On the Tate pairing associated to an isogeny between abelian varieties over pseudofinite field
In this note, we consider the Tate pairing associated to an isogeny between abelian varieties over pseudofinite field. P. Bruin [1] defined this pairing over finite field k: ker ˆφ(k) × coker(φ(k)) ⟶ k∗, and proved its perfectness over finite field. We prove perfectness of the Tate pairing associate...
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Інститут прикладної математики і механіки НАН України
2013
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| Назва видання: | Algebra and Discrete Mathematics |
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| Цитувати: | On the Tate pairing associated to an isogeny between abelian varieties over pseudofinite field / V. Nesteruk // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 1. — С. 103–106. — Бібліогр.: 8 назв. — англ. |
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irk-123456789-1523122019-06-10T01:26:22Z On the Tate pairing associated to an isogeny between abelian varieties over pseudofinite field Nesteruk, V. In this note, we consider the Tate pairing associated to an isogeny between abelian varieties over pseudofinite field. P. Bruin [1] defined this pairing over finite field k: ker ˆφ(k) × coker(φ(k)) ⟶ k∗, and proved its perfectness over finite field. We prove perfectness of the Tate pairing associated to an isogeny between abelian varieties over pseudofinite field, with help of the method, used by P. Bruin in the case of finite ground field [1]. 2013 Article On the Tate pairing associated to an isogeny between abelian varieties over pseudofinite field / V. Nesteruk // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 1. — С. 103–106. — Бібліогр.: 8 назв. — англ. 1726-3255 2010 MSC:12G99, 14H05, 14K02. http://dspace.nbuv.gov.ua/handle/123456789/152312 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this note, we consider the Tate pairing associated to an isogeny between abelian varieties over pseudofinite field. P. Bruin [1] defined this pairing over finite field k: ker ˆφ(k) × coker(φ(k)) ⟶ k∗, and proved its perfectness over finite field. We prove perfectness of the Tate pairing associated to an isogeny between abelian varieties over pseudofinite field, with help of the method, used by P. Bruin in the case of finite ground field [1]. |
| format |
Article |
| author |
Nesteruk, V. |
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Nesteruk, V. On the Tate pairing associated to an isogeny between abelian varieties over pseudofinite field Algebra and Discrete Mathematics |
| author_facet |
Nesteruk, V. |
| author_sort |
Nesteruk, V. |
| title |
On the Tate pairing associated to an isogeny between abelian varieties over pseudofinite field |
| title_short |
On the Tate pairing associated to an isogeny between abelian varieties over pseudofinite field |
| title_full |
On the Tate pairing associated to an isogeny between abelian varieties over pseudofinite field |
| title_fullStr |
On the Tate pairing associated to an isogeny between abelian varieties over pseudofinite field |
| title_full_unstemmed |
On the Tate pairing associated to an isogeny between abelian varieties over pseudofinite field |
| title_sort |
on the tate pairing associated to an isogeny between abelian varieties over pseudofinite field |
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Інститут прикладної математики і механіки НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/152312 |
| citation_txt |
On the Tate pairing associated to an isogeny between abelian varieties over pseudofinite field / V. Nesteruk // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 1. — С. 103–106. — Бібліогр.: 8 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT nesterukv onthetatepairingassociatedtoanisogenybetweenabelianvarietiesoverpseudofinitefield |
| first_indexed |
2025-07-13T02:48:25Z |
| last_indexed |
2025-07-13T02:48:25Z |
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1837498275056320512 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 16 (2013). Number 1. pp. 103 – 106
© Journal “Algebra and Discrete Mathematics”
On the Tate pairing associated to an isogeny
between abelian varieties over pseudofinite field
Volodymyr Nesteruk
Communicated by M. Ya. Komarnytskyj
Abstract. In this note, we consider the Tate pairing as-
sociated to an isogeny between abelian varieties over pseudofinite
field. P. Bruin [1] defined this pairing over finite field k: ker φ̂(k) ×
coker (φ(k)) −→ k∗, and proved its perfectness over finite field.
We prove perfectness of the Tate pairing associated to an isogeny
between abelian varieties over pseudofinite field, with help of the
method, used by P. Bruin in the case of finite ground field [1].
Introduction
P. Bruin [1] and E. Schaefer [7] shoved that the perfectness of the
Tate pairing and of the Frey-Rück pairing follows from that of the Tate
pairing associated to an isogeny between abelian varieties. The Tate
pairing may be defined over pseudofinite fields [5]. Recall that a field k is
called pseudofinite [2], if k is perfect, k has the unique extension of degree
n for each natural number n and every nonempty absolutely irreducible
variety over k, has a k-rational point.
The aim of this work is to prove that the Tate pairing ker φ̂(k) ×
coker (φ(k)) −→ k∗ associated to an isogeny φ of abelian varieties is
perfect over pseudofinite field k.
2010 MSC: 12G99, 14H05, 14K02.
Key words and phrases: pseudofinite field, isogeny, Tate pairing associated to
an isogeny.
104 On the Tate pairing associated to an isogeny.. .
1. Prerequisites
Let C be an absolutely irreducible projective curve defined over pseud-
ofinite field k, and k be algebraic closure of k, k∗ multiplicative group of
k, n is a positive integer, (n, char(k)) = 1 and µn(k) denotes the group of
n-th roots of unity in k
∗
, J(k) is the Jacobian of curve C over k, J[n](k)
denotes the subgroup of elements in J(k) of order dividing n. For divi-
sor classes x ∈ J[n](k) and y ∈ J(k)/nJ(k) there are coprime divisors
D and R such that x = [D] and y = [R] + nJ(k), and there exists a
function f ∈ k(C) such that (f) = nD. The Tate pairing is the pairing
tn(x, y) : J[n](k) × J(k)/nJ(k) −→ k∗, where tn(x, y) = f(R) [3].
Recall the concept of perfect pairing. Let A, B, C be abelian groups. A
pairing A × B → C is called perfect if the induced group homomorphisms
A → Hom(B, C) and B → Hom(A, C) are isomorphisms.
Let A, B be abelian varieties, defined over a field k. A homomor-
phism φ : A → B is called isogeny if it is surjective with finite kernel
ker φ [4, 8]. Recall that the degree degφ of isogeny φ : A → B is the index
[k(A) : φ k(B)], the degree of the corresponding function field extension
k(A)/k(B).
Clearly, the kernel ker φ of an isogeny is a finite abelian group and
satisfies the inequality |ker φ| ≤ deg φ.
Recall some principal properties of isogenies which will be used later.
For a homomorphisms φ : A → B of abelian varieties A, B the following
are equivalent: φ is an isogeny, dim A = dim B and φ is surjective, dim A =
dim B and ker φ is finite, φ is finite, flat, and surjective [4].
For any positive integer n, (n, chark) = 1, we have An(k) = ker (n :
A(k) → A(k)). Then if n = deg φ, so ker φ ⊆ An(k).
2. The Tate pairing associated to an isogeny φ
Let Gk be absolute Galois group of k, and D finite Gk-module. The
Cartier dual of D is the abelian group D∨ = Hom(D, k
∗
) with the Gk-
action given by
(σa)(x) = σ(a(σ−1x)),
where a ∈ D∨, σ ∈ Gk and x ∈ D. Let φ : A → B be an isogeny. Then
there is unique isogeny φ̂ : B → A, φ̂ ◦ φ = deg φ and φ̂ is called the dual
isogeny. Let ǫφ there canonical isomorphism from ker φ̂ to the Cartier
dual (ker φ)∨ of ker φ and φ(k) : A(k) → B(k) is homomorpfism induced
by φ. For x ∈ ker φ̂(k) = {b ∈ B(k)| φ̂(b) = 0}, y ∈ coker (φ(k)) =
V. Nesteruk 105
B(k)/φ(A(k)), we have (x, y) 7→ (ǫφx)(σa − a), where σ is the generator
of absolute Galois group Gk and a ∈ A(k), (φ(a) mod φ(A(k))) = y.
The Tate pairing associated to isogeny φ is the pairing
ker φ̂(k) × coker (φ(k)) −→ k∗, (1)
where (x, y) 7→ (ǫφx)(σa − a).
Lemma 1 ([6]). Let D be finite Gk-module. Then
|H0(Gk, D)| = |H1(Gk, D)|.
Applying the method, used by P. Bruin [1] in the case of finite ground
field, we prove the next theorem for pseudofinite field.
Theorem 1. Let φ be an isogeny between abelian varieties A, B over
pseudofinite field k. Let m be order of ker φ. Suppose that k contains m-th
roots of 1. Then the Tate pairing associated to φ is perfect.
Proof. Consider of the exact sequence of Gk-modules
0 → kerφ → A(k) → B(k) → 0.
This exact sequence gives us the following long exact sequence of coho-
mology groups
0 → H0(Gk, kerφ) → H0(Gk, A(k)) → H0(Gk, B(k))
→ H1(Gk, kerφ) → H1(Gk, A(k)) → H1(Gk, B(k)).
Hence,
0 → kerφ(k) → A(k)
φ
→ B(k) → H1(Gk, kerφ) → 0. (2)
It is known that the group H1(Gk, A(k)) = H1(Gk, B(k)) = 0, since k is
a pseudofinite field [5]. From (2) we have that
B(k)/φ(A(k))
∼
→ H1(Gk, kerφ). (3)
Thus (3), gives us the following description of coker (φ(k)),
coker (φ(k))
∼
→ H1(Gk, kerφ).
The exact sequence analogous (2) and the Lemma 1 allow to define the
perfect pairing
(ker φ)∨(k) × coker (φ(k)) → k∗.
Finally, taking into account the canonical isomorphism, ǫφ, we get that
this perfect pairing coincides with (1).
106 On the Tate pairing associated to an isogeny.. .
References
[1] P. Bruin, The Tate pairing for abelian varieties over finite fields, Journal de theorie
des nombres de Bordeaux, 23(2), 2011, 323-328.
[2] M. Fried, M. Jarden, Field arithmetic. Second edition. Ergebnisse der Mathematik
und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 11.
Springer-Verlag, Berlin, 2005.
[3] F. Hess, A Note on the Tate Pairing of Curves over Finite Fields, Archiv der
Mathematik 82, N.1, 2004, 28-32.
[4] J. S. Milne, Abelian Varieties, available at www.jmilne.org/math/, 2008.
[5] V. Nesteruk, On nondegeneracy of Tate product for curves over pseudofinite fields,
Visnyk of the Lviv Univ. Series Mechanics and Mathematics, Is. 72, 2010, 195-200
(in Ukrainian).
[6] V. Platonov, A. Rapinchuk, Algebraic groups and number theory, 1991 (in Russian).
[7] E. F. Schaefer, A new proof for the non-degeneracy of the Frey-Rück pairing and a
connection to isogenies over the base field. In: T. Shaska (editor), Computational
Aspects of Algebraic Curves (Conference held at the University of Idaho). Lecture
Notes Series in Computing 13. World Scientific Publishing, Hackensack, NJ, 2005,
1-12.
[8] J. H. Silverman, The arithmetic of elliptic curves. Graduate Texts in Mathematics,
106. Springer-Verlag, New York, 1986.
Contact information
V. Nesteruk Algebra and Logic Department, Mechanics and
Mathematics Faculty, Ivan Franko National Uni-
versity of L’viv, 1, Universytetska str., Lviv,
79000, Ukraine
E-Mail: volodymyr-nesteruk@rambler.ru
Received by the editors: 13.02.2012
and in final form 30.03.2013.
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