Serre's Conjecture II

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $F$ be a field and $\Gamma_F=\operatorname{Gal}(F^{\mathrm{sep}}/F)$ its absolute Galois group. For a prime $\ell\ne\operatorname{char}F$, $\operatorname{cd}_\ell(F)$ is the least $d$ such that continuous Galois cohomology $H^q(\Gamma_F,M)$ vanishes for $q>d$ and every discrete torsion $\ell$-primary $\Gamma_F$-module $M$. In characteristic zero, put $\operatorname{cd}(F)=\sup_\ell\operatorname{cd}_\ell(F)$.

For characteristic $p>0$, use the field-cohomological-dimension convention in Source S2, which also covers imperfect fields. Let $\Omega_F^j=\bigwedge_F^j\Omega^1_{F/\mathbb Z}$ be absolute differential forms, with exterior differential $d$, $\Omega_F^0=F$ and $\Omega_F^{-1}=0$. Define $H_p^{j+1}(F)$ to be the cokernel of the additive map \[ \wp:\Omega_F^j\longrightarrow\Omega_F^j/d\Omega_F^{j-1},\qquad x\,d\log y_1\wedge\cdots\wedge d\log y_j \longmapsto (x^p-x)\,d\log y_1\wedge\cdots\wedge d\log y_j, \] where $d\log y=dy/y$ and $y_i\in F^\times$. Set $\operatorname{cd}_p(F)$ to be the least $d\ge0$ such that $[F:F^p]\le p^d$ and $H_p^{d+1}(L)=0$ for every finite field extension $L/F$; use infinity if no such $d$ exists. Then $\operatorname{cd}(F)=\sup_\ell\operatorname{cd}_\ell(F)$, including this characteristic-prime term. For perfect fields this agrees with the usual absolute-Galois-group convention.

Let $G$ be a semisimple simply connected algebraic group over $F$: it is smooth, connected and affine, has no nontrivial connected solvable normal subgroup over an algebraic closure, and admits no nontrivial central isogeny from a connected semisimple group. The pointed set $H^1(F,G)$ consists of continuous cocycles $a:\Gamma_F\to G(F^{\mathrm{sep}})$ with $a_{\sigma\tau}=a_\sigma\sigma(a_\tau)$, modulo $a'_\sigma=g^{-1}a_\sigma\sigma(g)$. Its distinguished point is the class of the constant identity cocycle; thus $H^1(F,G)=\{1\}$ is a pointed-set assertion.

Statement recorded in the supplied source.

For every field $F$ with $\operatorname{cd}(F)\le2$ and every semisimple simply connected $F$-group $G$, is $H^1(F,G)=\{1\}$? Equivalently, does every principal homogeneous space (torsor) under $G$ have an $F$-rational point? A torsor is a nonempty variety on which $G$ acts freely and transitively after passing to an algebraic closure. The question includes every characteristic and every such group, rather than just the classical types. [S2]

Short English statement

Over every field of cohomological dimension at most two, must every torsor under a simply connected semisimple group have a rational point?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1468 (ALG-013). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Diego Izquierdo and Giancarlo Lucchini Arteche, Transfer Principles for Galois Cohomology and Serre’s Conjecture II, arXiv:2308.00903v3 (2025), Conjecture 1.2 and Section 2.1, Definition 2.1. https://arxiv.org/pdf/2308.00903v3.