On the Kolyvagin formula for elliptic curves with good reductions over pseudolocal fields

Author V. I. Nesteruk
Ivan Franko National University of Lviv

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)

Pages 16-20
Volume 39
Issue 1
Year 2013
Journal Matematychni Studii
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