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 -