Elliptic units, derived Heegner classes and the p-adic Birch and Swinnerton-Dyer quotient of CM elliptic curves of rank two

Abstract

Let $E\colon y^2=x^3-Dx$ have rank two and let $p\equiv1\pmod4$ be a prime of good reduction. We study the conjecture that the $p$-adic quotient $q_p$, the leading coefficient of the cyclotomic $p$-adic $L$-function divided by the Euler factor and the $p$-adic regulator, equals the complex quotient $q_\infty=L''(E,1)/(2Ω_E\mathrm{Reg}_\infty)$, and that the latter is rational. In rank one the analogous statement is a theorem of Perrin-Riou and Rubin, because a Heegner point computes both sides; in rank two there is no such point. We show that $q_p$ is the constant of the quadratic term of the two-variable Katz measure; this is how the conjecture arises, and it computes $q_p$ modulo any power of $p$ from critical Hecke $L$-values. We then show that $q_p$ is the coordinate of two global classes: Rubin's rank-two elliptic-unit class, whose position in $E(\mathbb{Q})\otimes\mathbb{Q}_p$ is $q_p$ times an explicit vector, and the derived Heegner class over an auxiliary imaginary quadratic field $F$, for which we prove an exact identity. As consequences, whenever such a field $F$ satisfying our hypotheses exists, $q_p$ is a square in $\mathbb{Q}_p$ times the product of the Tamagawa numbers of $E$, so that the unit $q_p/A_E$ of the $p$-adic Birch and Swinnerton-Dyer formula is a square when Sha is finite; if moreover the derived Heegner class has non-zero formal logarithm, the conjecture of Agboola and Castella for $E/F$ holds up to a $p$-adic unit, and with a theorem of Burungale, Castella and Skinner the anticyclotomic main conjecture holds integrally; and our conjecture is equivalent to the analytic form of theirs, with the products of anticyclotomic heights read sesquilinearly, which differs by a sign from its literal formulation. The conjecture is verified modulo $p^3$ for 5520 pairs $(E,p)$ with conductors up to $3.9\cdot10^{12}$, including curves with analytic Sha of order 4 and 9.