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Main conjectures for non-CM elliptic curves at good ordinary primes

Xiaojun Yan, Xiuwu Zhu

TL;DR

This work develops and proves new cases of Iwasawa main conjectures for non-CM elliptic curves at good ordinary primes by tying Selmer groups over $K_\infty$ to a network of $p$-adic $L$-functions (Perrin-Riou, Gr, MSD, and BDP) through Beilinson-Flach elements and explicit reciprocity laws. It leverages Hida theory to handle $\mathbb{I}$-adic families, Skinner-Urban’s results, and non-vanishing of $\mu$-invariants (via Hsieh) together with Kato’s Mazur conjecture to convert divisibility into equalities in many cases. Under the Heegner hypothesis and irreducibility of $\bar{\rho}_E|_{G_K}$, the authors establish that the relevant Selmer modules are torsion with characteristic ideals divisible by the corresponding $p$-adic $L$-functions, and in many settings obtain full equality upon localization. These results yield new instances of the $p$-part BSD formula and $p$-converse results for ranks $\le 1$, illustrating a deep connection between higher-dimensional Iwasawa theory, Beilinson-Flach elements, and arithmetic invariants of elliptic curves.

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve and $p > 2$ be a prime of good ordinary reduction for $E$. Assume that the residue representation associated with $(E, p)$ is irreducible. In this paper, we prove more cases on several Iwasawa main conjectures for $E$. As applications, we prove more general cases of $p$-converse theorem and $p$-part BSD formula when the rank is less than or equal to $1$.

Main conjectures for non-CM elliptic curves at good ordinary primes

TL;DR

This work develops and proves new cases of Iwasawa main conjectures for non-CM elliptic curves at good ordinary primes by tying Selmer groups over to a network of -adic -functions (Perrin-Riou, Gr, MSD, and BDP) through Beilinson-Flach elements and explicit reciprocity laws. It leverages Hida theory to handle -adic families, Skinner-Urban’s results, and non-vanishing of -invariants (via Hsieh) together with Kato’s Mazur conjecture to convert divisibility into equalities in many cases. Under the Heegner hypothesis and irreducibility of , the authors establish that the relevant Selmer modules are torsion with characteristic ideals divisible by the corresponding -adic -functions, and in many settings obtain full equality upon localization. These results yield new instances of the -part BSD formula and -converse results for ranks , illustrating a deep connection between higher-dimensional Iwasawa theory, Beilinson-Flach elements, and arithmetic invariants of elliptic curves.

Abstract

Let be an elliptic curve and be a prime of good ordinary reduction for . Assume that the residue representation associated with is irreducible. In this paper, we prove more cases on several Iwasawa main conjectures for . As applications, we prove more general cases of -converse theorem and -part BSD formula when the rank is less than or equal to .
Paper Structure (21 sections, 34 theorems, 108 equations)

This paper contains 21 sections, 34 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Strategy
  4. Acknowledgment
  5. Selmer groups
  6. Selmer groups for Iwasawa algebras
  7. Selmer groups for Hida families
  8. p-adic L-functions
  9. A three-variable p-adic L-function
  10. Two variable p-adic L-functions: type I
  11. Cyclotomic p-adic L-function
  12. Two variable p-adic L-functions: type II
  13. BDP p-adic L-function
  14. Main conjectures over $K_\infty$
  15. Two variable Iwasawa main conjectures
  16. ...and 6 more sections

Key Result

Theorem 1.2

Suppose that the Heegner hypothesis holds (in particular, $\operatorname{sign}(E/K) = -1$), and that $\bar{\rho}_E|_{G_K}$ is absolutely irreducible. Then: Moreover, if then Conjecture mainconj holds.

Theorems & Definitions (76)

  • Conjecture 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Lemma 2.2
  • proof
  • ...and 66 more