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arxiv: 2605.11100 · v1 · submitted 2026-05-11 · 🧮 math.NT · math.AG· math.KT

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The 2-part of the Bloch-Kato conjecture, and indivisibility results, for K₂ of some elliptic curves

Neil Dummigan , Vasily Golyshev , Rob de Jeu , Matt Kerr
Authors on Pith no claims yet

Pith reviewed 2026-05-13 00:51 UTC · model grok-4.3

classification 🧮 math.NT math.AGmath.KT
keywords Bloch-Kato conjectureK2elliptic curvestame symbolMerkur'ev-Suslin2-adic regulatorTamagawa factors
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The pith

An explicit K2 element for certain elliptic curves is proven indivisible by 2 in the tame symbol kernel.

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

The paper examines the 2-part of the Bloch-Kato conjecture for L(E_u,2) where E_u is y^2 = x(x+1)(x+u^2) for certain u. The authors compute 2-parts of Tamagawa factors and Galois invariants, and describe the 2-torsion in the Selmer group. They construct a specific element in the kernel of the tame symbol for K2 on an integral model of E_u that has non-vanishing real and 2-adic regulators. Using Merkur'ev-Suslin norm residue isomorphism, they prove this element is indivisible by 2 even modulo torsion, despite being divisible by 2 in the kernel on E_u itself. They also bound the 2-divisibility under the regulator and check the conjecture numerically in many cases.

Core claim

We construct a specific element in the kernel of the tame symbol for K2 on an integral model of E_u, with non-vanishing real and 2-adic regulators. Using techniques involving the norm residue isomorphism of Merkur'ev-Suslin, we prove indivisibility of this element by 2 in that kernel, even modulo torsion, even though it is explicitly divisible by 2 in the kernel of the tame symbol for K2 on E_u. We also bound the 2-divisibility of the images of these elements under the 2-adic regulator map.

What carries the argument

The Merkur'ev-Suslin norm residue isomorphism applied to a constructed K2 element on the integral model of E_u with non-vanishing regulators.

If this is right

  • The 2-part of the Bloch-Kato conjecture for these L-functions is investigated through the algebraic K2 data.
  • Explicit descriptions are given for the 2-torsion in the Selmer group H_f^1(Q, E_u[2^∞](-1)).
  • The 2-parts of the Tamagawa factors and Galois invariants are determined for the family.
  • Bounds are obtained on the 2-divisibility of the regulator images of the K2 elements.

Where Pith is reading between the lines

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

  • This approach might be adapted to prove results for other primes in the Bloch-Kato conjecture for similar curve families.
  • The numerical validations for many u suggest the conjecture holds in additional cases not covered by the proof.
  • Connections to Boyd's family via 2-isogeny could lead to comparisons with prior K2 computations.

Load-bearing premise

The chosen element in the kernel of the tame symbol on the integral model has non-vanishing real and 2-adic regulators and that the Merkur'ev-Suslin techniques apply directly to establish 2-indivisibility even after accounting for torsion.

What would settle it

A calculation for one of the E_u where the constructed K2 element turns out to be divisible by 2 in the kernel or where the Bloch-Kato predicted value does not match the computed K2 order.

read the original abstract

For certain integers $u$, we investigate the 2-part of the Bloch-Kato conjecture for $L(E_u,2)$, where $E_u: y^2=x(x+1)(x+u^2)$ is part of a (twisted) Legendre family that is 2-isogenous to a family studied by Boyd. For this, we first work out the corresponding 2-parts of the Tamagawa factors and Galois invariants. Then we give an explicit description of the 2-torsion in the Selmer group $H_f^1(\mathbb{Q},E_u[2^\infty](-1))$. We construct a specific element in the kernel of the tame symbol for $K_2$ on an integral model of $E_u$, with non-vanishing real and 2-adic regulators. Using techniques involving the norm residue isomorphism of Merkur'ev-Suslin, we prove indivisibility of this element by 2 in that kernel, even modulo torsion, even though it is explicitly divisible by 2 in the kernel of the tame symbol for $K_2$ on $E_u$. We also bound the 2-divisibility of the images of these elements under the 2-adic regulator map. Finally, in many cases we investigate numerically the validity of the 2-part of the Bloch-Kato conjecture.

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

0 major / 4 minor

Summary. The manuscript examines the 2-part of the Bloch-Kato conjecture for L(E_u,2) where E_u : y² = x(x+1)(x+u²) for certain integers u. It computes the 2-parts of the Tamagawa factors and Galois invariants, gives an explicit description of the 2-torsion in the Selmer group H_f¹(Q, E_u[2^∞](-1)), constructs a specific element in the kernel of the tame symbol for K_2 on an integral model of E_u with non-vanishing real and 2-adic regulators, proves 2-indivisibility of this element (even modulo torsion) via Merkur'ev-Suslin techniques despite explicit 2-divisibility in the tame-symbol kernel on E_u itself, bounds the 2-divisibility of the images under the 2-adic regulator, and provides numerical checks for the 2-part of the Bloch-Kato conjecture in many cases.

Significance. If the results hold, the work supplies explicit constructions and proofs supporting the 2-part of the Bloch-Kato conjecture for a twisted Legendre family of elliptic curves, including an indivisibility statement for a K_2 element that overcomes an apparent divisibility obstruction. The explicit Selmer-group description, regulator computations, and direct appeal to the norm-residue isomorphism constitute reproducible algebraic evidence. The numerical investigations add concrete supporting data for the conjecture.

minor comments (4)
  1. §3: the definition of the integral model of E_u and the precise choice of the K_2 element in the tame-symbol kernel should be stated with an explicit equation or generator before the regulator computations begin.
  2. The reference to the family studied by Boyd is mentioned but lacks a specific citation; adding the full bibliographic entry would improve traceability.
  3. In the numerical section, the tables reporting regulator values and 2-divisibility bounds should include the precision (e.g., number of p-adic digits) and the range of u tested to allow independent verification.
  4. The statement that the element is 'explicitly divisible by 2 in the kernel of the tame symbol for K_2 on E_u' would benefit from a short explicit computation or reference to the relevant equation showing the factor of 2.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and for recommending minor revision. No specific major comments were provided in the report, so we have no individual points to address. We will incorporate any minor editorial suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper constructs an explicit element in the kernel of the tame symbol for K2 on an integral model of Eu, computes its real and 2-adic regulators directly, describes the 2-Selmer group H_f^1(Q, Eu[2^∞](-1)) from the curve equation and Galois invariants, and applies the external Merkur'ev-Suslin norm-residue isomorphism to prove 2-indivisibility (even modulo torsion). None of these steps reduces the target indivisibility statement to a fitted parameter, a self-citation chain, or a definition that presupposes the result. Tamagawa factors and Galois invariants are computed from the Weierstrass model without reference to the K2 element's indivisibility. Numerical checks for the Bloch-Kato 2-part are presented separately as supporting evidence rather than as load-bearing input. The derivation is therefore self-contained against independent external theorems and explicit computations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard background results in Galois cohomology and algebraic K-theory; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math Norm residue isomorphism theorem of Merkur'ev-Suslin
    Invoked to prove the 2-indivisibility statement.

pith-pipeline@v0.9.0 · 5570 in / 1349 out tokens · 62747 ms · 2026-05-13T00:51:09.996968+00:00 · methodology

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Reference graph

Works this paper leans on

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