On the local-global divisibility over abelian varieties

Gillibert, Florence; Ranieri, Gabriele

doi:10.5802/aif.3179

Cited by 12 publications

(19 citation statements)

References 6 publications

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“…In the following proposition we put together the main results of [PRV1] and [PRV2] and we use some results of [GR2].…”

Section: Known Resultsmentioning

confidence: 99%

“…Many mathematicians got criterions for the validity of the local-global divisibility principle for many families of commutative algebraic groups, as algebraic tori ( [DZ1] and [Ill]), elliptic curves ( [Cre1], [Cre2], [DZ1], [DZ2], [DZ3], [GR1], [Pal1], [Pal2], [PRV1], [PRV2]), and very recently polarized abelian surfaces ( [GR2]) and GL 2 -type varieties ( [GR3]).…”

Section: Introductionmentioning

confidence: 99%

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Counterexamples to the local–global divisibility over elliptic curves

Ranieri

2017

Annali di Matematica

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“…In the following proposition we put together the main results of [PRV1] and [PRV2] and we use some results of [GR2].…”

Section: Known Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Counterexamples to the local–global divisibility over elliptic curves

Ranieri

2017

Annali di Matematica

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“…Proof In [, Section 5], it was proved that

H_{cyc}^{1} false(G_{2}, {false(double-struckZ / p^{2} double-struckZ false)}^{2} false) \neq 0 .

Let

N : = ker (π)

be the subgroup of matrices congruent to identity modulo

p^{2}

. By construction, we have an exact sequence

1 \overset{}{\to} N \overset{}{\to} G_{3} \overset{}{\to} G_{2} \overset{}{\to} 1 .

Let

M_{3} : = {(Z / p^{3} Z)}^{2}

and

M_{2} = M_{3} false[p^{2} false]

.…”

Section: Non‐direct Summandsmentioning

confidence: 99%

“…This implies that

H^{1} false(G_{2}, M_{2} false) = H_{cyc}^{1} false(G_{2}, M_{2} false)

, hence one can deduce from the previous discussion, combined with the fact that the inflation sends

H_{cyc}^{1} false(G_{2}, M_{2} false)

H_{cyc}^{1} false(G_{3}, M_{3} false)

, that

H_{cyc}^{1} false(G_{2}, {false(double-struckZ / p^{2} double-struckZ false)}^{2} false) = H_{cyc}^{1} false(G_{3}, {false(double-struckZ / p^{3} double-struckZ false)}^{2} false) .

Let

g : = (\begin{matrix} 1 & - 3 \\ 1 & - 2 \end{matrix}),

which has order 3. Because

g H_{2} = H_{2} g

(see [, Section 5]),

H_{2}

is a normal subgroup of

G_{2}

, and we have an exact sequence

1 \overset{}{\to} H_{2} \overset{}{\to} G_{2} \overset{}{\to} false⟨ g false⟩ \overset{}{\to} 1 .

p \neq 3

, given a cyclic subgroup

Γ ⩽ G_{2}

, its

p

‐Sylow subgroup is a subgroup of

H_{2}

, hence of the form

⟨ h

…”

Section: Non‐direct Summandsmentioning

confidence: 99%

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Selmer groups are intersection of two direct summands of the adelic cohomology

Gillibert

et al. 2019

Bull. London Math. Soc.

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We give a positive answer to a conjecture by Bhargava, Kane, Lenstra Jr., Poonen and Rains, concerning the cohomology of torsion subgroups of elliptic curves over global fields. This implies that, given a global field k and an integer n, for 100% of elliptic curves E defined over k, the nth Selmer group of E is the intersection of two direct summands of the adelic cohomology group H1false(boldA,E[n]false). We also give examples of elliptic curves for which the conclusion of this conjecture does not hold.

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Local-global questions for divisibility in commutative algebraic groups

Dvornicich

Paladino

2022

European Journal of Mathematics

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This is a survey focusing on the Hasse principle for divisibility of points in commutative algebraic groups and its relation with the Hasse principle for divisibility of elements of the Tate–Shavarevich group in the Weil–Châtelet group. The two local-global subjects arose as a generalization of some classical questions considered respectively by Hasse and Cassels. We describe the deep connection between the two problems and give an overview of the long-established results and the ones achieved during the last twenty years, when the questions were taken up again in a more general setting. In particular, by connecting various results about the two problems, we describe how some recent developments in the first of the two local-global questions imply an answer to Cassels’ question, which improves all the results published before about that problem. This answer is best possible over $${\mathbb Q}$$ Q . We also describe some links with other similar questions, for example the Support Problem and the local-global principle for existence of isogenies of prime degree in elliptic curves.

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On the local-global divisibility over abelian varieties

Cited by 12 publications

References 6 publications

Counterexamples to the local–global divisibility over elliptic curves

Counterexamples to the local–global divisibility over elliptic curves

Selmer groups are intersection of two direct summands of the adelic cohomology

Local-global questions for divisibility in commutative algebraic groups

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