Rational points on diagonal cubic surfaces

It is known that there are diagonal cubic surfaces defined over Q which do not contain rational points, although for each prime p they do contain p-adic points. The simplest example is

3X_{0}^{3} + 4X_{1}^{3} + 10X_{2}^{3} + 15X_{3}^{3} = 0.

It is natural to ask what condition additional to solubility in each p-adic field is needed to ensure solubility in Q; and there is strong numerical evidence that this condition is that there is no Brauer-Manin obstruction. However, one does not know how to exploit such a condition.

Provided one assumes the finiteness of the Tate-Šafarevič groups of elliptic curves, I shall show that there is a sufficient condition for solubility which is only slightly stronger than the vanishing of the Brauer-Manin obstruction, and which can be expressed in very straightforward terms.

Designing efficient Liapounov functions

Let ẋ_{i} = f_{i}(x_{1}, …, x_{n}) for 1 ≤ i ≤ n be a system of first order ordinary differential equations in R. A function V (x_{1}, …, x_{n}) is called a Liapounov function for the system if

W = ∑ \frac{∂V}{∂x_{i}} f_{i}

satisfies W < 0 whenever V ≥ 0. Liapounov’s theorem in its simplest form states that if V is a Liapounov function then (subject to some abuse of language in respect of behaviour at infinity) every trajectory eventually enters the set {V ≤ 0} and never thereafter leaves it.

The construction of efficient Liapounov functions is currently a hit-and-miss affair. This seminar is a contribution towards making the process more systematic.