The Birch and Swinnerton–Dyer formula for elliptic curves of analytic rank one

Abstract

Abstract. Let E/Q be a semistable elliptic curve such that ords=1L(E, s) = 1. We prove the p-part of the Birch and Swinnerton-Dyer formula for E/Q for each prime p ≥ 5 of good reduction such that E[p] is irreducible:This formula also holds for p = 3 provided ap(E) = 0 if E has supersingular reduction at p.

On the $p$-part of the Birch–Swinnerton-Dyer formula for multiplicative primes

“…With this result at hand, a simplified form (since N − = 1 here) of the arguments from [JSW15] in part (1) above lead to an equality in (1.1) taking K 1 = K, and so to the predicted order of…”

“…Similarly as in [JSW15], our proof of Theorem A uses anticyclotomic Iwasawa theory. In order to clarify the relation between the arguments in loc.cit.…”

“…When p is a prime of good reduction for E, Theorem A (in the stated level of generality) was first established by Jetchev-Skinner-Wan [JSW15]. (We should note that [JSW15, Thm.…”

“…(1) Exact lower bound on the predicted order of Ш(E/Q)[p ∞ ]. For this part of the argument, in [JSW15] one chooses a suitable imaginary quadratic field K 1 = Q( √ D 1 ) with L(E D 1 , 1) = 0; combined with the hypothesis that E has analytic rank one, it follows that E(K 1 ) has rank one and that #Ш(E/K 1 ) < ∞ by the work of Gross-Zagier and Kolyvagin. The lower bound…”

“…(2) Exact upper bound on the predicted order of Ш(E/Q)[p ∞ ]. For this second part of the argument, in [JSW15] one chooses another imaginary quadratic field K 2 = Q( √ D 2 ) (in general different from K 1 ) such that L(E D 2 , 1) = 0. Crucially, K 2 is chosen so that the associated N + (the product of the prime factors of N that are split or ramified in K 2 ) is as small as possible in a certain sense; this ensures optimality of the upper bound provided by Kolyvagin's methods:…”