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Non-vanishing of Kolyvagin systems and Iwasawa theory

Ashay Burungale, Francesc Castella, Giada Grossi, Christopher Skinner

TL;DR

This paper proves Kolyvagin's conjecture on the non-vanishing of the p-adic Heegner point Kolyvagin system for primes p of good ordinary reduction that split in an imaginary quadratic field K, under mild hypotheses; it also establishes a refined conjecture expressing the divisibility index in terms of Tamagawa numbers and extends the analysis to Kato’s Beilinson–Kato Kolyvagin system. The authors leverage anticyclotomic Iwasawa theory, Λ-adic Kolyvagin systems, and recent Main Conjectures to relate non-vanishing of base classes to p-adic L-values, and to translate these into precise Selmer-group information. A cyclotomic analogue is developed via Kato’s Euler system, proving non-vanishing of Kato’s Kolyvagin system under cyclotomic Main Conjectures and connecting to the MSD p-adic L-function. Overall, the work bridges Heegner-point and Kato-based Euler systems through a unified Iwasawa-theoretic framework, yielding concrete divisibility formulas and refined conjectures that illuminate the relationship between L-values, Tamagawa factors, and Selmer groups.

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve and $p$ an odd prime. In 1991 Kolyvagin conjectured that the system of cohomology classes for torsion quotients of the $p$-adic Tate module of $E$ derived from Heegner points over ring class fields of a suitable imaginary quadratic field $K$ (i.e., the Heegner point Kolyvagin system of $E/K$) is non-trivial. In this paper we prove Kolyvagin's conjecture when $p$ is a prime of good ordinary reduction for $E$ that splits in $K$. In particular, our results cover many cases where $p$ is an Eisenstein prime for $E$, complementing Wei Zhang's earlier results on the conjecture by a different approach. Our methods also yield a proof of a refinement of Kolyvagin's conjecture expressing the divisibility index of the Heegner point Kolyvagin system in terms of the Tamagawa numbers of $E$, as conjectured by Wei Zhang in 2014, as well as proofs of analogous results for the Kolyvagin system obtained from Kato's Euler system.

Non-vanishing of Kolyvagin systems and Iwasawa theory

TL;DR

This paper proves Kolyvagin's conjecture on the non-vanishing of the p-adic Heegner point Kolyvagin system for primes p of good ordinary reduction that split in an imaginary quadratic field K, under mild hypotheses; it also establishes a refined conjecture expressing the divisibility index in terms of Tamagawa numbers and extends the analysis to Kato’s Beilinson–Kato Kolyvagin system. The authors leverage anticyclotomic Iwasawa theory, Λ-adic Kolyvagin systems, and recent Main Conjectures to relate non-vanishing of base classes to p-adic L-values, and to translate these into precise Selmer-group information. A cyclotomic analogue is developed via Kato’s Euler system, proving non-vanishing of Kato’s Kolyvagin system under cyclotomic Main Conjectures and connecting to the MSD p-adic L-function. Overall, the work bridges Heegner-point and Kato-based Euler systems through a unified Iwasawa-theoretic framework, yielding concrete divisibility formulas and refined conjectures that illuminate the relationship between L-values, Tamagawa factors, and Selmer groups.

Abstract

Let be an elliptic curve and an odd prime. In 1991 Kolyvagin conjectured that the system of cohomology classes for torsion quotients of the -adic Tate module of derived from Heegner points over ring class fields of a suitable imaginary quadratic field (i.e., the Heegner point Kolyvagin system of ) is non-trivial. In this paper we prove Kolyvagin's conjecture when is a prime of good ordinary reduction for that splits in . In particular, our results cover many cases where is an Eisenstein prime for , complementing Wei Zhang's earlier results on the conjecture by a different approach. Our methods also yield a proof of a refinement of Kolyvagin's conjecture expressing the divisibility index of the Heegner point Kolyvagin system in terms of the Tamagawa numbers of , as conjectured by Wei Zhang in 2014, as well as proofs of analogous results for the Kolyvagin system obtained from Kato's Euler system.
Paper Structure (37 sections, 33 theorems, 212 equations)

This paper contains 37 sections, 33 theorems, 212 equations.

Table of Contents

  1. Main results
  2. Strategy
  3. Relation to previous works
  4. Heegner points and anticyclotomic Iwasawa theory
  5. The Kolyvagin system of Heegner points
  6. Selmer structures
  7. The Kolyvagin system of Heegner points
  8. The $\Lambda$-adic Kolyvagin system
  9. Kolyvagin system for anticyclotomic twists
  10. Anticyclotomic Iwasawa Main Conjecture
  11. $p$-adic $L$-function
  12. $\Lambda$-adic formula of Bertolini--Darmon--Prasanna formula: A consequence
  13. Anticyclotomic Selmer group
  14. The Main Conjecture
  15. The Kolyvagin system bound with error term
  16. ...and 22 more sections

Key Result

Theorem A

Let $E/\mathbb{Q}$ be an elliptic curve, and let $p$ be an odd prime of good ordinary reduction for $E$. Let $K$ be a quadratic imaginary field satisfying eq:intro-Heeg, eq:intro-disc, eq:intro-tor, and such that $p$ splits in $K$. Assume that the rational anticyclotomic Main Conjecture conj:anticyc In particular, $\{\kappa_{n}^{\rm Heeg}\}\neq 0$ in both of the following cases:

Theorems & Definitions (85)

  • Theorem A: Kolyvagin's conjecture
  • Corollary A
  • Remark
  • Theorem B: Refined Kolyvagin's conjecture
  • Remark
  • Theorem C: Non-vanishing of Kato's Kolyvagin system
  • Corollary C
  • Remark 1.1.1
  • Definition 1.1.2
  • Remark 1.1.3
  • ...and 75 more