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

Abstract

Let $E$ be a semistable elliptic curve over $\mathbb{Q}$. 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\[\mathrm{rank}_{\mathbb{Z}} E(\mathbb{Q}) =1\quad \mathrm{and}\quad \#\Sha(E) \lt \infty X(E)\implies \mathrm{ord}_{s=1}L(E,s) = 1.\] 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 $f$ and $H^1_f(\mathbb{Q},V)$, where $V$ is the $p$-adic Galois representation associated with $f$, that ensure that $\mathrm{ord}_{s=1}L(f,s) = 1$. 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.