Optimal embeddings for maximal orders of central simple algebras of degree 3 over number fields

Technical Summary: Optimal Embeddings for Maximal Orders of Central Simple Algebras of Degree 3

Problem Statement
This paper addresses the problem of integral embeddings of orders into central simple algebras (CSAs) over number fields. While the local-global principle holds for field extensions embedding into CSAs (via class field theory), it fails for integral embeddings. Specifically, an order $S$ of a field extension $K/F$ of degree $n$ may not embed optimally into all maximal orders of a CSA $B$ of degree $n$.

Previous work by Chevalley, Chinburg, Friedman, Chan, Xu, Linowitz, Shemanske, and Voight established "selectivity" criteria for $n=2$ (quaternion algebras) and for Eichler orders. This paper extends the study of optimal selectivity to the case where $B$ is a central simple algebra of degree 3 over a number field $F$, and $K/F$ is a degree 3 extension. The primary goal is to determine exactly when an order $S \subset K$ cannot be optimally embedded into all maximal orders of $B$, and to quantify the proportion of maximal orders that do admit such embeddings.

Methodology
The authors employ a combination of global class field theory and detailed local analysis:

1. Global Framework (Selectivity Sandwich):
   The paper utilizes the "selectivity sandwich" framework adapted from Voight's work on quaternions. It relates the existence of optimal embeddings to the reduced norm of a specific set of elements $E$ in the algebra. The key object is the group $F^\times \text{Nr}(\mathbb{E})$ within the idele group $\mathbb{A}_F^\times$.

   • The paper establishes that if $K$ is not contained in a specific class field $H_{GN(\mathcal{O})}$ associated with the maximal order $\mathcal{O}$, then $S$ embeds into all maximal orders in the genus.
   • If $K \subseteq H_{GN(\mathcal{O})}$, the problem reduces to analyzing the index of $F^\times \text{Nr}(\mathbb{E})$ relative to $F^\times \text{Nr}(\mathbb{K}^\times)$.

2. Local Analysis (Degree 3):
   A significant portion of the work is dedicated to computing local embedding numbers $m(S_p, \mathcal{O}_p)$ for degree 3 algebras over local fields $F_p$.

   • Normal Case: The authors analyze embeddings into $M_3(R_p)$ and division algebras. They derive explicit criteria for optimality using matrix reductions modulo the maximal ideal.
   • Special Optimal Embeddings: Unlike the degree 2 case, the authors identify a phenomenon specific to degree 3: "special optimal embeddings." These occur when the reduction of the embedding matrices falls into a specific Jordan block form (Case 3 in Lemma 3.3) where the standard conjugation argument to the regular representation fails.
   • Split and Inert Cases: The paper computes the structure of the reduced norm set $\text{Nr}(E_p)$ for both split ($K_p \cong F_p \times F_p \times F_p$) and inert (unramified extension) cases. They determine that the number of conjugacy classes of optimal embeddings is either 1 or 2, depending on the discriminant of the order $S_p$.

3. Synthesis:
   By combining the local results via the Artin map, the authors relate the global selectivity set to the discriminant ideal of the order $S$.

Key Contributions and Results

• The Selectivity Set $D(S)$: The paper defines a "selectivity set" $D(S)$ for an order $S$ in a degree 3 unramified extension $K/F$. This set is a subset of the Galois group $\text{Gal}(K/F)$, constructed via the restriction map from the ideal class group of $F$ to the Galois group, applied to the prime factors of the square root of the discriminant of $S$.
  $$D(S) = \{ \rho([\mathfrak{p}^k]) : \mathfrak{p}^k \parallel \sqrt{\text{disc}(S)}, \mathfrak{p}^{k+1} \nmid \sqrt{\text{disc}(S)} \}$$

• Main Theorem (Theorem 4.1): The paper establishes a necessary and sufficient condition for an order $S$ to be "selective" (i.e., not embeddable into all maximal orders):
  $S$ cannot be optimally embedded into every maximal order of $B$ if and only if:

  1. $K/F$ is everywhere unramified.
  2. $B \cong M_3(F)$ (i.e., $B$ is split).
  3. $D(S) \neq \text{Gal}(K/F)$.

  In the case where these conditions hold, the paper proves that $S$ can be optimally embedded into exactly $|D(S)|/3$ of the isomorphism classes of maximal orders in the genus. Since $|\text{Gal}(K/F)| = 3$, this implies the embedding is possible in either $1/3$ or $2/3$ of the classes, depending on the size of $D(S)$.

• Refinement of Previous Results: The paper provides a counter-example to the direct application of selectivity results from degree 2 or previous higher-degree generalizations. Specifically, it demonstrates an order $S_2$ that, according to previous non-optimal embedding criteria (Linowitz and Shemanske), would embed into all maximal orders, but fails to admit an optimal embedding into all of them.

Significance
The paper claims to establish the precise selectivity condition for optimal embeddings in central simple algebras of degree 3. Its significance lies in:

1. Generalization: Extending the theory of optimal selectivity from quaternion algebras (degree 2) to degree 3, a non-trivial step due to the increased complexity of matrix similarity classes and the emergence of "special" embeddings.
2. Exact Quantification: Moving beyond binary existence results to provide an exact count ($1/3$ or $2/3$) of the maximal orders admitting optimal embeddings, governed explicitly by the arithmetic of the discriminant of the order.
3. Correction of Intuition: Highlighting that results for non-optimal embeddings or lower degrees do not automatically generalize to optimal embeddings in degree 3, necessitating the specific local analysis of "special" cases presented in the paper.

The work relies on standard notation from class field theory and the theory of orders in CSAs, building directly upon the foundational work of Eichler, Voight, and Linowitz/Shemanske.