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

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 $langle cdot,,cdot rangle colon mathrm{E}(K)/mmathrm{E}(K) times mathrm{H}^1(G_K, mathrm{E}(overline{K}))_mlongrightarrow mathbb{Z}/mmathbb{Z}$ 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.