On the Brauer group of diagonal quartic surfaces

Ieronymou, Evis; Skorobogatov, Alexei N.; Zarhin, Yuri G.

doi:10.1112/jlms/jdq083

Cited by 36 publications

(24 citation statements)

References 21 publications

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“…If we want V to have the form (6) Here V 2 and V 8 are the same, but by considering both we can confine ourselves to the study of points for which the X i are integers with X 0 , X 2 odd. In Bright's table V 4 is case A75, and Ieronymou, Skorobogatov and Zarhin [Ieronymou et al 2011] have shown that it has no transcendental Brauer-Manin obstruction either. V 2 and V 8 are instances of Bright's case A104.…”

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confidence: 99%

Density of rational points on certain surfaces

Swinnerton‐Dyer

2013

Algebra Number Theory

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Let V be a nonsingular projective surface defined over ‫ޑ‬ and having at least two elliptic fibrations defined over ‫;ޑ‬ the most interesting case, though not the only one, is when V is a K3 surface with these properties. We also assume that V ‫)ޑ(‬ is not empty. The object of this paper is to prove, under a weak hypothesis, the Zariski density of V ‫)ޑ(‬ and to study the closure of V ‫)ޑ(‬ under the real and the p-adic topologies. The first object is achieved by the following theorem:Let V be a nonsingular surface defined over ‫ޑ‬ and having at least two distinct elliptic fibrations. There is an explicitly computable Zariski closed proper subset X of V defined over ‫ޑ‬ such that if there is a pointThe methods employed to study the closure of V ‫)ޑ(‬ in the real or p-adic topology demand an almost complete knowledge of V ; a typical example of what they can achieve is as follows. Let V c be 1. Introduction. Let V be a nonsingular projective surface defined over ‫ޑ‬ and having at least two elliptic fibrations defined over ‫;ޑ‬ the most interesting case, though not the only one, is when V is a K3 surface with these properties. (By an elliptic fibration we mean a fibration by curves of genus 1; we do not assume that these curves have distinguished points, so they need not be elliptic curves in the number-theoretic sense.) We also assume that V ‫)ޑ(‬ is not empty. The object of this paper is to prove, under a weak hypothesis, the Zariski density of V ‫)ޑ(‬ and to study the closure of V ‫)ޑ(‬ under the real and the p-adic topologies. These results can be found in Section 2; Sections 3 and 4 contain applications and examples. Note that the geometric terminology in this paper is that of Weil.For the real and Zariski topologies, such results have been proved for a particular family of such surfaces in [Logan et al. 2010], and for one such surface already in [Swinnerton-Dyer 1968]; see Section 3. I am indebted to the referee for drawing

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confidence: 99%

Density of rational points on certain surfaces

Swinnerton‐Dyer

2013

Algebra Number Theory

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“…A naive extrapolation suggests that this requires a search for points up to height 10 15 . Further, one has to compute the transcendental Brauer-Manin obstruction in the case when [ISZ,Corollary 3.3] is not applicable.…”

Section: Resultsmentioning

confidence: 99%

Rational points on diagonal quartic surfaces

Elsenhans

2012

Math. Comp.

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“…An easy way to see this is to use [17,Theorem 1.4], which says that if A is odd torsion and fixed by the covering automorphism z → −z, then it comes from a class in Br(Y ).…”

Section: Supersingular Reduction and Pic(x)mentioning

confidence: 99%

Brauer–Manin obstructions on degree 2 K3 surfaces

Corn

Nakahara

2018

Res. number theory

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We analyze the Brauer-Manin obstruction to rational points on the K3 surfaces over Q given by double covers of P 2 ramified over a diagonal sextic. After finding an explicit set of generators for the geometric Picard group of such a surface, we find two types of infinite families of counterexamples to the Hasse principle explained by the algebraic Brauer-Manin obstruction. The first type of obstruction comes from a quaternion algebra, and the second type comes from a 3-torsion element of the Brauer group, which gives an affirmative answer to a question asked by Ieronymou and Skorobogatov.

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On the Brauer group of diagonal quartic surfaces

Cited by 36 publications

References 21 publications

Density of rational points on certain surfaces

Density of rational points on certain surfaces

Rational points on diagonal quartic surfaces

Brauer–Manin obstructions on degree 2 K3 surfaces

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