On a filtration of  for an abelian variety

Gazaki, Evangelia

doi:10.1112/s0010437x14007453

Cited by 8 publications

(39 citation statements)

References 15 publications

(27 reference statements)

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“…We start by reviewing the construction of the filtration

{false{F^{r} C H_{0} (A) false}}_{r \geq 0}

C H_{0} false(A false)

constructed in .…”

Section: Zero‐cycles On Abelian Varietiesmentioning

confidence: 99%

“…We are now ready to review the construction of

F^{r} C H_{0} false(A false)

. In [, Proposition 3.1] we defined for every

r \geq 0

a canonical homomorphism

{normalΦ}_{r} : C H_{0} false(A false) \to S_{r} false(k; A false),

with

{normalΦ}_{0} = deg

. We then defined ([, Definition 3.2]) the filtration

F^{r}

as follows,

F^{0} = C H_{0} false(A false), and for r \geq 1, F^{r} C H_{0} false(A false) : = \cap_{j = 0}^{r - 1} ker (Φ_{j}) .

It follows by the definition that for every

r \geq 0

we have an injection,

{normalΦ}_{r} : F^{r} / F^{r + 1} ↪ S_{r} false(k; A false) .

Moreover, we showed the following properties.…”

Section: Zero‐cycles On Abelian Varietiesmentioning

confidence: 99%

“…Remark We note that for a semi‐abelian variety G , the group

H_{0}^{sing} false(G false) \otimes Q

has been previously studied by Sugiyama (), who obtained a similar isomorphism for the Pontryagin filtration

{false{I^{r} \otimes double-struckQ false}}_{r \geq 0}

(see Definition ). Our focus both in and in the current paper is an integral study of the subject, providing some evidence towards the Beilinson conjectures. In fact, it was recently established by Kahn and Yamazaki () that the Somekawa K‐group

K (k; {scriptF}_{1}, \dots, {scriptF}_{r})

attached to more general coordinates than just semi‐abelian varieties (namely homotopy invariant Nisnevich sheaves with transfers) has the expected motivic realization …”

Section: Introductionmentioning

confidence: 99%

“…In we constructed a candidate for such an integral filtration

{false{F^{r} false}}_{r \geq 0}

for the Chow group

C H_{0} false(A false)

of zero‐cycles on an abelian variety A over a field k . This decreasing filtration has the property that for every

r \geq 0

, there is an isomorphism

\begin{matrix} true \\ right F^{r} / F^{r + 1} \otimes double-struckZ ([], \frac{1}{r!}) ≃ S_{r} (k; A) \otimes double-struckZ ([], \frac{1}{r!}), \end{matrix}

where

S_{r} false(k; A false)

is the symmetric quotient of the Somekawa K‐group

K (k; A, \dots, A)

attached to r copies of A .…”

Section: Introductionmentioning

confidence: 99%

“…In Section 3, imitating the method of , we construct, for a semi‐abelian variety G over a perfect field k , a decreasing filtration,

F^{0} \supset F^{1} \supset \dots \supset F^{r} \supset \dots

H_{0}^{sing} false(G false)

such that the successive quotients

F^{r} / F^{r + 1}

can be understood by the Somekawa K‐group,

S_{r} false(k; G false)

. Namely, we prove the following theorem.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Some results about zero‐cycles on abelian and semi‐abelian varieties

Gazaki

2019

Mathematische Nachrichten

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In this short note we extend some results obtained in [7]. First, we prove that for an abelian variety A with good ordinary reduction over a finite extension of Qp with p an odd prime, the Albanese kernel of A is the direct sum of its maximal divisible subgroup and a torsion group. Second, for a semi‐abelian variety G over a perfect field k, we construct a decreasing integral filtration false{Frfalse}r≥0 of Suslin's singular homology group, H0singfalse(Gfalse), such that the successive quotients are isomorphic to a certain Somekawa K‐group.

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“…We start by reviewing the construction of the filtration

{false{F^{r} C H_{0} (A) false}}_{r \geq 0}

C H_{0} false(A false)

constructed in .…”

Section: Zero‐cycles On Abelian Varietiesmentioning

confidence: 99%

“…We are now ready to review the construction of

F^{r} C H_{0} false(A false)

. In [, Proposition 3.1] we defined for every

r \geq 0

a canonical homomorphism

{normalΦ}_{r} : C H_{0} false(A false) \to S_{r} false(k; A false),

with

{normalΦ}_{0} = deg

. We then defined ([, Definition 3.2]) the filtration

F^{r}

as follows,

F^{0} = C H_{0} false(A false), and for r \geq 1, F^{r} C H_{0} false(A false) : = \cap_{j = 0}^{r - 1} ker (Φ_{j}) .

It follows by the definition that for every

r \geq 0

we have an injection,

{normalΦ}_{r} : F^{r} / F^{r + 1} ↪ S_{r} false(k; A false) .

Moreover, we showed the following properties.…”

Section: Zero‐cycles On Abelian Varietiesmentioning

confidence: 99%

“…Remark We note that for a semi‐abelian variety G , the group

H_{0}^{sing} false(G false) \otimes Q

has been previously studied by Sugiyama (), who obtained a similar isomorphism for the Pontryagin filtration

{false{I^{r} \otimes double-struckQ false}}_{r \geq 0}

K (k; {scriptF}_{1}, \dots, {scriptF}_{r})

attached to more general coordinates than just semi‐abelian varieties (namely homotopy invariant Nisnevich sheaves with transfers) has the expected motivic realization …”

Section: Introductionmentioning

confidence: 99%

“…In we constructed a candidate for such an integral filtration

{false{F^{r} false}}_{r \geq 0}

for the Chow group

C H_{0} false(A false)

of zero‐cycles on an abelian variety A over a field k . This decreasing filtration has the property that for every

r \geq 0

, there is an isomorphism

\begin{matrix} true \\ right F^{r} / F^{r + 1} \otimes double-struckZ ([], \frac{1}{r!}) ≃ S_{r} (k; A) \otimes double-struckZ ([], \frac{1}{r!}), \end{matrix}

where

S_{r} false(k; A false)

is the symmetric quotient of the Somekawa K‐group

K (k; A, \dots, A)

attached to r copies of A .…”

Section: Introductionmentioning

confidence: 99%

“…In Section 3, imitating the method of , we construct, for a semi‐abelian variety G over a perfect field k , a decreasing filtration,

F^{0} \supset F^{1} \supset \dots \supset F^{r} \supset \dots

H_{0}^{sing} false(G false)

such that the successive quotients

F^{r} / F^{r + 1}

can be understood by the Somekawa K‐group,

S_{r} false(k; G false)

. Namely, we prove the following theorem.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Some results about zero‐cycles on abelian and semi‐abelian varieties

Gazaki

2019

Mathematische Nachrichten

Self Cite

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Divisibility results for zero-cycles

Gazaki

Hiranouchi

2021

European Journal of Mathematics

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Torsion phenomena for zero-cycles on a product of curves over a number field

Gazaki,

Love

2024

Res. number theory

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For a smooth projective variety X over an algebraic number field k a conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map of X is a torsion group. In this article we consider a product $$X=C_1\times \cdots \times C_d$$ X = C 1 × ⋯ × C d of smooth projective curves and show that if the conjecture is true for any subproduct of two curves, then it is true for X. For a product $$X=C_1\times C_2$$ X = C 1 × C 2 of two curves over $$\mathbb {Q} $$ Q with positive genus we construct many nontrivial examples that satisfy the weaker property that the image of the natural map $$J_1(\mathbb {Q})\otimes J_2(\mathbb {Q})\xrightarrow {\varepsilon }{{\,\textrm{CH}\,}}_0(C_1\times C_2)$$ J 1 ( Q ) ⊗ J 2 ( Q ) → ε CH 0 ( C 1 × C 2 ) is finite, where $$J_i$$ J i is the Jacobian variety of $$C_i$$ C i . Our constructions include many new examples of non-isogenous pairs of elliptic curves $$E_1, E_2$$ E 1 , E 2 with positive rank, including the first known examples of rank greater than 1. Combining these constructions with our previous result, we obtain infinitely many nontrivial products $$X=C_1\times \cdots \times C_d$$ X = C 1 × ⋯ × C d for which the analogous map $$\varepsilon $$ ε has finite image.

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On a filtration of for an abelian variety

Cited by 8 publications

References 15 publications

Some results about zero‐cycles on abelian and semi‐abelian varieties

Some results about zero‐cycles on abelian and semi‐abelian varieties

Divisibility results for zero-cycles

Torsion phenomena for zero-cycles on a product of curves over a number field

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