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

Let K be an imaginary quadratic field and p = 5 a rational prime inert in K. For a Q-curve E with complex multiplication by O-K and good reduction at p, K. Rubin introduced a p-adic L-function 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 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 Z(p)-extension ?(8) of the unramified quadratic extension of Q(p). Along the way, we present a theory of local points over ?(8) of the Lubin-Tate formal group of height 2 for the uniformizing parameter -p.