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On the Kolyvagin formula for elliptic curves with good reductions over pseudolocal fields |
| Author |
V. I. Nesteruk
volodymyr-nesteruk@rambler.ru
Ivan Franko National University of Lviv
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| Abstract |
We consider the relationships between the local Artin mapp $\theta \colon K^*
\to \mathrm{Gal}(K^{ab}/K)$ and the Hilbert symbol
$(\cdot\,,\cdot)\colon K^*/K^{*m} \times K^*/K^{*m} \longrightarrow \mu_m$
for a general local field, as well as between the Tate pairing and the
Weil pairing for elliptic curves with good reductions over
pseudolocal fields
(complete discretely valued fields with pseudofinite residue fields).
It is known that the Weil pairing $\{ \cdot\,,\cdot\}\colon
\mathrm{E}(\overline{K})_m \times \mathrm{E}(\overline{K})_m
\longrightarrow \mu_m $ and the Tate pairing $\langle \cdot\,,\cdot
\rangle \colon \mathrm{E}(K)/m\mathrm{E}(K) \times
\mathrm{H}^1(G_K, \mathrm{E}(\overline{K}))_m\longrightarrow
\mathbb{Z}/m\mathbb{Z}$
satisfy
$\zeta^{\langle c_1, c_2 \rangle}=\{e_1,e_2\}$,
where $\mathrm{E}$ is an elliptic curve with good reduction over
local field and $\zeta$ an appropriate $m^{th}$ root of 1. This is
Kolyvagin's formula. It is
proved that the same holds true for elliptic curves with good reductions
over pseudolocal fields. |
| Keywords |
pseudolocal field; general local field; elliptic curve; local Artin map; Hilbert symbol; Tate pairing; Weil pairing; Kolyvagin formula |
| Reference |
1. Fesenko I.B., Vostokov S.V.
Local Fields and Their Extensions.
Transl. Math. Monogr., Amer. Math. Soc., Second Edition. - 2001,
V.121. - 353p.
2. Milne J.S.
Class Field Theory. - Available at www.jmilne.org/math/ - 2008. - 287p.
3. Papikian M.
On Tate Local Duality.
Seminar ``Kolyvagin's Application of Euler Systems to Elliptic
curves'',
Massachusetts Institute of Technology - 2000, preprint.
4. Serre J.P. Corps locaux. - Paris: Hermann, 1968. - 246p.
5. Lang S. Fundamentals of Diophantine Geometry. Springer-Verlag: Berlin-Heidelberg-New York-Tokyo, 1983. - 370p.
6. Nesteruk V.I. On nondegeneracy of Tate pairing for elliptic curves with good reduction over pseudolocal field// Applied problems of mechanics and mathematics. - 2010. - V.8. - P. 37-40. (in Ukrainian)
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| Pages |
16-20 |
| Volume |
39 |
| Issue |
1 |
| Year |
2013 |
Journal |
Matematychni Studii |
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