TY - JOUR

T1 - A converse to a theorem of Gross, Zagier, and Kolyvagin

AU - Skinner, Christopher

N1 - Funding Information:
for helpful conversations. This work was partially supported by grants from the National Science Foundation, including DMS-0701231 and DMS-0758379. The first version of this paper was written while the author was a visitor in the School of Mathematics at the Institute for Advanced Study.
Publisher Copyright:
© 2020. Department of Mathematics, Princeton University

PY - 2020/3

Y1 - 2020/3

N2 - 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.

AB - 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.

KW - Birch-swinnerton-dyer conjecture

KW - Heegner points

KW - Iwasawa theory

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U2 - 10.4007/annals.2020.191.2.1

DO - 10.4007/annals.2020.191.2.1

M3 - Article

AN - SCOPUS:85129978811

SN - 0003-486X

VL - 191

SP - 329

EP - 354

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 2

ER -