A polynomial relation f(x, y) = 0 in two variables defines a curve C. If the coefficients of the polynomial are rational numbers then one can ask for solutions of the equation f(x, y) = 0 with x, y ∈ Q, in other words for rational points on the curve. If we consider a non-singular projective model C of the curve then over C it is classified by its genus. Mordell conjectured, and in 1983 Falting...

This essay starts by first explaining, for elliptic curves defined over Q, the statement of the conjecture of Birch and Swinnerton-Dyer. Alongside, it contains a discussion of some results that have been proved in the direction of the conjecture, such as the theorem of Kolyvagin-Gross-Zagier and the weak parity theorem of Tim and Vladimir Dokchitser. The second, third and fourth part of the ess...

In this talk I shall attempt to introduce some of the main features of the Birch and Swinnerton-Dyer conjecture, (BSD). The congruent number problem, deciding whether an integer D is the area of a right angle triangle with rational sides, is not easy. It turns out that the problem is equivalent to finding out if a certain elliptic curve has an infinite number of rational points. In 1983 Tunnell...

We provide a lower bound for the coherence of the homotopy commutativity of the Brown–Peterson spectrum, BP , at a given prime p and prove that it is at least (2p2 + 2p− 2)–homotopy commutative. We give a proof based on Dyer–Lashof operations that BP cannot be a Thom spectrum associated to n–fold loop maps to BSF for n = 4 at 2 and n = 2p + 4 at odd primes. Other examples where we obtain estima...

Let E be an optimal elliptic curve of conductor N , such that the L-function LE(s) of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N split. The Gross-Zagier theorem gives a formula that expresses the Birch and Swinnerton-Dyer conjectural order the Shafarevich-Tate group of E over K as a rational number. We extract an integer factor from...

Michael Dyer Artificial Intelligence Lab Computer Science Department UCLA Abst rac t This paper diacusses a method for learning thematic level structures, i.e. abstract plan/goal combinations, by observing the bad planning behavior of narrative characters. The learning method discussed is a one-trial, schema acquisition method, which is similar to DeJong's [DeJong, 1983]. The method uses constr...

We prove that almost any pair of real numbers α, β, satisfies the following inhomogeneous uniform version of Littlewood’s conjecture: ∀γ, δ ∈ R, lim inf |n|→∞ |n| 〈nα− γ〉〈nβ − δ〉 = 0, (0.1) where 〈·〉 denotes the distance from the nearest integer. The existence of even a single pair that satisfies (0.1), solves a problem of Cassels [Ca] from the 50’s. We then prove that if 1, α, β span a totally...

High deer numbers in northern Fiordland in the 1960s significantly changed forest understorey composition. The density of woody plants in the understorey was reduced in some areas by as much as 50%, and preferred plants became less abundant than those seldom eaten. However, the impact of deer and wapiti varied between forest types. Seral communities and silver beech forests at low altitudes in ...