Recent workers [1, 3] have proved density theorems about the rational points on K3 surfaces of the form

$$V:\;X_0^4+cX_1^4=X_2^4+cX_3^4$$

Let V be a nonsingular cubic surface defined over the finite field Fq. It is well known that the number of points on V satisfies #V(Fq) = q2 + nq + 1 where −2 ≤ n ≤ 7 and that n = 6 is impossible; see for example [1], Table 1. Serre has asked if these bounds are best possible for each q. In this paper I shall show that this is so, with three exceptions:

Let Γ be an elliptic curve defined over Q, all of whose 2-division points are rational, and let Γb be its quadratic twist by b. Subject to a mild additional condition on Γ, we find the limit of the probability distribution of the dimension of the 2-Selmer group of Γb as the number of prime factors of b increases; and we show that this distribution depends only on whether the 2-Selmer group of Γ has odd or even dimension.

Let $V$ be a complete nonsingular projective surface defined over an algebraic number field $k$, such that the Picard variety of $V$ is trivial and Pic($\bar{V}$) is torsion-free. Since our main interest is in necessary conditions for $V(k)$ not to be empty, we shall further assume that $V(k_v)$ is non-empty for every completion $k_v$ of $k$. We do not assume that $\bar{V}$ is rational, and indeed the case which primarily interests us is when $V$ is a K3 surface. Our objective is to describe effective ways of computing the Brauer–Manin obstructions to the existence of points on $V$ defined over $k$. Write \[ \hbox{Br}^0 (V) = \{\hbox{Ker}(\hbox{Br}(V)\longrightarrow\hbox{Br}(\bar{V}))\}/\{{\rm Im}({\rm Br}(k)\longrightarrow\hbox{Br}(V))\};\] then it is known that $\hbox{Br}^0(V)$ is isomorphic to ${\rm H}^1(k, \hbox{Pic}(\bar{V}))$, though to the best of our knowledge there is in general no known algorithm for computing either $\hbox{Br}^0(V)$ or this isomorphism. It is generally believed that $\hbox{Br}^0(V)$ contains all the information about the Brauer group Br$(V)$ which is useful in this context. Most of this paper is concerned with computing groups isomorphic to $\hbox{Br}^0(V)$, and with describing in terms of these groups the Brauer–Manin obstructions coming from elements of $\hbox{Br}^0(V)$.

2000 Mathematical Subject Classification: 10B10.

The object of this paper is to find all the irreducible algebraic surfaces which (for special values of the parameters b, r, s) are invariant under the Lorenz system

x˙ = X(x, y, z) = s(y−x), y˙ = Y(x, y, z) = rx−y−xz, ż = Z(x, y, z) =−bz+xy. (1)

It is customary in considering the Lorenz system to require the parameters b, r, s to be all strictly positive; however for this particular problem we shall follow previous practice in only imposing the condition s ≠ 0. (If s = 0 the equations are trivially integrable and x is constant on any trajectory; thus x should be regarded as a parameter and the question discussed in this paper ceases to be a natural one.)

The main part of the paper finds necessary conditions for solubility of a pencil of curves of genus 1, each of which is a 2-covering of an elliptic curve with at least one 2-division point. As in previous work, these are proved subject to Schinzel's Hypothesis and to the finiteness of the Tate-\u{S}afarevi\u{c} group of elliptic curves defined over a number field. It thus generalizes earlier work of Colliot-Thélène, Skorobogatov and the second author.

The final section gives necessary conditions (though of a rather ugly nature) for the solubility of a Del Pezzo surface of degree 4.

2000 Mathematical Subject Classification: 11D25.

The author investigates the solubility in rationals of equations of the form $$a_0X_0^4 + a_1X_1^4 + a_2X_2^4 + a_3X_3^4 = 0$$where $a_0a_1a_2a_3$ is a square, building on the ideas which Colliot-Thène, Skorobogatov and he have developed; see{\em Invent. Math.} 134 (1998) 579--650. He obtains sufficient conditions for solubility, which appear to be related to the absence of a Brauer--Manin obstruction. This represents the first large family of K3 surfaces which almost satisfy theHasse principle, in the sense that the auxiliary condition which ensures that local solubility everywhere implies global solubility is nearly always satisfied. 1991 Mathematics Subject Classification: 10B10.

In an earlier paper [2], I gave explicit rules for calculating the Brauer group of a Del Pezzo surface of degree 3 or 4. Recent developments have made it desirable to extend these to the construction of the corresponding Azumaya algebras, so as to be able to calculate the associated Brauer–Manin obstruction to weak approximation or even to the Hasse principle. That is the main purpose of this note, which needs to be read in conjunction with [2]. The special case of diagonal cubic surfaces has already been treated in [1], though the process described there is not an algorithm in the strict sense; however, in numerical cases it is probably more efficient than the algorithm described in this note would be.

1. Let V be a non-singular rational surface defined over an algebraic number field k. There is a standard conjecture that the only obstructions to the Hasse principle and to weak approximation on V are the Brauer–Manin obstructions. A prerequisite for calculating these is a knowledge of the Brauer group of V; indeed there is one such obstruction, which may however be trivial, corresponding to each element of Br V/Br k. Because k is an algebraic number field, the natural injection

is an isomorphism; so the first step in calculating the Brauer–Manin obstruction is to calculate the finite group H1 (k), Pic .

The Hopf bifurcation theorem describes the creation of a limit cycle from an isolated singular point of a system of first-order differential equations depending on a parameter. This paper describes a method for determining explicitly a range of values of the parameter throughout which the Hopf configuration continues to exist; only the three-dimensional case is described in this paper, but the method can be generalized.