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Skinner · converse to Gross–Zagier–Kolyvagin
Annals of Mathematics 191 (2020), 329–354. Introduction also states the classical low-analytic-rank theorem.
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For E/Q with ord_(s=1)L(E,s) equal to 0 or 1, rank E(Q) equals that analytic rank and Sha(E/Q) is finite.
Passage ID: birch-and-swinnerton-dyer-conjecture:bsd-analytic-rank-at-most-one:v1
Source item: Introduction, journal page 330 (PDF page 2), classical theorem
Source record: skinner-annals-2020@annals-191-2-2020
For semistable E/Q with nonsplit multiplicative reduction at an odd prime or split multiplicative reduction at two odd primes, algebraic rank 1 and finite Sha imply analytic rank 1.
Passage ID: birch-and-swinnerton-dyer-conjecture:bsd-rank-one-converse-finite-sha:v1
Source item: Abstract and Theorem A, journal pages 329–331
Source record: skinner-annals-2020@annals-191-2-2020
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Problem publication v3
Source assessment
Proved
Rank equality at analytic rank 0 or 1
Gross–Zagier and Kolyvagin, with modularity for all E/Q: rank equality and finite Sha in the low-analytic-rank regime.
Source date Mar 1, 2020 · Observed Oct 10, 2026
Source assessment
Proved
Rank-one converse assuming finite Sha
Skinner’s converse in the stated semistable family; a proved implication with a finiteness assumption.
Source date Mar 1, 2020 · Observed Oct 10, 2026
Question
Birch and Swinnerton-Dyer conjecture
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Manuscript / exposition · external
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Resource revision: Annals 191 (2020), 329–354
Recorded authorship
Christopher Skinner
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Skinner · converse to Gross–Zagier–Kolyvagin
Skinner · converse to Gross–Zagier–Kolyvagin
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