A converse to a theorem of Gross, Zagier, and Kolyvagin

Let E be a semistable elliptic curve over Q. We prove that if E has non-split multiplicative reduction at at least one odd prime or split multiplicative reduction at at least two odd primes, then (Formula Presented) We also prove the corresponding result for the abelian variety associated with a weight 2 newform f of trivial character. These, and other related results, are consequences of our main theorem, which establishes criteria for (Formula Presented) where V is the p-adic Galois representation associated with f, that ensure that (Formula Presented) The main theorem is proved using the Iwasawa theory of V over an imaginary quadratic field to show that the p-adic logarithm of a suitable Heegner point is non-zero.