Essential Dimension of CSAs

• Essential dimension is a numerical invariant that measures the minimum number of independent parameters required to define central simple algebras over a field.
• It encapsulates complex relationships among field theory, cohomology, and invariant theory, particularly addressing challenges in bad characteristic cases.
• Explicit constructions using symbol algebras establish sharp upper and lower bounds that advance parameterization and classification strategies for these algebras.

The essential dimension of central simple algebras is a fundamental invariant measuring the minimal number of independent parameters required to define a family of such algebras over a given field. For central simple algebras of fixed degree and exponent, the essential dimension encodes a deep relationship between cohomological, field-theoretic, and algebraic properties of the algebras, especially in the "bad characteristic" case where the characteristic divides the degree. Particular attention has been devoted to prime power degree and prime power exponent cases, the behavior of symbol algebras, and the refinement of upper and lower bounds for these invariants. The essential dimension has connections to invariant theory, Galois cohomology, and the structure theory of algebraic groups.

1. Central Simple Algebras and the Essential Dimension Framework

A central simple algebra (CSA) over a field $F$ is a finite-dimensional associative $F$-algebra $A$ with center exactly $F$ and with no nontrivial two-sided ideals. The degree of $A$ is $\deg A=\sqrt{\dim_F A}$, and its exponent is its order in the Brauer group $\operatorname{Br}(F)$.

For any algebraic group $G$ or more generally a functor $\mathcal{F}:\mathrm{Fields}_F\to \mathrm{Sets}$, the essential dimension $\ed(\mathcal{F})$ is defined as the minimal integer $d$ such that every element $a\in\mathcal{F}(K)$ descends to a subfield $L\subseteq K$ of transcendence degree at most $d$ over $F$. For $G$, $\ed(G):=\ed(H^1(-,G))$. The essential $p$-dimension $\ed_p(\mathcal{F})$ is defined analogously, allowing for descent up to finite extensions of degree prime to $p$.

A central simple algebra of degree $n$ and exponent dividing $m$ corresponds functorially to elements of $H^1(-,GL_n/\mu_m)$, and thus the essential dimension of the classifying functor

$\Alg_{n,m}(K) = \{\text{isomorphism classes of CSA of deg %%%%23%%%%, exp%%%%24%%%% over %%%%25%%%%} \}$

equals $\ed(GL_n/\mu_m)$.

2. Prime Power Degree and Symbol Algebras

For central simple algebras with degree $p^r$ and exponent dividing $p^s$ over a field of characteristic $p$, symbol algebras provide a crucial structure. A symbol $p$-algebra over $E$ is given by

$$[a,b)_p = E\langle u,v\rangle/(u^p-u-a,\; v^p-b,\; vu-uv-v)$$

and has degree $p$ and exponent dividing $p$. The functor $\Dec_{p^r}$ sends $K$ to isomorphism classes of tensor products of $r$ such symbol algebras over $K$, so each $A\in\Dec_{p^r}(K)$ has degree $p^r$ and exponent dividing $p$.

These explicit constructions give rise to "universal" families whose parameter count yields the upper bound $\ed(\Dec_{p^r})\le r+1$ when $|F|>p^r$ and $\operatorname{char} F=p$ (Baek, 2010, Chapman et al., 18 Nov 2025). The reduction in parameters is achieved via Artin–Schreier theory and the essential dimension $1$ result for $(\mathbb{Z}/p\mathbb{Z})^r$-torsors in positive characteristic.

3. Lower and Upper Bounds: Degree/Exponent Structure

The essential dimension of the functor $\Alg_{p^r,p^s}$ with $1\leq s<r$ is subject to strong cohomological lower bounds:

• If $F$ is of char $p$ and $1\leq s<r$, $\ed_p(\Alg_{p^r,p^s})\geq 3$, by an argument involving Tsen's theorem (which kills Brauer groups for fields of transcendence degree $1$) and de Jong's index-exponent theorem for transcendence degree $2$ (Baek, 2010).
• For indecomposable algebras of degree $p^{\ell m}$ and exponent $p^m$ over a perfect field, the essential $p$-dimension satisfies $\ed_p(\Alg_{p^{\ell m},p^m})\ge \ell+1$, matching the generic symbol length for the corresponding Kato–Milne cohomology group (Chapman et al., 2019, Chapman et al., 18 Nov 2025).

Upper bounds are typically constructed via explicit parameterizations:

• For $\Dec_{p^r}$, the explicit chain construction (see Section 4) demonstrates that the essential dimension equals $r+1$.
• For certain small degrees: e.g., for $\Alg_{4,2}$ in characteristic $2$, the exact value is $3$; for $\Alg_{8,2}$ in char $2$, Baek constructs a 10-parameter universal CSA, giving $\ed(\Alg_{8,2})\le 10$ (Baek, 2010), and Chapman further reduces this to $\le 9$ via improvements in the parameter count (Chapman, 2022).

4. Explicit Constructions and Cohomological Techniques

Explicit parameterizations of universal families are central to upper bound proofs:

• The $r$-fold tensor of $p$-symbol algebras can be rewritten as a "chain" $$[\alpha_1,\alpha_2)_p\otimes[\alpha_2,\alpha_3)_p\cdots$$ yielding $\ed(\Alg_{p^r,p})\le r+1$ (Chapman et al., 18 Nov 2025).
• In characteristic $2$, a generic degree-$8$, exponent-$2$ division algebra is presented using three Artin–Schreier extensions forming a maximal subfield and reduced step by step using chain and corestriction arguments to a 9-parameter formula for a universal algebra (Chapman, 2022).
• For $\Alg_{4,2}$, detailed analysis shows every degree-$4$, exponent-$2$ algebra is a biquaternion algebra, whose explicit presentation as a tensor of two $2$-symbol algebras descends to transcendence degree $3$ (Chapman et al., 18 Nov 2025, Baek, 2010).

Key tools include:

• Cohomological invariants: Kato–Milne cohomology groups $H_{p^m}^{n+1}(F)$, symbol length bounds, and corestriction maps.
• The chain lemma for quaternion algebras in characteristic $2$, used to relate corestriction vanishing to parameter reductions.
• Invariants from Pfister forms in certain cases, especially for degree $4$ algebras.

5. Main Results, Special Values, and Tables

Combining these methods, the precise or best-known values in characteristic $p>0$ (with $p$ dividing the degree) are summarized:

Algebraic Class	Lower Bound	Upper Bound	Exact Value(s)
$\Alg_{p^n,p}$	$n+1$	$n+1$	$n+1$
$\Alg_{4,2}$ (char $2$)	$3$	$3$	$3$
$\Alg_{8,2}$ (char $2$)	$4$ [McKinnie]	$9$ [Chapman]	$4\leq \text{ed}\leq 9$
$\Alg_{4,4}$ (char $2$)	$4$	$5$	$4\leq \text{ed}\leq 5$

For general degree/exponent, upper bounds like $\ed(\Alg_{n,n})\le n^2-3n+1$ and, for even $n>4$, $\ed(\Alg_{2n,2})\le 2n^2-3n-6$ are achieved via generically free representations of the normalizer of a torus in $PGL_n$ (Garibaldi et al., 2015, Chapman et al., 18 Nov 2025).

6. Open Problems and Frontiers

Several questions remain open or only partially resolved:

• Determining $\ed(\Alg_{8,2})$ in characteristic $2$ precisely remains an outstanding problem ($4 \leq \ed \leq 9$).
• Whether the sharp essential dimension for sequences of linked cyclic algebras equals the twofold case for higher $p$ is unresolved.
• For Karpenko's indecomposable algebras, it is unknown if their essential dimension can exceed the lower bound $n+1$ for $\Alg_{p^n,p}$.
• The relationship between the essential dimension of $\Alg_{n,m}$ and $\SL_n/\mu_m$ in bad characteristic is not settled.

Further, the connection to error-correcting codes appears in the computation of essential dimension for tuples $(A_1,\dots,A_r)$ of mutually constrained CSAs via consideration of the weight profile of codes associated to central subgroups in $(GL_{n_1}\times\cdots\times GL_{n_r})/C$ (Cernele et al., 2014).

7. Broader Implications and Related Structures

The theory of essential dimension for central simple algebras in bad characteristic integrates methods from algebraic geometry, Galois cohomology, representation theory, and quadratic form theory. Advances on lower and upper bounds have impacted both the understanding of the structure of division algebras and parametrization problems in algebraic groups. The essential dimension remains a key invariant in classification problems, guiding the search for generic division algebras, the structure of classifying varieties, and the analysis of cohomological invariants in both classical and wild ramification settings (Chapman et al., 18 Nov 2025, Baek, 2010, Chapman, 2022, Chapman et al., 2019, Cernele et al., 2014).