Kato-Kuzumaki's properties for function fields over higher local fields
Felipe Gambardella
TL;DR
The paper advances the Kato–Kuzumaki program for function fields of curves over $d$-local fields by (i) computing Tate–Shafarevich groups via the special fibre combinatorics of curve models, and (ii) proving that Milnor $K$-theory in degree $d{+}1$ is generated by norm images from suitable extensions for hypersurfaces of bounded degree. The authors establish a concrete torsion control: for unramified base extensions, the quotient $K_{d+1}(K)/iglra N_{d+1}(L/K),N_{d+1}(Z/K)igrra$ is torsion with exponent governed by Euler characteristics, leading to the main cohomological and diophantine consequences. As a corollary, the function field $K$ of a curve over such a $d$-local field satisfies the $C_{d+1}^{d+1}$ property, and under ramification constraints they obtain sharper $C_{j+1}^{d+1}$ bounds with explicit torsion exponents. The results push toward a broader understanding of cohomological dimension for arithmetic-geometric fields via motivic and duality techniques, with concrete implications for norm surjectivity and Diophantine characterizations of rational points.
Abstract
Let $k$ be a $d$-local field such that the corresponding $1$-local field $k^{(d-1)}$ is a $p$-adic field and $C$ a curve over $k$. Let $K$ be the function field of $C$. We prove that for each $n,m \in \mathbf{N}$, and hypersurface $Z$ of $\mathbf{P}^n_K$ with degree $m$ such that $m^{d+1} \leq n$, the $(d+1)$-th Milnor $\mathrm{K}$-theory group is generated by the images norms of finite extension $L$ of $K$ such that $Z$ admits an $L$-point. Let $j \in \{1,\cdots , d\}$. When $C$ admits a point in an extension $l/k$ that is not $i$-ramified for every $i \in \{1, \cdots, d-j\}$ we generalise this result to hypersurfaces $Z$ of $\mathbf{P}_K^n$ with degree $m$ such that $m^{j+1} \leq n$. \par In order to prove these results we give a description of the Tate-Shafarevich group $\Sha^{d+2}(K,\mathbf{Q}/\mathbf{Z}(d+1))$ in terms of the combinatorics of the special fibre of certain models of the curve $C$.