Curves with prescribed period and index over local fields

Sharif, Shahed

doi:10.1016/j.jalgebra.2007.02.020

Cited by 6 publications

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References 6 publications

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“…For example, for genus 1 curves over local fields, the period and index are always equal [10], and over R, the index of any curve must divide 2. One may then ask what triples (g, P, I) actually occur as the genus, period, and index of a curve over a fixed K. Over local fields of characteristic not 2, the problem is solved in [18]. We consider the g = 1 case over number fields and prove a full converse: Theorem 1.2.…”

Section: Introductionmentioning

confidence: 99%

Period and index of genus one curves over global fields

Sharif

2011

Math. Ann.

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Section: Introductionmentioning

confidence: 99%

Period and index of genus one curves over global fields

Sharif

2011

Math. Ann.

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“…The divisor D is Galois-equivariant on X . As N K /Q p (π ) = 1/ p is trivial in E(Q p ) = Q * p / p Z , the same argument as in Lemma 8 of [21] shows that D is principal. Therefore we can find an f such that div( f ) = D, which is the one we were looking for.…”

Section: Proposition 34mentioning

confidence: 71%

“…We finally show that it is possible to choose f in such a way that the order of [Alb 1 Y ] is p. As in [21], for this it is sufficient to find an f whose number of ramification points is 2mp with m odd (this yields a Y of genus mp + 1 by the Hurwitz formula). To get an example with m = 1, we proceed as in [21]. Define a divisor D on X = E by…”

Section: Proposition 34 Let P Be An Odd Prime Number There Exists a C...mentioning

confidence: 94%

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Galois sections for abelianized fundamental groups

Harari

Szamuely

2009

Math. Ann.

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Given a smooth projective curve $X$ of genus at least 2 over a number field $k$, Grothendieck's Section Conjecture predicts that the canonical projection from the \'etale fundamental group of $X$ onto the absolute Galois group of $k$ has a section if and only if the curve has a rational point. We show that there exist curves where the above map has a section over each completion of $k$ but not over $k$. In the appendix Victor Flynn gives explicit examples in genus 2. Our result is a consequence of a more general investigation of the existence of sections for the projection of the \'etale fundamental group `with abelianized geometric part' onto the Galois group. We give a criterion for the existence of sections in arbitrary dimension and over arbitrary perfect fields, and then study the case of curves over local and global fields more closely. We also point out the relation to the elementary obstruction of Colliot-Th\'el\`ene and Sansuc.Comment: This is the published version, except for a characteristic 0 assumption added in Section 5 which was unfortunately omitted there. Thanks to O. Wittenberg for noticing i

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“…This method is essentially the same as that used in the local case in [Sha07]. However, the situation in the local case is easier and much more explicit, since we are able to make use of Tate curves.…”

Section: Outline Of Proofmentioning

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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Curves with prescribed period and index over local fields

Cited by 6 publications

References 6 publications

Period and index of genus one curves over global fields

Period and index of genus one curves over global fields

Galois sections for abelianized fundamental groups

Period and index for higher genus curves

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