Period, index and potential, III

Clark, Pete L.; Sharif, Shahed

doi:10.2140/ant.2010.4.151

Cited by 26 publications

(31 citation statements)

References 23 publications

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“…Recall that the index of a variety X over a field is the gcd of the degrees of the closed points on X. By work of Clark and Sharif [CS10] we can find a torsor V under an elliptic curve E/k of period N and index N 2 . Moreover, the proof of loc.…”

Section: 2mentioning

confidence: 99%

Degree and the Brauer–Manin obstruction

Creutz

Viray

2018

Alg. Number Th.

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Let X ⊂ P n k be a smooth projective variety of degree d over a number field k and suppose that X is a counterexample to the Hasse principle explained by the Brauer-Manin obstruction. We consider the question of whether the obstruction is given by the d-primary subgroup of the Brauer group, which would have both theoretic and algorithmic implications. We prove that this question has a positive answer in the case of torsors under abelian varieties, Kummer surfaces and (conditional on finiteness of Tate-Shafarevich groups) bielliptic surfaces. In the case of Kummer surfaces we show, more specifically, that the obstruction is already given by the 2-primary torsion, and indeed that this holds for higher-dimensional Kummer varieties as well. We construct a conic bundle over an elliptic curve that shows that, in general, the answer is no.Question 1.1. Suppose that X ֒→ P n is embedded as a subvariety of degree d in projective space. Does the d-primary subgroup of Br X capture the Brauer-Manin obstruction to rational points on X?More intrinsically, let us say that degrees capture the Brauer-Manin obstruction on X if the d-primary subgroup of Br X captures the Brauer-Manin obstruction to rational points on X for all integers d that are the degree of some k-rational globally generated ample line bundle on X. Since any such line bundle determines a degree d morphism to projective space and conversely, it is clear that the answer to Question 1.1 is affirmative when degrees capture the Brauer-Manin obstruction.1.1. Summary of results. In general, the answer to Question 1.1 can be no (see the discussion in §1.2). However, there are many interesting classes of varieties for which the answer is yes. We prove that degrees capture the Brauer-Manin obstruction for torsors under abelian varieties, for Kummer surfaces, and, assuming finiteness of Tate-Shafarevich groups of elliptic curves, for bielliptic surfaces. We also deduce (from various results appearing in the literature) that degrees capture the Brauer-Manin obstruction for all geometrically rational minimal surfaces.Assuming finiteness of Tate-Shafarevich groups, one can deduce the result for torsors under abelian varieties rather easily from a theorem of Manin (see Remark 4.4 and Proposition 4.9). In §4 we unconditionally prove the following much stronger result.Theorem 1.2. Let X be a k-torsor under an abelian variety, let B ⊂ Br X be any subgroup, and let d be any multiple of the period of X. In particular, d could be taken to be the degree of a k-rational globally generated ample line bundle.This not only shows that degrees capture the Brauer-Manin obstruction (apply the theorem with B = Br X), but also that the Brauer classes with order relatively prime to d cannot provide any obstructions to the existence of rational points. Remark 1.3. As one ranges over all torsors of period d under all abelian varieties over number fields, elements of arbitrarily large order in of (Br V )[d ∞ ] are required to produce the obstruction (see Proposition 4.10).We use Theorem 1.2 to ded...

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Section: 2mentioning

confidence: 99%

Degree and the Brauer–Manin obstruction

Creutz

Viray

2018

Alg. Number Th.

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“…A slightly weaker version of the above was proved in [CS10], and was used to construct Shafarevich-Tate groups with arbitrarily high p-rank.…”

Section: Period and Indexmentioning

confidence: 99%

Period and index for higher genus curves

Sharif

2018

Journal of Number Theory

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Period and indexLet K be a field; usually, we will consider it to be a number field. By a curve C over K, we shall mean a smooth projective geometrically integral curve. The period and index of C over K are two integer invariants which measure the failure of C to have rational points. Specifically, the index is the gcd of degrees of all effective divisors D ∈ Div C-that is, effective divisors which are rational over K. Equivalently, the index is the gcd of degreesσP is a rational effective divisor, where σ ranges over the embeddings of L into an algebraic closure of K; and conversely any minimal rational effective divisor is of this form. Also observe that if C already has K-points, then its index is 1.The period is less stringent: here we look at the smallest positive degree of rational divisor classes; these are given by divisors which are linearly equivalent to their Galois conjugates. To see that the two invariants need not be the same, consider any conic over R without rational points, say the curve with affine piece x 2 + y 2 = −1. Certainly the index is 2, but as our curve is genus 0, there is a single divisor class of degree 1; namely, the class of a point. Therefore this class must be rational over R, and so the period is 1.As a rational divisor automatically belongs to a rational divisor class, we have P | I. Furthermore, the canonical class gives a rational divisor of degree 2(g − 1), so I | 2(g − 1). Lichtenbaum in [Lic69] found further conditions on the possible values of the period and index: Theorem 1.1 (Theorem 8,[Lic69]). Let C/K be a curve over a field with genus g, period P , and index I. Then2 , and(ii) if either 2(g − 1)/I or P is even, then I | P 2 .We say a triple of integers (g, P, I) is admissible if they satisfy the divisibility conditions of Lichtenbaum's theorem; that is, if they are possible values for the genus, period, and index of a curve. The goal of this paper is to determine whether every admissible triple indeed occurs as the invariants of some curve over a number field. Our main result is the following: 1

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“…Also, in a recent preprint by Clark and Sharif [3] it is shown that for any E/Q the Tate-Shafarevich group can get arbitrarily large over extensions F/Q of degree p, not necessarily Galois. Remark 1.3.…”

Section: Remark 12mentioning

confidence: 99%