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arxiv: 2310.06813 · v2 · submitted 2023-10-10 · 🧮 math.NT

Anticyclotomic Iwasawa theory of abelian varieties of GL₂-type at non-ordinary primes II

Ashay Burungale , K\^az{\i}m B\"uy\"ukboduk , Antonio Lei This is my paper

Pith reviewed 2026-05-24 06:12 UTC · model grok-4.3

classification 🧮 math.NT
keywords anticyclotomic Iwasawa theoryHeegner pointsmain conjecturessupersingular primesp-converse theoremselliptic curvesGL2-type abelian varietiesEuler systems
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The pith

The paper proves a p-converse to Gross-Zagier and Kolyvagin for semistable elliptic curves at supersingular primes p≥5.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper generalizes plus/minus Heegner point main conjectures in the anticyclotomic Z_p-extension of an imaginary quadratic field K to elliptic curves and GL2-type abelian varieties at non-ordinary primes. When p splits in K, it formulates and proves Sprung-type main conjectures under some conditions. When p is inert in K, it formulates plus/minus Heegner point main conjectures relying on prior work and proves the minus version for semistable curves. A sympathetic reader cares because the inert case yields a p-converse theorem that controls the Mordell-Weil rank and the p-part of the Shafarevich-Tate group via Heegner points when the analytic rank is one.

Core claim

The central claim is that the minus main conjecture among the plus/minus Heegner point main conjectures holds for semistable elliptic curves E/Q with good supersingular reduction at p≥5 when p is inert in K; this is proved using Howard's bipartite Euler systems together with Zhang's resolution of Kolyvagin's conjecture and the cyclotomic main conjecture at non-ordinary primes, and it produces the stated p-converse to the Gross-Zagier and Kolyvagin theorem.

What carries the argument

Howard's framework of bipartite Euler systems, which carries the argument by relating the Heegner point main conjectures to known cyclotomic results and Kolyvagin's theorem.

If this is right

  • The p-converse theorem holds: if the analytic rank of E over K is one then the Mordell-Weil rank is one and the p-primary part of Sha(E/K) is finite.
  • Sprung-type main conjectures hold for GL2-type abelian varieties at non-ordinary primes when p splits in K, under the stated conditions.
  • The minus main conjecture holds in the inert case for semistable curves, complementing the split-case results of Castella-Wan.
  • The results rely on and extend the cyclotomic main conjecture at non-ordinary primes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The methods could be tested numerically by verifying the main conjecture predictions for specific semistable curves at small supersingular primes such as p=5 or p=7.
  • If the semistable hypothesis can be relaxed using the same Euler-system techniques, the p-converse would apply to a larger class of curves.
  • The bipartite Euler system framework may connect to similar Iwasawa-theoretic statements for higher-dimensional abelian varieties of GL2-type.

Load-bearing premise

The semistable hypothesis on the elliptic curve is required for the proofs in both the split and inert cases to go through.

What would settle it

An explicit semistable elliptic curve E with good supersingular reduction at some p≥5, root number -1 over K, analytic rank one, but Mordell-Weil rank greater than one or infinite p-part of Sha, would falsify the p-converse.

read the original abstract

Let $E/\mathbb{Q}$ an elliptic curve with good supersingular reduction at a prime $p\geq 5$, and $K$ an imaginary quadratic field such that the root number of $E$ over $K$ equals $-1$. When $p$ splits in $K$, Castella and Wan formulated the plus/minus Heegner point main conjectures for $E$ along the anticyclotomic $\mathbb{Z}_p$-extension of $K$, and proved them for semistable curves. We generalize their results to two settings: 1. For $p$ split in $K$, we formulate Sprung-type main conjectures for $\mathrm{GL}_2$-type abelian varieties at non-ordinary primes and prove them under some conditions. 2. For $p$ inert in $K$, we formulate, relying on the work of the first-named author with Kobayashi and Ota, plus/minus Heegner point main conjectures for elliptic curves, and prove the minus main conjecture for semistable curves. The latter yields a $p$-converse to the Gross--Zagier and Kolyvagin theorem for semistable elliptic curves $E$ at supersingular primes $p\geq 5$, complementing the pioneering $p$-converse theorems of Skinner and Zhang. Our method relies on Howard's framework of bipartite Euler systems, Zhang's resolution of Kolyvagin's conjecture and the recent proof of cyclotomic main conjecture at non-ordinary primes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript generalizes plus/minus Heegner-point main conjectures in the anticyclotomic Iwasawa theory of elliptic curves and abelian varieties of GL₂-type at good supersingular primes p ≥ 5. For p split in the imaginary quadratic field K it formulates Sprung-type main conjectures and proves them under unspecified conditions; for p inert in K it formulates plus/minus conjectures (relying on prior work of the first author with Kobayashi–Ota) and proves the minus conjecture for semistable curves. The latter is used to obtain a p-converse to the Gross–Zagier–Kolyvagin theorem, complementing Skinner–Zhang.

Significance. If the derivations are complete, the work supplies the first p-converse statements at supersingular primes in the inert case and extends the plus/minus formalism to GL₂-type abelian varieties, using Howard’s bipartite Euler systems, Zhang’s Kolyvagin resolution, and the cyclotomic main conjecture at non-ordinary primes. These are concrete advances in the non-ordinary Iwasawa theory of Heegner points.

major comments (3)
  1. [§3] §3 (inert-case formulation): the minus main conjecture is stated to follow from the plus/minus setup of the first author–Kobayashi–Ota together with Howard’s bipartite Euler system and Zhang’s resolution; the manuscript must explicitly verify that the local conditions at the supersingular prime p (inert in K) produce the required signed Heegner classes without additional restrictions on the semistable reduction or on the choice of the anticyclotomic extension.
  2. [Theorem 5.3] Theorem 5.3 (p-converse): the deduction that the minus main conjecture implies the p-converse to Gross–Zagier–Kolyvagin for semistable E at supersingular p ≥ 5 is load-bearing; the argument must confirm that the Euler-system classes remain non-trivial after the sign change and that the Kolyvagin system does not vanish for the same reason as in the ordinary case.
  3. [§2.2] §2.2 (Sprung-type conjectures): the formulation for GL₂-type abelian varieties at split primes invokes an unspecified set of “some conditions”; these conditions must be stated explicitly and shown to be satisfied by the semistable elliptic curves treated in the inert case, or the two settings cannot be compared.
minor comments (2)
  1. [Abstract] The abstract and introduction should list the precise hypotheses (e.g., semistable at all primes dividing the conductor, root number −1 over K) under which each main conjecture is proved.
  2. [§2–§3] Notation for the plus/minus Selmer groups and the signed Heegner points should be uniform between the split and inert sections to avoid confusion when the two settings are compared.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We are grateful to the referee for their thorough review and valuable suggestions. We address each of the major comments below, indicating the changes we will implement in the revised manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (inert-case formulation): the minus main conjecture is stated to follow from the plus/minus setup of the first author–Kobayashi–Ota together with Howard’s bipartite Euler system and Zhang’s resolution; the manuscript must explicitly verify that the local conditions at the supersingular prime p (inert in K) produce the required signed Heegner classes without additional restrictions on the semistable reduction or on the choice of the anticyclotomic extension.

    Authors: We agree that an explicit verification would strengthen the presentation. In the revised version, we will insert a detailed paragraph or subsection in §3 explaining how the local conditions at the supersingular prime p, for semistable reduction and the standard anticyclotomic extension, yield the signed Heegner classes as defined in the plus/minus setup of the first author with Kobayashi and Ota. This verification relies on the compatibility already established in that prior work and does not impose additional restrictions. revision: yes

  2. Referee: [Theorem 5.3] Theorem 5.3 (p-converse): the deduction that the minus main conjecture implies the p-converse to Gross–Zagier–Kolyvagin for semistable E at supersingular p ≥ 5 is load-bearing; the argument must confirm that the Euler-system classes remain non-trivial after the sign change and that the Kolyvagin system does not vanish for the same reason as in the ordinary case.

    Authors: We will revise the proof of Theorem 5.3 to include explicit confirmation that the Euler-system classes remain non-trivial after the sign change, adapting the non-vanishing arguments from the ordinary case to the supersingular setting using the minus main conjecture. Additionally, we will clarify that the Kolyvagin system does not vanish for reasons analogous to those in the ordinary case, drawing on Zhang's resolution of Kolyvagin's conjecture. revision: yes

  3. Referee: [§2.2] §2.2 (Sprung-type conjectures): the formulation for GL₂-type abelian varieties at split primes invokes an unspecified set of “some conditions”; these conditions must be stated explicitly and shown to be satisfied by the semistable elliptic curves treated in the inert case, or the two settings cannot be compared.

    Authors: The phrase 'some conditions' in the abstract and §2.2 refers to the technical assumptions needed for the formulation of Sprung-type main conjectures, including the existence of the relevant p-adic L-functions and their compatibility with the cyclotomic main conjecture at non-ordinary primes. In the revision, we will explicitly list these conditions in §2.2 and demonstrate that they hold for the semistable elliptic curves considered in the inert case, thereby allowing a direct comparison between the split and inert settings. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation builds on external frameworks

full rationale

The paper formulates and proves main conjectures in new settings (Sprung-type for split case; plus/minus for inert case) and derives the p-converse from the minus conjecture proof. The inert-case formulation cites prior work by the first author with Kobayashi and Ota, but this is a standard reference for setup rather than a load-bearing reduction of the current proofs. The method explicitly invokes Howard's bipartite Euler systems, Zhang's Kolyvagin resolution, and the cyclotomic main conjecture at non-ordinary primes as independent inputs. No equations, fitted parameters, or self-definitional steps are exhibited that reduce the claimed results to the paper's own inputs by construction. The derivation is therefore self-contained against the cited external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are identifiable from the given text.

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Reference graph

Works this paper leans on

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