On Iwasawa main conjectures for elliptic curves at supersingular primes: Beyond the case ap = 0

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Abstract

We reduce the chromatic Iwasawa main conjecture for elliptic curves to the conjectured existence of certain Beilinson–Flach classes in the supersingular case. This generalizes the method of Wan in which ap=0 was assumed; the main innovation of this paper consists in the new 2-dimensional technique to overcome this condition. We also derive the 3-part of the leading term formula in the Birch and Swinnerton-Dyer conjecture for analytic rank 0 or 1 from the main conjecture. Another consequence is that among those elliptic curves with a prescribed number of solutions modulo any fixed prime, infinitely many would satisfy the full Birch and Swinnerton-Dyer conjecture.

Original languageEnglish (US)
Article number109741
JournalAdvances in Mathematics
Volume449
DOIs
StatePublished - Jul 2024

Keywords

  • Birch and Swinnerton-Dyer
  • Elliptic curve
  • Iwasawa theory

ASJC Scopus subject areas

  • General Mathematics

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