Abstract. Let r > 0 be an integer. We present a sufficient condition for an abelian variety A over a number field k to have infinitely many quadratic twists of rank at least r, in terms of the density properties of rational points on the Kummer variety Km(A r ) of the r-fold product of A with itself.One consequence of our results is the following. Fix an abelian variety A over k, and suppose that for some r > 0 the Brauer-Manin obstruction to weak approximation on the Kummer variety Km(A r ) is the only one. Then A has a quadratic twist of rank at least r. Hence if the Brauer-Manin obstruction is the only one to weak approximation on all Kummer varieties, then ranks of twists of any positive-dimensional abelian variety are unbounded.

Introduction Historical background The main results Overview of the paper Guide Notation Chapter 1. The curve, explicit divisors, and relations 1.1. A generalization of the Legendre curve 1.2. Explicit points and the visible subgroup 1.3. Relations 1.4. Torsion 1.5. First main theorem 1.6. Complement: Other curves Chapter 2. Descent calculations 2.1. The isogeny φ 2.2. The homomorphism (x − T ) 2.3. The image of (x − T ) 2.4. Proof of the main theorem Chapter 3. Minimal regular model, local invariants, and domination by a product of curves 3.1. Models 3.2. Local invariants of the Néron model 3.3. Domination by a product of curves Chapter 4. Heights and the visible subgroup 4.1. Height pairing 4.2. A group-theoretic pairing 4.3. Structure of the visible subgroup 4.4. Discriminants Chapter 5. The L-function and the BSD conjecture 5.1. The L-function 5.2. The conjecture of Birch and Swinnerton-Dyer for J 5.3. Elementary calculation of the L-function 5.4. Ranks Chapter 6. Analysis of J[p] and NS(X d ) tor 6.1. Kodaira-Spencer and p-torsion iii iv CONTENTS 6.2. Néron-Severi of X d is torsion-free Chapter 7. Index of the visible subgroup and the Tate-Shafarevich group 7.1. Visible versus Mordell-Weil 7.2. Tamagawa number 7.3. Application of the BSD formula Chapter 8. Monodromy of ℓ-torsion and decomposition of the Jacobian 8.1. Statement of results 8.2. New and old 8.3. Endomorphism rings 8.4. The Λ-module structure of J[ℓ] 8.5. Monodromy of J[λ] 8.6. Independence 8.7. Conclusion Appendix A.

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