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Optimal embeddings for maximal orders of central simple algebras of degree 3 over number fields
Pith reviewed 2026-05-07 13:30 UTC · model grok-4.3
The pith
An order S in a cubic extension K of a number field F can be optimally embedded into all maximal orders of a degree-3 central simple algebra B only in special cases; otherwise the embeddings occur in exactly one-third or two-thirds of the B
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Let B be a central simple algebra of degree 3 over a number field F and let K/F be a cubic extension. For any order S of K, the paper determines the exact arithmetic conditions under which S cannot be optimally embedded into any maximal order of B, and shows that in all other cases S embeds optimally into precisely one-third or precisely two-thirds of the isomorphism classes of maximal orders of B.
What carries the argument
The 1/3–2/3 partitioning of the isomorphism classes of maximal orders of B according to whether they admit an optimal embedding of the given order S, governed by local embedding obstructions at finitely many primes.
If this is right
- There exist explicit conditions, readable from local invariants, under which no maximal order of B admits an optimal embedding of S.
- In the generic case the density of maximal orders containing S is a positive rational number 1/3 or 2/3.
- The result distinguishes the two possible non-trivial densities without computing the full class set of maximal orders.
- The same arithmetic data that controls the density also determines whether S fails to embed anywhere.
Where Pith is reading between the lines
- The fractional counts suggest that the class set of maximal orders behaves like a torsor under a group of order dividing 3 with respect to the embedding problem.
- Similar density statements may exist for central simple algebras of higher degree, although the paper restricts attention to degree 3.
- One could test the result computationally by enumerating maximal orders in quaternion or cubic division algebras over small number fields and checking the observed embedding proportions.
- The criterion supplies a new tool for counting solutions to embedding problems in the adelic description of maximal orders.
Load-bearing premise
The isomorphism classes of maximal orders of B can be partitioned into well-defined thirds according to the embedding behavior of S, with all obstructions captured by the standard arithmetic data of S and B.
What would settle it
Pick an explicit cubic field K over Q, a degree-3 central simple algebra B over Q, and an order S in K; count the actual number of isomorphism classes of maximal orders of B and check whether the number that admit an optimal embedding of S is exactly 0, exactly one-third, or exactly two-thirds of the total.
read the original abstract
Let $B$ be a central simple algebra of degree 3 over a number field $F$ and $K/F$ be a finite extension of degree 3. For an order $S$ of $K$, we determine exactly when $S$ cannot be optimally embedded into all maximal orders of $B$. Moreover, we further determine exactly when $S$ can be optimally embedded into $\frac{1}{3}$ isomorphism classes of maximal orders of $B$ and $\frac{2}{3}$ isomorphism classes of maximal orders of $B$ in the rest of cases.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript determines exactly when an order S in a cubic extension K/F cannot be optimally embedded into all maximal orders of a central simple algebra B of degree 3 over F. In the remaining cases, it classifies the situations in which S embeds optimally into precisely 1/3 or 2/3 of the isomorphism classes of maximal orders of B.
Significance. If the stated proportions are rigorously established, the work supplies a complete arithmetic criterion for optimal embeddings of cubic orders into maximal orders of degree-3 central simple algebras. This would be a useful addition to the literature on embedding problems and class sets in non-commutative arithmetic, with the explicit 1/3 and 2/3 counts suggesting an underlying group action or equidistribution that could be applied to counting or density questions.
major comments (1)
- [Abstract (main theorem)] The central claim that the proportion is exactly 1/3 or 2/3 requires that the finite set of isomorphism classes of maximal orders of B admits a natural partition into three equal parts on which the optimal-embedding condition is constant. The manuscript must explicitly construct or prove the existence of such a partition (for instance via a free action of a group of order 3 arising from a cubic residue symbol, narrow class group element, or automorphism of the Brauer class) and show that the embedding obstruction is invariant on the orbits. If the action is not free or the obstruction fails to be equivariant, other proportions are possible and the 'exactly' statement does not hold.
Simulated Author's Rebuttal
We thank the referee for the careful reading of the manuscript and for highlighting the need to make the partition into three equal parts explicit in the proof of the main theorem. We address this point below and have incorporated the requested clarification into the revised version.
read point-by-point responses
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Referee: [Abstract (main theorem)] The central claim that the proportion is exactly 1/3 or 2/3 requires that the finite set of isomorphism classes of maximal orders of B admits a natural partition into three equal parts on which the optimal-embedding condition is constant. The manuscript must explicitly construct or prove the existence of such a partition (for instance via a free action of a group of order 3 arising from a cubic residue symbol, narrow class group element, or automorphism of the Brauer class) and show that the embedding obstruction is invariant on the orbits. If the action is not free or the obstruction fails to be equivariant, other proportions are possible and the 'exactly' statement does not hold.
Authors: We agree that the original manuscript would benefit from an explicit construction of the partition to rigorously justify the exact proportions. The proportions arise from the local embedding conditions at places dividing the discriminant of B and the action of a group of order 3 on the set of maximal orders. In the revised manuscript we have added a new subsection (Section 3.2) that constructs this action explicitly: it is induced by the automorphism of the Brauer class [B] in Br(F)[3] combined with the cubic residue symbol of the extension K/F. We prove that the action is free whenever B is a division algebra (the case where the set of isomorphism classes is non-trivial) and that the optimal-embedding obstruction, which is determined by whether certain local norms lie in the image of the reduced norm map, is constant on each orbit. This equivariance follows from the compatibility of the local Hilbert symbols with the global Brauer class. The revised proof of the main theorem now invokes this partition directly, confirming that the embedding condition holds on exactly one or two of the three orbits in the remaining cases. We believe this fully resolves the concern. revision: yes
Circularity Check
No circularity: derivation rests on independent arithmetic invariants
full rationale
The paper determines exact conditions for optimal embeddings of an order S into maximal orders of a central simple algebra B of degree 3, including cases where the proportion of isomorphism classes is exactly 1/3 or 2/3. The abstract and context indicate these proportions arise from partitioning via the given arithmetic data (discriminants, local invariants, ramification) without any quoted reduction of the output proportions to fitted parameters, self-definitions, or load-bearing self-citations. The central claims are presented as structural results on embeddings and class partitions, with no evidence that the 1/3 or 2/3 counts are forced by construction from the inputs themselves. This is the normal case of a self-contained number-theoretic argument.
Axiom & Free-Parameter Ledger
Reference graph
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