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The anticyclotomic main conjectures for elliptic curves

Massimo Bertolini, Matteo Longo, Rodolfo Venerucci

TL;DR

The article proves the anticyclotomic Main Conjectures for a modular elliptic curve $E$ over the anticyclotomic $\mathbf{Z}_p$-extension $K_\infty/K$ in both good ordinary and supersingular cases. It unifies the definite and indefinite Iwasawa theories by leveraging Heegner/Gross points, level-raising, and Euler-system techniques to relate $p$-adic $L$-functions $L_p^\varepsilon(f)$ to Selmer-ideals, yielding divisibilities and, in non-exceptional situations, equalities (the DAMC and IAMC). The work develops a robust framework of $\varepsilon$-Selmer groups, local-global control theorems, and reciprocity laws that connect algebraic invariants (Selmer groups, Shafarevich–Tate groups, regulators) to analytic data (central and derivative $L$-values) via Gross-type and Kolyvagin-system arguments. The results have significant implications for explicit BSD-type formulas in the anticyclotomic setting and for understanding the arithmetic of elliptic curves in $p$-adic families, including server-side implications for computational Iwasawa theory and L-functions in the anticyclotomic direction.

Abstract

The goal of this article is to obtain a proof of the Main conjectures of Iwasawa theory for rational elliptic curves over anticyclotomic extensions of imaginary quadratic fields, under mild arithmetic assumptions, both in the case where the rational prime $p$ is good ordinary or supersingular.

The anticyclotomic main conjectures for elliptic curves

TL;DR

The article proves the anticyclotomic Main Conjectures for a modular elliptic curve over the anticyclotomic -extension in both good ordinary and supersingular cases. It unifies the definite and indefinite Iwasawa theories by leveraging Heegner/Gross points, level-raising, and Euler-system techniques to relate -adic -functions to Selmer-ideals, yielding divisibilities and, in non-exceptional situations, equalities (the DAMC and IAMC). The work develops a robust framework of -Selmer groups, local-global control theorems, and reciprocity laws that connect algebraic invariants (Selmer groups, Shafarevich–Tate groups, regulators) to analytic data (central and derivative -values) via Gross-type and Kolyvagin-system arguments. The results have significant implications for explicit BSD-type formulas in the anticyclotomic setting and for understanding the arithmetic of elliptic curves in -adic families, including server-side implications for computational Iwasawa theory and L-functions in the anticyclotomic direction.

Abstract

The goal of this article is to obtain a proof of the Main conjectures of Iwasawa theory for rational elliptic curves over anticyclotomic extensions of imaginary quadratic fields, under mild arithmetic assumptions, both in the case where the rational prime is good ordinary or supersingular.
Paper Structure (56 sections, 32 theorems, 241 equations)

This paper contains 56 sections, 32 theorems, 241 equations.

Table of Contents

  1. Introduction
  2. Special points on Shimura curves and Gross curves
  3. Shimura curves and Gross curves
  4. Hecke operators
  5. The Jacquet--Langlands correspondence
  6. Special points
  7. Compatible sequences of special points
  8. Admissible primes and raising the level
  9. Eigenforms of level $(N^{+},N^{-})$
  10. Admissible primes
  11. Level raising
  12. $p$-adic $L$-functions and special values formulae
  13. Modular parametrisations
  14. The ordinary case
  15. The supersingular case
  16. ...and 41 more sections

Key Result

Lemma 3.1

Let $\ell$ be an admissible prime relative to $(f,K)$, and let $K_{\ell}/\mathbf{Q}_{\ell}$ be the completion of $K$ at the unique prime dividing $\ell$$($so that $K_{\ell}=\mathbf{Q}_{\ell^{2}}$ is the quadratic unramified extension of $\mathbf{Q}_{\ell}$$)$. There is a decomposition of $R[G_{K_{\e

Theorems & Definitions (74)

  • Remark 1.2
  • Definition 1.3
  • Remark 1.4
  • Lemma 3.1
  • Theorem 3.3
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 64 more