p-ADIC L-FUNCTIONS AND RATIONAL POINTS ON CM ELLIPTIC CURVES AT INERT PRIMES

Abstract

Abstract Let K be an imaginary quadratic field and $p\geq 5$ a rational prime inert in K . For a $\mathbb {Q}$ -curve E with complex multiplication by $\mathcal {O}_K$ and good reduction at p , K. Rubin introduced a p -adic L -function $\mathscr {L}_{E}$ which interpolates special values of L -functions of E twisted by anticyclotomic characters of K . In this paper, we prove a formula which links certain values of $\mathscr {L}_{E}$ outside its defining range of interpolation with rational points on E . Arithmetic consequences include p -converse to the Gross–Zagier and Kolyvagin theorem for E . A key tool of the proof is the recent resolution of Rubin’s conjecture on the structure of local units in the anticyclotomic ${\mathbb {Z}}_p$ -extension $\Psi _\infty $ of the unramified quadratic extension of ${\mathbb {Q}}_p$ . Along the way, we present a theory of local points over $\Psi _\infty $ of the Lubin–Tate formal group of height $2$ for the uniformizing parameter $-p$ .