Given an L-function, one of the most important questions concerns its vanishing at the central point; for example, the Birch and Swinnerton-Dyer conjecture states that the order of vanishing there of an elliptic curve L-function equals the rank of the Mordell-Weil group. The Katz and Sarnak Density Conjecture states that this and other behavior is well-modeled by random matrix ensembles. This c...

In a few earlier papers ([8], [9], [10]) attention was called to the striking parallel between the ideas surrounding the well-known conjecture of Birch and Swinnerton-Dyer for elliptic curves, and the mysterious section conjecture of Grothendieck [6] that concerns hyperbolic curves. We wish to explain here some preliminary ideas for ‘effective non-abelian descent’ on hyperbolic curves equipped ...

The Weil conjectures constitute one of the central landmarks of 20th century algebraic geometry: not only was their proof a dramatic triumph, but they served as a driving force behind a striking number of fundamental advances in the field. The conjectures treat a very elementary problem: how to count the number of solutions to systems of polynomial equations over finite fields. While one might ...

To Takayuki Oda on his 60th birthday.

1. Quick Review of Elliptic Curves 2 2. Elliptic Curves over C 4 3. Elliptic Curves over Local Fields 6 4. Elliptic Curves over Number Fields 12 5. Elliptic Curves with Complex Multiplication 15 6. Descent 22 7. Elliptic Units 27 8. Euler Systems 37 9. Bounding Ideal Class Groups 43 10. The Theorem of Coates and Wiles 47 11. Iwasawa Theory and the “Main Conjecture” 50 12. Computing the Selmer G...

A positive integer D is called a ‘congruent number’ if there exists a right triangle with rational sidelengths with area D. Over the centuries there have been many investigations attempting to classify the congruent numbers, but little was known until Tunnell [T] brilliantly applied a tour de force of methods and provided a conditional solution to this problem. It turns out that a square-free i...

Let K be a number field, and let A/K be an abelian variety. Let c denote the product of the Tamagawa numbers of A/K, and let A(K)tors denote the finite torsion subgroup of A(K). The quotient c/|A(K)tors| is a factor appearing in the leading term of the L-function of A/K in the conjecture of Birch and Swinnerton-Dyer. We investigate in this article possible cancellations in this ratio. Precise r...

We review some of Kolyvagin’s results and conjectures about elliptic curves, then make a new conjecture that slightly refines Kolyvagin’s conjectures. We introduce a definition of finite index subgroups Wp ⊂ E(K), one for each prime p that is inert in a fixed imaginary quadratic field K. These subgroups generalize the group ZyK generated by the Heegner point yK ∈ E(K) in the case ran = 1. For a...

Fix an elliptic curve E/Q, and assume the generalized Riemann hypothesis for the L-function L(E D , s) for every quadratic twist E D of E by D ∈ Z. We combine Weil's explicit formula with techniques of Heath-Brown to derive an asymptotic upper bound for the weighted moments of the analytic rank of E D. It follows from this that, for any unbounded increasing function f on R, the analytic rank an...