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On anticyclotomic Euler and Kolyvagin systems

Luca Mastella, Francesco Zerman

TL;DR

This work develops a unified, axiomatic treatment of anticyclotomic Euler systems for $p$-adic Galois representations of rank 2, and shows how to derive a universal (modified) Kolyvagin system from such Euler data. The authors introduce a precise descent mechanism, via derivative classes and local-global compatibilities, to produce a modified universal KS in both the classical and anticyclotomic Iwasawa-theoretic settings; Shapiro’s lemma is used to pass to the anticyclotomic twist. The framework applies to Heegner points and cycles, modular forms, and Hida families, yielding Selmer-group structure results and instances of Iwasawa main conjecture type divisibilities. Overall, the paper provides a versatile, general method to translate anticyclotomic Euler systems into robust Kolyvagin-system machinery across a broad range of arithmetic objects, with concrete applications to elliptic curves, modular forms, and families therein.

Abstract

We introduce an axiomatization of the notion of ( $p$-complete) anticyclotomic Euler system for a wide class of Galois representations, including those attached to a cuspidal eigenform and to a Hida family of modular forms. Under a minimal set of assumptions, we show how to build from these data a universal Kolyvagin system for the representation and for its anticyclotomic twist. Eventually, we recover some applications to the structure of Selmer groups and Iwasawa main conjectures and we review a few concrete examples of these abstract notions that can be found in the literature.

On anticyclotomic Euler and Kolyvagin systems

TL;DR

This work develops a unified, axiomatic treatment of anticyclotomic Euler systems for -adic Galois representations of rank 2, and shows how to derive a universal (modified) Kolyvagin system from such Euler data. The authors introduce a precise descent mechanism, via derivative classes and local-global compatibilities, to produce a modified universal KS in both the classical and anticyclotomic Iwasawa-theoretic settings; Shapiro’s lemma is used to pass to the anticyclotomic twist. The framework applies to Heegner points and cycles, modular forms, and Hida families, yielding Selmer-group structure results and instances of Iwasawa main conjecture type divisibilities. Overall, the paper provides a versatile, general method to translate anticyclotomic Euler systems into robust Kolyvagin-system machinery across a broad range of arithmetic objects, with concrete applications to elliptic curves, modular forms, and families therein.

Abstract

We introduce an axiomatization of the notion of ( -complete) anticyclotomic Euler system for a wide class of Galois representations, including those attached to a cuspidal eigenform and to a Hida family of modular forms. Under a minimal set of assumptions, we show how to build from these data a universal Kolyvagin system for the representation and for its anticyclotomic twist. Eventually, we recover some applications to the structure of Selmer groups and Iwasawa main conjectures and we review a few concrete examples of these abstract notions that can be found in the literature.
Paper Structure (35 sections, 37 theorems, 107 equations)

This paper contains 35 sections, 37 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Acknowledgments
  4. Euler systems and Kolyvagin systems
  5. The arithmetic picture
  6. Selmer structures
  7. The finite-singular isomorphism
  8. Action of the Frobenius
  9. Hypotheses
  10. Anticyclotomic Euler systems
  11. Universal Kolyvagin systems
  12. Kolyvagin primes and main result
  13. Dictionary
  14. Derivative classes and local properties
  15. From Euler systems to Kolyvagin systems
  16. ...and 20 more sections

Key Result

Theorem A

If $\{\mathbf{c}(n)\}_{n\in\mathcal{N}}$ is an anticyclotomic Euler system for $\mathcal{T}$, then there is a subset $\mathcal{N}'$ of $\mathcal{N}$ and a modified universal Kolyvagin system $\{\boldsymbol{\kappa}(n)_\mathfrak{s} : \mathfrak{s} \in \mathbb{Z}_{>0}^2, n\in\mathcal{N}'\}$ for $\mathc

Theorems & Definitions (110)

  • Theorem A: Theorem \ref{['th:euler-to-kolyvagin-system']}
  • Theorem B: Theorem \ref{['th:Iwasawa-euler-to-kolyvagin-system']}
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Lemma 2.8
  • ...and 100 more