Arithmetic of diagonal quartic surfaces, II

The author investigates the solubility in rationals of equationsof the form where a₀a₁a₂a₃is a square, building on the ideas which Colliot-Thène,Skorobogatov and he have developed; see Invent. Math. 134 (1998)579–650. He obtains sufficient conditions for solubility,which appear to be related to the absence of a Brauer–Maninobstruction. This represents the first large family of K3 surfaceswhich almost satisfy the Hasse principle, in the sense thatthe auxiliary condition which ensures that local solubilityeverywhere implies global solubility is nearly always satisfied.1991 Mathematics Subject Classification: 10B10.     

The number of lattice points on a convex curve

If C is a strictly convex plane curve of length l, it has been known for a long time that the number of integer lattice points on C is $\text{O(l}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}\text{)}$ and the exponent is best possible. In this paper, it is shown that the exponent can be decreased by imposing suitable smoothness conditions on C; in particular, if C has a continuous third derivative with a sensible bound, the best possible value of the exponent lies between $\text{3}\text{5}$ and $\text{1}\text{3}$ inclusive.     

Solubility of Certain Pencils of Curves of Genus 1, and of the Intersection of two Quadrics in P4

The main part of the paper finds necessary conditions for solubilityof a pencil of curves of genus 1, each of which is a 2-coveringof an elliptic curve with at least one 2-division point. Asin previous work, these are proved subject to Schinzel's Hypothesisand to the finiteness of the Tate-afarevi group of ellipticcurves defined over a number field. It thus generalizes earlierwork of Colliot-Thélène, Skorobogatov and thesecond author. The final section gives necessary conditions (though of a ratherugly nature) for the solubility of a Del Pezzo surface of degree4. 2000 Mathematical Subject Classification: 11D25.     

Some third-order ordinary differential equations

The paper deals with periodic orbits in three systems of ordinarydifferential equations. Two of the systems, the Falkner–Skanequations and the Nosé equations, do not possess fixedpoints, and yet interesting dynamics can be found. Here, periodicorbits emerge in bifurcations from heteroclinic cycles, connectingfixed points at infinity. We present existence results for suchperiodic orbits and discuss their properties using careful asymptoticarguments. In the final part results about the Nosé equationsare used to explain the dynamics in a dissipative perturbation,related to a system of dynamo equations.     

Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points

Inventiones mathematicae -     

The Falkner-Skan Equation II: Dynamics and the Bifurcations of P- and Q-Orbits

The Falkner-Skan equation is a reversible three-dimensional system of ordinary differential equations with two distinguished straight-line trajectories which form a heteroclinic loop between fixed points at infinity. We showed in the previous paper (1995, J. Differential Equations119, 336-394) that at positive integer values of the parameter λ there are bifurcations creating large sets of periodic and other interesting trajectories. Here we show that all but two of these trajectories are destroyed in another sequence of bifurcations as λ→∞, and by considering topological invariants and orderings on certain manifolds we obtain unusually detailed information about the sequences of bifurcations which can occur.     

Arithmetic of Weil curves

Inventiones mathematicae -