Iwasawa Main Conjecture for Supersingular Elliptic Curves and BSD conjecture

Abstract

In this paper we prove the $\pm$-main conjecture of Iwasawa theory formulated by Kobayashi for elliptic curves with supersingular reduction at an odd prime $p$ such that $a_p=0$, using a key new observation that it can be reduced to another Iwasawa-Greenberg main conjecture, which is more accessible and proved here as a first step. Then we develop some generalized $\pm$ local theory and deduce the main conjecture. The argument uses in an essential way the recent study on explicit reciprocity law for Beilinson-Flach elements by Kings-Loeffler-Zerbes. We also prove as corollaries the $p$-part of the BSD formula at supersingular primes when the analytic rank is $0$ or $1$. The main result enables us to present in the Appendix a number of explicit infinite families of elliptic curves without complex multiplications for which we can now prove the full Birch-Swinnerton-Dyer conjecture. No such infinite families of curves without complex multiplication were known previously.

arXiv commentThe preprint is superseded by part of the new preprint 2409.01350

References in corpus (7)

• Rankin--Eisenstein classes and explicit reciprocity laws
• Euler systems for Rankin--Selberg convolutions of modular forms
• On anticyclotomic mu-invariants of modular forms
• Indivisibility of Heegner points in the multiplicative case
• P-adic integration on ray class groups and non-ordinary p-adic L-functions
• Rankin-Selberg Euler systems and p-adic interpolation
• Iwasawa Main Conjecture for Rankin-Selberg $p$-adic $L$-functions: Non-Ordinary Case

Cited by in corpus (17)

• Statistics for Iwasawa invariants of elliptic curves
• On the Iwasawa main conjectures for modular forms at non-ordinary primes
• Euler systems with local conditions
• Arithmetic of p-irregular modular forms: families and p-adic L-functions
• On fine Selmer groups and the greatest common divisor of signed and chromatic $p$-adic $L$-functions
• On the 2-part of the Birch and Swinnerton-Dyer conjecture for quadratic twists of elliptic curves
• Special Values of the Zeta Function of an Arithmetic Surface
• On fine Selmer groups and signed Selmer groups of elliptic modular forms
• On the refined conjectures on Fitting ideals of Selmer groups of elliptic curves with supersingular reduction
• Iwasawa theory for Symmetric Square of non-$p$-ordinary eigenforms
• The Birch--Swinnerton-Dyer exact formula for quadratic twists of elliptic curves
• Codimension two cycles in Iwasawa theory and elliptic curves with supersingular reduction
• On the algebraic functional equation of the eigenspaces of mixed signed Selmer groups of elliptic curves with good reduction at primes above $p$
• Iwasawa-Greenberg main conjecture for non-ordinary modular forms and Eisenstein congruences on $\mathrm{GU}(3,1)$
• On the parity of supersingular Weil polynomials
• Generalized Birch lemma and the 2-part of the Birch and Swinnerton-Dyer conjecture for certain elliptic curves
• Plus and minus logarithms and Amice transform