Splitting p-primary cohomology classes of tori in characteristic p
Zev Rosengarten
TL;DR
The paper analyzes splitting p‑primary cohomology classes of tori over global function fields of characteristic p, addressing both bounded‑degree separable p‑primary extensions and containment within a given p‑primary extension. It develops strengthened Grunwald–Wang type results and two exhaustion lemmas to reduce infinite data to finite subextensions, enabling quantitative splitting statements. The main achievement is a uniform bound: any p^n‑torsion class on a torus splitting over a degree‑m field can be killed by a solvable separable p‑primary extension of degree at most (p^n)^{1+ c m log(m)^3}, with c universal. These results extend to ell‑primary cases and to both i = 1,2 cohomology, contributing to the period–index landscape for tori in function field settings and providing tools for local‑global approximation of p‑primary extensions.
Abstract
We prove that $p$-primary cohomology classes of a torus $T$ over a global function field of characteristic $p$ may be split by suitable separable $p$-primary extensions. More precisely, we show that such cohomology classes will split in any ``large'' $p$-primary extension (and in fact, prove the same for $\ell$-primary classes over ``large'' $\ell$-primary extensions for every prime $\ell$, including $\ell \neq \mathrm{char}(K)$), and we prove that $p^n$-torsion classes may be split by a (solvable) separable $p$-primary extension of degree $\leq (p^n)^{1+cm\mathrm{log}(m)^3}$ for an explicitly computable universal constant $c > 0$, where $m$ is the degree of a finite Galois extension splitting the torus $T$. Along the way, we also prove Grunwald-Wang type results of independent interest which allow one to approximate a given finite list of abelian $p$-primary local extensions of places of a global function field by a suitable global extension.