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

TL;DR

This paper establishes new criteria linking the reduction types of elliptic curves and associated abelian varieties to the order of vanishing of their L-functions at s=1, using Iwasawa theory and Heegner points.

Contribution

It provides a converse to a classical theorem, connecting reduction properties and rank with the order of L-functions at s=1, via new Iwasawa-theoretic methods.

Findings

Proves that certain reduction conditions imply L(E,1)=0 of order exactly one.

Establishes criteria for the order of vanishing of L-functions for associated abelian varieties.

Uses Iwasawa theory of Galois representations and Heegner points to derive main results.

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 and if the rank of $E(\mathbb{Q})$ is one and the Tate-Shafarevich group of $E$ has finite order, then $\mathrm{ord}_{s=1}L(E,s)=1$. We also prove the corresponding result for the abelian variety associated with a weight two 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.