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

We consider the {relationships between the} local Artin map$ heta colon K^* o mathrm{Gal}(K^{ab}/K)$ {and the} Hilbert symbol $(cdot,,cdot)colon K^*/K^{*m} imes K^*/K^{*m} longrightarrow mu_m$ 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). % Abstract (in English) It is known that the Weil pairing ${ cdot,,cdot}colon mathrm{E}(overline{K})_m imes mathrm{E}(overline{K})_m longrightarrow mu_m $ and the Tate pairing $langle cdot,,cdot angle colon mathrm{E}(K)/mmathrm{E}(K) imes mathrm{H}^1(G_K, mathrm{E}(overline{K}))_mlongrightarrow mathbb{Z}/mmathbb{Z}$ satisfy $zeta^{langle c_1, c_2 angle}={e_1,e_2}$, 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.