By Bjorn Poonen, Yuri Tschinkel

One of many nice successes of 20th century arithmetic has been the striking qualitative knowing of rational and quintessential issues on curves, gleaned partially throughout the theorems of Mordell, Weil, Siegel, and Faltings. It has develop into transparent that the research of rational and quintessential issues has deep connections to different branches of arithmetic: advanced algebraic geometry, Galois and ,tale cohomology, transcendence thought and diophantine approximation, harmonic research, automorphic kinds, and analytic quantity concept. this article, which makes a speciality of larger dimensional forms, presents accurately such an interdisciplinary view of the topic. it's a digest of analysis and survey papers by means of top experts; the e-book files present wisdom in higher-dimesional mathematics and offers symptoms for destiny examine. it will likely be invaluable not to basically to practitioners within the box, yet to a large viewers of mathematicians and graduate scholars with an curiosity in mathematics geometry. participants contain: 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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Some K3 surfaces contain one or more pencils of curves of genus 1, and these pencils may even be of the kind discussed in the previous section; but one should not confine one’s attention to K3 surfaces with this additional property. For the time being, there is merit in concentrating on diagonal quartics V : a0 X04 + a1 X14 + a2 X24 + a3 X34 = 0, (12) because these contain few enough parameters to make systematic numerical experimentation possible. However, the number theory of such surfaces may be exceptional, because the geometry certainly is.

For this machinery to have any chance of working, we must be able to implement the 2-descent on Jλ in a manner which is uniform in λ. This more or less requires Jλ to have its 2-division points defined over Q(λ) and therefore to have the form Y 2 = (X − c1 (λ))(X − c2 (λ))(X − c3 (λ)) (9) where the ci (λ) are in Q(λ); but an additional trick, given in [2], enables the method to be used even if Jλ has just one 2-division point in Q(λ). 22 The details of this method are unattractive, but the strategy is as follows.

A variant of the method above can be applied to diagonal cubic surfaces V : a0 X03 + a1 X13 = a2 X23 + a3 X33 , (10) subject to one very counterintuitive condition, which is that K, the field of definition of V , must not contain the primitive cube roots of unity. Write V in the form a0 X03 + a1 X13 = λY 3 , a2 X23 + a3 X33 = λY 3 (11) where λ is at √ our disposal. We now have two pencils of curves of genus 1, each of which is a −3-covering of its Jacobian; and we have to apply the method simultaneously to both curves.

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