZ 2 + bZ 3, (1) where the parameters a, b ∈ Z satisfy the condition ∆: = −2 12 (a 3 + 27b 2) ̸ = 0. The Diophantine theory studies the rational solutions (X, Y, Z) ∈ Q3 of equation (1). It is convenient to ignore the trivial solution (0, 0, 0) and to identify solutions if they differ by multiplication by a non-zero scalar. Solutions to (1) are thus viewed as points in the projective plane P2(Q). Let E(Q) ⊂ P2(Q) denote this solution set. More generally, if F is any field, let E(F) ⊂ P2(F) be the corresponding set of solutions with values in F. It is identified with the set of (x, y) ∈ F 2 satisfying the associated affine equation y 2 = 4x 3 + ax + b, (2) together with the “point at infinity ” corresponding to (X, Y, Z) = (0, 1, 0). Among all the projective 1 curves over Q, the elliptic ones are worthy of special consideration, because they alone are algebraic groups: the set E(Q) This is a transcription of the author’s Coxeter-James lecture given at the CMS Winter meeting in Kingston in December 1998. It is a pleasure to thank Massimo Bertolini and Adrian Iovita for many fruitful exchanges over the years, and the Canadian Mathematical Society for its invitation to deliver the Coxeter-James lecture. 1 I.e., defined by a system of homogeneous equations. 1 is equipped with a binary composition law E(Q) × E(Q) − → E(Q) defined by a system of polynomials with rational coefficients, making E(Q) into a commutative group, with identity element the distinguished point at infinity. The same set of polynomials endows E(F) with a natural addition law 2, admitting a simple geometric description in terms of the chord and tangent method: viewing points in E(F) as points on the affine plane by equation (2), one simply sets P + Q + R = 0 whenever P, Q, and R lie on the same line. (See for example [ST], ch. I.) The Diophantine study of E is facilitated and enriched by the presence of this extra structure. The group E(C) is isomorphic to the quotient of C by a lattice Λ. For a suitable Λ, the inverse isomorphism sends z ∈ C to (℘(z), ℘ ′ (z)) ∈ E(C), where ℘(z) = 1 z2 ∑