Exact 2-primary essential dimensions after Chernousov–Serre: the value of $\operatorname{ed}(E_8;2)=\operatorname{ed}(\mathrm{HSpin}_{16};2)$, the equality $\operatorname{ed}(f)=\operatorname{ed}(f;2)$ for quadratic forms, and $\operatorname{ed}(\mathrm{Spin}_n)$ for $11\le n\le 14$ in characteristic 2

Chernousov and Serre (2006) introduced the orthogonal-representation method: lower bounds on the essential dimension at the prime 2 of an algebraic group, obtained by twisting a quadratic form by a torsor under an elementary abelian 2-subgroup and applying Stiefel-Whitney classes. They applied it to spin, half-spin, projective orthogonal, and certain exceptional groups, proved $\operatorname{ed}(f)=\operatorname{ed}(f;2)$ for square-free monomial quadratic forms $f$, and listed the unresolved cases of their program. Later literature resolved most of them: $\operatorname{ed}(\mathrm{Spin}_n)$ is now known exactly in every characteristic for $n\ge 15$ and in characteristic $\ne 2$ for all $n$, the method was extended to characteristic 2, and $\operatorname{ed}(E_8;2)$ was identified with $\operatorname{ed}(\mathrm{HSpin}_{16};2)$ and bounded below by 56. The cases that remain open are: the exact value of $\operatorname{ed}(E_8;2)=\operatorname{ed}(\mathrm{HSpin}_{16};2)$, currently pinched between 56 and 120; whether $\operatorname{ed}(f)=\operatorname{ed}(f;2)$ holds for every quadratic form $f$; and the four characteristic-2 values $\operatorname{ed}(\mathrm{Spin}_n)$ for $11\le n\le 14$.

Background

Let $k$ be a base field and $\mathcal{F}$ a functor from field extensions of $k$ to sets. The essential dimension $\operatorname{ed}(\mathcal{F})$ is the smallest integer $d$ such that every element of $\mathcal{F}(K)$, for every finitely generated extension $K/k$, is pulled back from some element of $\mathcal{F}(K_0)$ for an intermediate field $K_0\subset K$ of transcendence degree at most $d$ over $k$: informally, the number of algebraically independent parameters needed to define an object of the kind $\mathcal{F}$ classifies. For an algebraic group $G$ over $k$ one takes $\mathcal{F}(K)=H^1(K,G)$, the set of isomorphism classes of $G$-torsors (principal homogeneous $G$-spaces), and writes $\operatorname{ed}(G)$. For a prime $p$, the $p$-essential dimension $\operatorname{ed}(G;p)$ is defined by the same minimization allowing descent only up to field extensions of degree prime to $p$; always $\operatorname{ed}(G;p)\le\operatorname{ed}(G)$. The basic example: over fields of characteristic $\ne 2$, $n$-dimensional quadratic forms are $O(n)$-torsors, $\operatorname{ed}(O(n))=n$ and $\operatorname{ed}(SO(n))=n-1$; for a single quadratic form $f$ over $K$, $\operatorname{ed}(f)$ (resp. $\operatorname{ed}(f;2)$) is the least transcendence degree of a field over which a form isomorphic to $f$ (resp. to $f$ after a prime-to-2 extension of $K$) is defined.

The groups involved: $\mathrm{Spin}_n$ is the simply connected double cover of the special orthogonal group $SO_n$; for even $n$, the quotient $\mathrm{HSpin}_n=\mathrm{Spin}_n/\mu_2$ determined by one half-spin representation is the half-spin (semispinor) group; $E_8$ is the largest exceptional simple algebraic group (adjoint and simply connected forms coincide, since the center is trivial). The Borel-de Siebenthal maximal-rank subgroup theorem embeds $\mathrm{HSpin}_{16}$ in $E_8$. An action of $G$ on an affine variety is generically free if the stabilizer of a general point is trivial; a generically free $G$-action on a variety of dimension $d$ yields the standard upper bound $\operatorname{ed}(G)\le d-\dim G$. A monomial quadratic form is a diagonal form $\langle a_1,\dots,a_n\rangle$ whose entries are monomials in $m$ algebraically independent variables, the exponent vector of each entry being read modulo 2 as an element of $(\mathbb{Z}/2)^m$; the form is square-free when no two entries have the same exponent vector. For square-free monomial forms, Chernousov–Serre proved that $\operatorname{ed}(f)=\operatorname{ed}(f;2)$, both equal to the $\mathbb{F}_2$-rank of the span of the exponent vectors that occur with odd multiplicity.

Before 2006, lower bounds on $\operatorname{ed}(G;p)$ came mainly from fixed-point techniques valid in characteristic 0. Chernousov–Serre's orthogonal-representation method works over arbitrary base fields of characteristic $\ne 2$: given a representation $\rho\colon G\to O(V,q)$ and a torsor $\theta_A\in H^1(K,A)$ under an elementary abelian 2-subgroup $A\subset G$ (a subgroup isomorphic to $(\mathbb{Z}/2)^s$ for some $s$), one twists $(V,q)$ by the image of $\theta_A$; Stiefel-Whitney classes (the mod-2 characteristic classes of quadratic forms) certify that the resulting monomial twisted forms are incompressible (not definable over any field of strictly smaller transcendence degree), giving bounds such as $\operatorname{ed}(G;2)\ge r+1$ for split simple $G$ of adjoint type, where $r$ is the rank of a maximal elementary abelian 2-subgroup of $G$; $\operatorname{ed}(\mathrm{Spin}_n;2)\ge\lfloor n/2\rfloor$ for $n\ge 11$, improved by 1 when $n\equiv -1,0,1\pmod 8$; $\operatorname{ed}(\mathrm{HSpin}_n;2)\ge n/2+1$ when $8\mid n$; and analogues for $\mathrm{PSO}_n$, $2.E_7$ (the simply connected double cover of the adjoint group of type $E_7$), and $\mathrm{PGL}_n$. The method is genuinely 2-primary and yields no bound at all for groups of type $E_6$. In particular it gives $\operatorname{ed}(E_8;2)\ge 9$.

The subsequent history reshaped this picture. Brosnan–Reichstein–Vistoli (2010) and, for the remaining congruence class, Chernousov–Merkurjev (2014) proved that for $n\ge 15$ in characteristic 0 the exact values are exponential, for example $\operatorname{ed}(\mathrm{Spin}_n)=2^{(n-1)/2}-n(n-1)/2$ for odd $n$ and $\operatorname{ed}(\mathrm{Spin}_{16})=24$; Garibaldi–Guralnick (2017) extended these computations to all characteristics $\ne 2$ by group-scheme generic-freeness arguments, and Totaro (2019) determined $\operatorname{ed}(\mathrm{Spin}(n))$ over fields of characteristic 2 for $n\ge 15$ and $n\le 10$, with the same values. For $n\le 14$ in characteristic $\ne 2$ the values are known from a table of Rost reproduced in Garibaldi's memoir (2009): $\operatorname{ed}(\mathrm{Spin}_{11}),\dots,\operatorname{ed}(\mathrm{Spin}_{14})=5,6,6,7$. Babic–Chernousov (2015) extended the orthogonal-representation method itself to characteristic 2, proving $\operatorname{ed}(G)\ge r+1$ for adjoint simple $G$ there. For the exceptional end, Ofek proved $\operatorname{ed}(E_8;2)=\operatorname{ed}(\mathrm{HSpin}_{16};2)$ over base fields of characteristic $\ne 2,3$: the inequality $\le$ because every $E_8$-torsor admits, after a field extension of degree prime to 2, a reduction of structure to $\mathrm{HSpin}_{16}$ (a rewriting of the torsor as one under the subgroup, using the Borel-de Siebenthal embedding), and $\ge$ by an anisotropic $\mathrm{HSpin}_{16}$-torsor (one admitting no reduction to a proper parabolic subgroup) over a 2-closed field (a field all of whose finite extensions have degree a power of 2); his loop-torsor technique (decompositions of torsors over iterated Laurent-series fields, after Gille–Pianzola) improves the lower bound to 56. The upper bound remains the known estimate $\operatorname{ed}(E_8;2)\le 120$ recorded in Merkurjev's survey. Ofek–Reichstein (2025) developed a valuation-theoretic formula computing $\operatorname{ed}(\alpha)=\operatorname{ed}(\alpha;p)$ for cohomology classes $\alpha$ over strictly Henselian fields, but their Remark 12.1 shows the Rost-invariant route cannot reproduce the exponential spinor bounds, indicating the remaining $E_8$ gap needs different ideas.

Problem Statement

Determine the surviving open cases of the 2-primary essential-dimension program of Chernousov–Serre (2006): the exact parameter count for torsors under the groups of the spin family, under the exceptional group $E_8$, and for the quadratic forms these groups classify. Concretely, over base fields of the stated characteristics:

(a) Prove matching lower and upper bounds establishing $\operatorname{ed}(E_8;2)=\operatorname{ed}(\mathrm{HSpin}_{16};2)=N$ for the split groups over base fields of characteristic $\ne 2,3$. Currently $56\le\operatorname{ed}(E_8;2)=\operatorname{ed}(\mathrm{HSpin}_{16};2)\le 120$; the lower bound 56 is due to Ofek and the upper bound 120 is recorded in Merkurjev's survey. Here the lower bound is certified by an explicit incompressible $E_8$- or $\mathrm{HSpin}_{16}$-torsor (one not definable over any field of strictly smaller transcendence degree), and the upper bound by a construction descending every such torsor to at most $N$ parameters.

(b) Decide whether $\operatorname{ed}(f)=\operatorname{ed}(f;2)$ for every quadratic form $f$ over a field of characteristic $\ne 2$: either prove the equality in general (it is known for square-free monomial forms and for all $f$ with $\operatorname{ed}(f)\le 2$), or exhibit an explicit form $f$ over an explicit finitely generated field extension, with proof, such that $\operatorname{ed}(f;2)<\operatorname{ed}(f)$.

(c) Compute $\operatorname{ed}(\mathrm{Spin}_n)$ over fields of characteristic 2 for $n=11,12,13,14$. In characteristic $\ne 2$ these four values are $5,6,6,7$; in characteristic 2 the values are known for $n\le 10$ and $n\ge 15$ and coincide with those, and Totaro explicitly poses the four remaining cases.

These components are the remaining entries of a single table: the 2-primary essential dimensions of the groups classifying quadratic forms and of the exceptional group attached to the half-spin family. They share one obstruction: no known construction (for upper bounds) or incompressibility argument (for lower bounds) closes the remaining gaps.

References

1. Vladimir Chernousov and Jean-Pierre Serre, Lower bounds for essential dimensions via orthogonal representations, Journal of Algebra 305 (2006), no. 2, 1055–1070, DOI 10.1016/j.jalgebra.2005.10.032, https://doi.org/10.1016/j.jalgebra.2005.10.032 (source paper).
2. Patrick Brosnan, Zinovy Reichstein, and Angelo Vistoli, Essential dimension, spinor groups, and quadratic forms, Annals of Mathematics (2) 171 (2010), no. 1, 533–544, DOI 10.4007/annals.2010.171.533, https://doi.org/10.4007/annals.2010.171.533 (preprint arXiv:0907.0922, https://arxiv.org/abs/0907.0922).
3. Vladimir Chernousov and Alexander Merkurjev, Essential dimension of spinor and Clifford groups, Algebra & Number Theory 8 (2014), no. 2, 457–472, DOI 10.2140/ant.2014.8.457, https://doi.org/10.2140/ant.2014.8.457.
4. Skip Garibaldi, Cohomological invariants: exceptional groups and spin groups, Memoirs of the American Mathematical Society 200 (2009), no. 937, DOI 10.1090/memo/0937, https://doi.org/10.1090/memo/0937 (Section 23 tabulates essential dimensions of spin and exceptional groups in characteristic not 2, reproducing Rost's table).
5. Skip Garibaldi and Robert M. Guralnick, Spinors and essential dimension, Compositio Mathematica 153 (2017), no. 3, 535–556, DOI 10.1112/S0010437X16008162, https://doi.org/10.1112/S0010437X16008162 (preprint arXiv:1601.00590, https://arxiv.org/abs/1601.00590; version 2 contains an appendix by A. Premet on the generic stabilizer in $\mathrm{HSpin}_{16}$ in characteristic 2).
6. Burt Totaro, Essential dimension of the spin groups in characteristic 2, Commentarii Mathematici Helvetici 94 (2019), no. 1, 1–20, DOI 10.4171/cmh/452, https://doi.org/10.4171/cmh/452 (preprint arXiv:1701.05959, https://arxiv.org/abs/1701.05959).
7. Antonio Babic and Vladimir Chernousov, Lower bounds for essential dimensions in characteristic 2 via orthogonal representations, Pacific Journal of Mathematics 279 (2015), no. 1–2, 37–63, DOI 10.2140/pjm.2015.279.37, https://doi.org/10.2140/pjm.2015.279.37.
8. Zinovy Reichstein, Essential dimension, in Proceedings of the International Congress of Mathematicians (Hyderabad, August 19–27, 2010), Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 162–188, https://personal.math.ubc.ca/~reichst/icm4-25-10.pdf.
9. Danny Ofek, Lower bounds on the essential dimension of reductive groups, preprint arXiv:2411.03789 (2024, version 3 of 2025), to appear in Annales scientifiques de l'École normale supérieure, https://arxiv.org/abs/2411.03789.
10. Danny Ofek and Zinovy Reichstein, Essential dimension of cohomology classes via valuation theory, Israel Journal of Mathematics (2025), DOI 10.1007/s11856-025-2861-8, https://doi.org/10.1007/s11856-025-2861-8 (preprint arXiv:2401.08532, https://arxiv.org/abs/2401.08532).
11. Alexander S. Merkurjev, Essential dimension: a survey, Transformation Groups 18 (2013), no. 2, 415–481, DOI 10.1007/s00031-013-9216-y, https://doi.org/10.1007/s00031-013-9216-y (records the upper bound $\operatorname{ed}(E_8;2)\le 120$).
12. Fatma Kader Bingöl, Adam Chapman, and Ahmed Laghribi, Essential dimension of sequences of quadratic Pfister forms, Proceedings of the American Mathematical Society 154 (2026), no. 6, 2337–2344, DOI 10.1090/proc/17601, https://doi.org/10.1090/proc/17601 (preprint arXiv:2505.22237, https://arxiv.org/abs/2505.22237).
13. Roland Lötscher and Mark MacDonald, The slice method for G-torsors, Advances in Mathematics 320 (2017), 329–360, DOI 10.1016/j.aim.2017.08.042, https://doi.org/10.1016/j.aim.2017.08.042.
14. Philippe Gille and Arturo Pianzola, Torsors, reductive group schemes and extended affine Lie algebras, Memoirs of the American Mathematical Society 226 (2013), no. 1063, DOI 10.1090/s0065-9266-2013-00679-x, https://doi.org/10.1090/s0065-9266-2013-00679-x (the monograph introducing loop torsors and their decompositions, the tool behind Ofek's lower bound).