By Bjorn Poonen, Yuri Tschinkel

One of many nice successes of 20th century arithmetic has been the notable qualitative figuring out of rational and indispensable issues on curves, gleaned partially during the theorems of Mordell, Weil, Siegel, and Faltings. It has develop into transparent that the research of rational and necessary issues has deep connections to different branches of arithmetic: complicated algebraic geometry, Galois and ,tale cohomology, transcendence idea and diophantine approximation, harmonic research, automorphic kinds, and analytic quantity concept. this article, which makes a speciality of better dimensional types, presents accurately such an interdisciplinary view of the topic. it's a digest of study and survey papers via prime experts; the booklet files present wisdom in higher-dimesional mathematics and offers symptoms for destiny learn. will probably be worthwhile not to purely to practitioners within the box, yet to a large viewers of mathematicians and graduate scholars with an curiosity in mathematics geometry. members comprise: P. Swinnerton-Dyer * B. Hassett * Yu. Tschinkel * J. Shalika * R. Takloo-Bighash * J.-L. Colliot-Th,lSne * A. de Jong * Ph. Gille * D. Harari * J. Harris * B. Mazur * W. Raskind * J. Starr * T. Wooley

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By Faltings’ theorem, C(K) is then finite; thus each point of V (K) lies on one of a finite set of fibres, and it is enough to study these. In contrast, we know nothing about the case when C is elliptic. The surfaces with κ = −∞ are precisely the ruled surfaces — that is, those which are birationally equivalent over C to P1 × C for some curve C. Among these, by far the most interesting are the rational surfaces, which are birationally equivalent to P2 over C. Surfaces with κ = 0 fall into four families: • Abelian surfaces.

But K3 surfaces pose new problems — and not ones on which any practicable amount of computation is likely to shed light. If V is a K3 surface, then we have to study not N (H, V ) but N (H, U ) where U is obtained from V by deleting the curves of genus 0 on V defined over Q, of which there may be an infinite number. One can expect that N (H, U ) ∼ A(log H)c for some constants A and c; and it seems reasonable to hope that c will be a half-integer. The surface (13) suggests that we can have c = 0, and it must be certain (though perhaps difficult to prove) that c can sometimes be strictly positive.

Let V be a variety defined over Q; then R-equivalence is defined as the finest equivalence relation such that two points given by the same parametric solution are equivalent. Alternatively, it is the finest equivalence relation such that for any map f : P1 → V and points P1 , P2 , all defined over Q, the points f (P1 ) and f (P2 ) are equivalent. A good deal is known about R-equivalence on cubic surfaces; in particular, it is shown in [43] that the closure of an R-equivalence class in V (A) is computable, and that the closures of two Requivalence classes are either the same or disjoint.