On tate duality for Jacobian varieties

Takao Yamazaki

doi:10.1016/S0022-314X(02)00067-7

On tate duality for Jacobian varieties

Takao Yamazaki

SCI - Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let A be an abelian variety over a p-adic field k and A^t its dual. The group of k-rational point A(k) has a p-adic decreasing filtration U A(k). When A = J is a Jacobian variety, we give a precise description of the exact annihilator of Uⁿ A(k) with respect to the Tate pairing A(k) × H¹ (k, A^t) → ℚ/ℤ. As an application, we give another proof of the result of McCallum in the special case A = J, which says that Uⁿ A(k) annihilates ker(H¹ (k, A^t) → H¹ (k′, A^t)) whenever k′/k is a finite extension of conductor ≤n.

Original language	English
Pages (from-to)	298-306
Number of pages	9
Journal	Journal of Number Theory
Volume	99
Issue number	2
DOIs	https://doi.org/10.1016/S0022-314X(02)00067-7
Publication status	Published - 2003 Apr 1

Keywords

Abelian variety
Brauer group
Galois cohomology
Wild ramification
p-adic field

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/S0022-314X(02)00067-7

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title = "On tate duality for Jacobian varieties",

abstract = "Let A be an abelian variety over a p-adic field k and At its dual. The group of k-rational point A(k) has a p-adic decreasing filtration U A(k). When A = J is a Jacobian variety, we give a precise description of the exact annihilator of Un A(k) with respect to the Tate pairing A(k) × H1 (k, At) → ℚ/ℤ. As an application, we give another proof of the result of McCallum in the special case A = J, which says that Un A(k) annihilates ker(H1 (k, At) → H1 (k′, At)) whenever k′/k is a finite extension of conductor ≤n.",

keywords = "Abelian variety, Brauer group, Galois cohomology, Wild ramification, p-adic field",

author = "Takao Yamazaki",

note = "Funding Information: This note is written while the author is a visitor at Utrecht University, whose hospitality is gratefully appreciated. He is supported by JSPS Postdoctoral Fellowships for Research Abroad. ",

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T1 - On tate duality for Jacobian varieties

AU - Yamazaki, Takao

N1 - Funding Information: This note is written while the author is a visitor at Utrecht University, whose hospitality is gratefully appreciated. He is supported by JSPS Postdoctoral Fellowships for Research Abroad.

PY - 2003/4/1

Y1 - 2003/4/1

N2 - Let A be an abelian variety over a p-adic field k and At its dual. The group of k-rational point A(k) has a p-adic decreasing filtration U A(k). When A = J is a Jacobian variety, we give a precise description of the exact annihilator of Un A(k) with respect to the Tate pairing A(k) × H1 (k, At) → ℚ/ℤ. As an application, we give another proof of the result of McCallum in the special case A = J, which says that Un A(k) annihilates ker(H1 (k, At) → H1 (k′, At)) whenever k′/k is a finite extension of conductor ≤n.

AB - Let A be an abelian variety over a p-adic field k and At its dual. The group of k-rational point A(k) has a p-adic decreasing filtration U A(k). When A = J is a Jacobian variety, we give a precise description of the exact annihilator of Un A(k) with respect to the Tate pairing A(k) × H1 (k, At) → ℚ/ℤ. As an application, we give another proof of the result of McCallum in the special case A = J, which says that Un A(k) annihilates ker(H1 (k, At) → H1 (k′, At)) whenever k′/k is a finite extension of conductor ≤n.

KW - Abelian variety

KW - Brauer group

KW - Galois cohomology

KW - Wild ramification

KW - p-adic field

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