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We say that a field $K$ is $C_{m}$ if it satisfies the following property: for every positive integer $n$ and every sequence of positive integers $(d_{1},\dotsc,d_{r})$ satisfying $d_{1}^{m} + \dotsb + d_{r}^{m} \le n$, every sequence of $(F_{1},\dotsc,F_{r})$ of homogeneous polynomials in $K[x_{0},\dotsc,x_{n}]$ with $\deg F_{i} = d_{i}$ has a common nontrivial zero in $K^{n+1} \setminus \{0\}$.

Are local fields $C_{2}$?

Motivation: I was reading the paper "Period-index bounds for arithmetic threefolds" (link) by Antieau, Auel, Ingalls, Krashen, Lieblich; (one case of) their version of the period-index conjecture (Conjecture 1.3) says that $\mathrm{ind} | \mathrm{per}^{n+1}$ for classes in $\operatorname{Br}(K)$ where $K$ is a field of transcendence degree $n$ over a local field $k$. For $n=0$, this is by Albert, Brauer, Hasse, Noether; for $n=1$, this is by Saltman and Parimala-Suresh; the authors of the above paper address the $n=2$ case for $p$-adic local fields and prove the slightly weaker bound $\mathrm{ind} | \mathrm{per}^{4}$ for classes whose period is prime to $6p$. Another version of the period-index conjecture (see here) says that if $K$ is a $C_{m}$-field then $\mathrm{ind} | \mathrm{per}^{m-1}$ for every class in $\operatorname{Br}(K)$, and I was wondering if the first conjecture can be viewed as a special case of the second.

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    $\begingroup$ Note that the standard definition of $C_m$ is that a single homogeneous homogeneous polynomial of degree $d$ in $>d^m$ variables has a nontrivial zero. What you define is usually called, I think, $C'_m$, although I am easily lost in a maze of twisty little definitions, all alike. (Since $C'_m$ is stronger, of course, Terjanian's counterexample in my answer still answers your question.) $\endgroup$
    – Gro-Tsen
    Commented Nov 9, 2018 at 21:47

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No: see Guy Terjanian, "Un contre-example à une conjecture d'Artin", C. R. Acad. Sci. Paris Sér. A–B 262 (1966) A612 for an example of homogeneous form of degree $4$ in $18$ variables over the $2$-adics that has non non-trivial zero, showing that $\mathbb{Q}_2$ is not $C_2$ (Artin had conjectured that the $\mathbb{Q}_p$ were $C_2$).

PS: On the other hand, the function fields of curves over finite fields are $C_2$: I'm not sure whether this is due to Lang or Nagata (and there may be subtleties between $C_2$ and $C'_2$), but see Serre, Galois Cohomology (2d ed, 1995), chapter II, §4.5, remark (b).

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    $\begingroup$ Certainly Gro-Tsen knows this, but for others, Artin's conjecture is "approximately true". James Ax and Simon Kochen proved that for every $(d_1,\dots,d_r)$ and $n$ such that $d_1^2 + \dots + d_r^2 \leq n$, there exists an integer $p_0$ such that for every local $p$-adic field with $p\geq p_0$, every $(d_1,\dots,d_r)$-complete intersection in $\mathbb{P}^n$ over that field does have a rational point. $\endgroup$
    – Jason Starr
    Commented Nov 9, 2018 at 21:44
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    $\begingroup$ Regarding the PS, Tsen did prove a general "transition theorem", although it is not precisely Lang's result: a function field of transcendence degree $r$ over a field satisfying $C_n$, resp. $C'_n$, satisfies $C_{n+r}$, resp. $C'_{n+r}$. Tsen's work was published in China during World War II and was not rediscovered until the 1970s. $\endgroup$
    – Jason Starr
    Commented Nov 11, 2018 at 11:10

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