Finite abelian subgroups of algebraic groups

Technical Summary: Finite Abelian Subgroups of Algebraic Groups

Problem Statement
The paper investigates the structure of finite abelian $k$-subgroups $A$ within an algebraic group $G$ over an algebraically closed field $k$, specifically focusing on the case where the order of $A$ is not divisible by the characteristic of $k$. The central question is to quantify "how far" an arbitrary finite abelian subgroup is from being toral (i.e., contained in a maximal torus). This problem extends classical work by Borel (1961) on compact Lie groups and torsion primes. While previous results, such as those by Reichstein and Youssin (2001), established that for a $p$-subgroup $A$, the index $[A : A \cap T]$ divides the Grothendieck torsion index $t(G)$, these results were limited to characteristic 0 and often required $A$ to be a $p$-group. This work seeks to generalize these structural results to arbitrary finite abelian subgroups and positive characteristic (provided the characteristic does not divide the group order), and to apply these structural insights to the splitting of torsors over fields of iterated Laurent series.

Methodology
The authors employ a combination of algebraic group theory, Galois cohomology, and the theory of torsors over fields of iterated Laurent series ($k_r = k((t_1))\dots((t_r))$).

1. Reduction to Torsors: The authors utilize the correspondence between finite abelian subgroups and torsors. Specifically, they relate the structure of $A$ to the splitting behavior of $G$-torsors induced by $A$-torsors over $k_r$.
2. Valuative Criteria and Properness: A key technical tool involves the use of the valuative criterion for properness. By considering the extension of valuations from $k_r$ to the function field of a connected $A$-torsor, the authors demonstrate that if a torsor splits, the subgroup $A$ must be contained in a Borel subgroup, and subsequently in a maximal torus, under specific conditions.
3. Galois Theory of Iterated Laurent Series: The paper leverages the structure of the absolute Galois group of $k_r$, specifically its maximal prime-to-$l$ quotient (where $l$ is the characteristic exponent), to analyze the splitting fields of torsors. This allows for the translation of geometric splitting conditions into group-theoretic constraints on the Galois groups of field extensions.
4. Depth and Torsion Indices: The authors introduce and utilize the concept of the depth of a finite abelian subgroup $A$, defined as the greatest common divisor of the indices $[A : A \cap T]$ over all maximal tori $T$. They relate this to variants of the Grothendieck torsion index, denoted $t_2(G)$ and $t_3(G)$, defined via torsors over iterated Laurent series and the depths of subgroups, respectively.
5. Case-by-Case Analysis for $E_8$: For the exceptional group $E_8$, the authors perform a detailed classification of maximal finite abelian subgroups (referencing Draper and Elduque) and compute their depths using Lie algebra decompositions and coding theory (specifically self-orthogonal binary codes associated with spin groups).

Key Contributions and Results

• Generalization of Reichstein-Youssin (Theorem 1.1): The paper proves that for any finite abelian subgroup $A \subset G$ with $\text{char}(k) \nmid |A|$, there exists a maximal torus $T$ such that the index $[A : A \cap T]$ divides the Grothendieck torsion index $t(G)$. Crucially, this result holds in positive characteristic (as long as it does not divide $|A|$) and does not require $A$ to be a $p$-group. This answers a question posed by Totaro and Wang.
• Splitting over Iterated Laurent Series (Theorem 1.2): The authors establish a precise relationship between the splitting of a $G$-torsor induced by an $A$-torsor over $k_r$ and the structure of $A$. Specifically, if a finite extension $L/k_r$ splits the induced torsor, the index $[A : A \cap T]$ divides the degree $[L : k_r]$. Furthermore, under specific tower conditions, the quotient $A/(A \cap T)$ is shown to be isomorphic to both a quotient and a subgroup of the Galois group of the splitting extension.
• Variants of the Torsion Index (Theorem 1.3): The paper defines $t_2(G)$ and $t_3(G)$ and proves that under "good" characteristic assumptions, $t_2(G) = t_3(G)$. For the exceptional group $E_8$, the authors compute that $t_2(E_8) = 60$. This result is significant because it salvages Tits' "hypothèse optimiste" (which conjectured $t(E_8) = 60$) by showing that while the full torsion index $t(E_8)$ is much larger ($2^{63} \cdot 3^2 \cdot 5 \cdot 7 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 37 \cdot 41 \cdot 43 \cdot 47 \cdot 53 \cdot 59 \cdot 61 \cdot 67 \cdot 71 \cdot 73 \cdot 79 \cdot 83 \cdot 89 \cdot 97 \cdot 101 \cdot 103 \cdot 107 \cdot 109 \cdot 113 \cdot 127 \cdot 131 \cdot 137 \cdot 139 \cdot 149 \cdot 151 \cdot 157 \cdot 163 \cdot 167 \cdot 173 \cdot 179 \cdot 181 \cdot 191 \cdot 193 \cdot 197 \cdot 199 \cdot 211 \cdot 223 \cdot 227 \cdot 229 \cdot 233 \cdot 239 \cdot 241 \cdot 251 \cdot 257 \cdot 263 \cdot 269 \cdot 271 \cdot 277 \cdot 281 \cdot 283 \cdot 293 \cdot 307 \cdot 311 \cdot 313 \cdot 317 \cdot 331 \cdot 337 \cdot 347 \cdot 349 \cdot 353 \cdot 359 \cdot 367 \cdot 373 \cdot 379 \cdot 383 \cdot 389 \cdot 397 \cdot 401 \cdot 409 \cdot 419 \cdot 421 \cdot 431 \cdot 433 \cdot 439 \cdot 443 \cdot 449 \cdot 457 \cdot 461 \cdot 463 \cdot 467 \cdot 479 \cdot 487 \cdot 491 \cdot 499 \cdot 503 \cdot 509 \cdot 521 \cdot 523 \cdot 541 \cdot 547 \cdot 557 \cdot 563 \cdot 569 \cdot 571 \cdot 577 \cdot 587 \cdot 593 \cdot 599 \cdot 601 \cdot 607 \cdot 613 \cdot 617 \cdot 619 \cdot 631 \cdot 641 \cdot 643 \cdot 647 \cdot 653 \cdot 659 \cdot 661 \cdot 673 \cdot 677 \cdot 683 \cdot 691 \cdot 701 \cdot 709 \cdot 719 \cdot 727 \cdot 733 \cdot 739 \cdot 743 \cdot 751 \cdot 757 \cdot 761 \cdot 769 \cdot 773 \cdot 787 \cdot 797 \cdot 809 \cdot 811 \cdot 821 \cdot 823 \cdot 827 \cdot 829 \cdot 839 \cdot 853 \cdot 857 \cdot 859 \cdot 863 \cdot 877 \cdot 881 \cdot 883 \cdot 887 \cdot 907 \cdot 911 \cdot 919 \cdot 929 \cdot 937 \cdot 941 \cdot 947 \cdot 953 \cdot 967 \cdot 971 \cdot 977 \cdot 983 \cdot 991 \cdot 997 \dots$), the variant $t_2(G)$ is exactly 60.
• Zero-Cycles vs. Rational Points (Theorem 1.5): The paper provides a positive answer to a question raised by Totaro regarding zero-cycles on $G$-homogeneous spaces over fields of iterated Laurent series. It proves that if a $G$-torsor over $k_r$ has a zero-cycle of degree $d$, it possesses a closed point of degree dividing $d$, provided the characteristic is "good" for $G$. This resolves counterexamples that exist over other fields.
• Splitting by Genus 1 Curves (Theorem 1.8): The authors generalize previous negative results on splitting Brauer classes by genus 1 curves. They prove that for certain algebraic groups $G$ containing elementary abelian $p$-subgroups of sufficiently large "depth" (rank of the non-toral part), there exist $G$-torsors over $k_r$ that cannot be split by any torsor under a $g$-dimensional abelian variety (including genus 1 curves). This is achieved by establishing a lower bound on the rank of the Galois group required to split such torsors, which exceeds the maximum possible $p$-rank of the Galois group of a splitting field provided by a genus 1 curve.

Significance
The paper claims significance in three primary areas:

1. Structural Theory: It sharpens the understanding of finite abelian subgroups in algebraic groups by removing the $p$-group restriction and extending validity to positive characteristic, thereby unifying and strengthening classical results of Borel and Reichstein-Youssin.
2. Torsion Indices: It introduces and computes the variant torsion index $t_2(G)$, demonstrating that it can be strictly smaller than the classical Grothendieck torsion index $t(G)$. The computation $t_2(E_8) = 60$ is presented as a partial vindication of Tits' "hypothèse optimiste," suggesting that while the full torsion index is large, the "effective" index for torsors over iterated Laurent series is much smaller.
3. Applications to Splitting Fields: The results provide definitive answers to questions regarding the splitting of torsors over fields of iterated Laurent series (Theorem 1.5) and the limitations of splitting torsors using curves of genus 1 (Theorem 1.8). These applications connect the abstract structure of algebraic groups to concrete problems in the arithmetic of fields and the geometry of homogeneous spaces.

The authors explicitly state that their work relies on the structural properties of torsors and the specific arithmetic of iterated Laurent series fields, avoiding the use of resolution of singularities (which was required in previous characteristic 0 proofs) to achieve broader applicability.