Linkage of Pfister forms over $\mathbb{C}(x_1,\ldots,x_n)$

TL;DR

This paper constructs a specific set of Pfister forms over the complex rational function field that do not share a common factor, answering a previously open question negatively.

Contribution

It demonstrates the existence of $n$-fold Pfister forms over $ ext{C}(x_1, ext{...},x_n)$ without a common $(n-1)$-fold factor, using valuation theory and recent bilinear form results.

Findings

Existence of such Pfister forms over complex rational function fields.

Negative answer to Becher's question about common factors.

Application of dyadic valuation and symmetric bilinear form theory.

Abstract

In this note, we prove the existence of a set of $n$-fold Pfister forms of cardinality $2^n$ over $\mathbb{C}(x_1,\dots,x_n)$ which do not share a common $(n-1)$-fold factor. This gives a negative answer to a question raised by Becher. The main tools are the existence of the dyadic valuation on the complex numbers and recent results on symmetric bilinear over fields of characteristic 2.

Equations

$$(x_1,x_2),(x_1,x_2+1),(x_2,x_1+1),(x_2,x_1{x}_2+1).$$

$$WF\supset IF\supset I^{2}F\supset \dots$$

$$⟨⟨{\alpha }_1,\dots ,{\alpha }_n⟩⟩=⟨1,-{\alpha }_1⟩\otimes \cdots \otimes ⟨1,-{\alpha }_n⟩.$$

$${\phi }_d=⟨⟨{\alpha }_1,\dots ,\widehat{{\alpha }_{\ell }},\dots ,{\alpha }_n⟩⟩\otimes ⟨⟨1+{\alpha }^d⟩⟩,$$

$${\phi }_0=⟨⟨{\alpha }_1,\dots ,{\alpha }_n⟩⟩.$$

$${\phi }_d^{\prime }(t_d,t_d)=-\alpha \text{for d∈2n}$$

$${\phi }_d^{\prime }(t_d,t_d)={\phi }_0^{\prime }(t_0,t_0)\text{for d∈2n∖{0}.}$$

$$\min \{v(u_{i,d})\mid i=1,\dots ,2^n-1,d\in 2^n\}=0.$$

$${\overline{\phi }}_d^{\prime }(\overline{u_d},\overline{u_d})={\overline{\phi }}_0^{\prime }(\overline{u_0},\overline{u_0})\text{for d∈2n∖{0}.}$$

Full text

The second author acknowledges support from the Fonds de la Recherche Scientifique–FNRS under grant n∘ J.0159.19.

The field $\mathbb{C}(x_{1},x_{2})$ of rational functions in two indeterminates over the field of complex numbers is known to be a $C_{2}$-field in the sense of Lang (see [4, Section 97]). It follows that every quadratic form in five variables over $\mathbb{C}(x_{1},x_{2})$ is isotropic, which implies that any two quaternion algebras over $\mathbb{C}(x_{1},x_{2})$ share a common maximal subfield, see [6, Th. X.4.20]. Fields with this property are said to be linked. It was noticed by Becher in [1] and by Chapman–Dolphin–Leep in [3, Cor. 5.3] that the following stronger property holds: $\mathbb{C}(x_{1},x_{2})$ is $3$-linked in the sense that any three quaternion algebras over $\mathbb{C}(x_{1},x_{2})$ share a common maximal subfield. Comparison with the case of number fields, which are $m$-linked for every integer $m$ by the local-global principle (see [6, Ex. X.5.12A]), suggests to ask whether there exists an upper bound on the integer $m$ for which $\mathbb{C}(x_{1},x_{2})$ is $m$-linked. We prove below:

[TABLE]

The arguments apply to a more general linkage question raised by Becher [1]. Given a field $F$, the Witt ring $WF$ of (Witt classes of) symmetric bilinear forms over $F$ has a natural filtration by the powers of the maximal ideal $IF$ of even-dimensional forms:

[TABLE]

Each $I^{n}F$ is generated by (bilinear) $n$-fold Pfister forms, i.e., forms of the shape

[TABLE]

For $m$, $n\geq 2$, we say that $I^{n}F$ is $m$-linked if every $m$ bilinear $n$-fold Pfister forms over $F$ share a common $(n-1)$-fold factor. If $\operatorname{char}(F)\neq 2$, quadratic forms can be identified with their symmetric bilinear polar forms, and in particular the $2$-fold Pfister forms are the norm forms of quaternion algebras, hence $F$ is $m$-linked in the sense discussed above if and only if $I^{2}F$ is $m$-linked. Becher raised the following question:

This question was answered in the negative for fields $F$ of $\operatorname{char}(F)=2$ in [2]. In this note, we shall show how Becher’s question can be answered also in the case of $\operatorname{char}(F)=0$ using the main result of [2] on symmetric bilinear forms over fields of characteristic $2$ and the existence of a dyadic valuation on $\mathbb{C}$:

Proofs

Notation 1**.**

For a given integer $n\geq 2$, let $\mathbb{2}^{n}=\{0,1\}^{\times n}$, and write $\mathbb{0}=(0,\dots,0)\in\mathbb{2}^{n}$. Given a sequence $\alpha_{1}$, …, $\alpha_{n}$ in the multiplicative group of a field $F$ and $\mathbb{d}=(d_{1},\ldots,d_{n})\in\mathbb{2}^{n}$, let $\alpha^{\mathbb{d}}=\prod_{i=1}^{n}\alpha^{d_{i}}\in F^{\times}$. If $\mathbb{d}\neq\mathbb{0}$, let

[TABLE]

where $\ell$ is the minimal index in $\{1,\ldots,n\}$ for which $d_{\ell}\neq 0$, and let

[TABLE]

The following result is from [2, Th. 3.3]:

Proposition 2**.**

Suppose $\operatorname{char}(F)=2$ and $\alpha_{1}$, …, $\alpha_{n}$ are $2$-independent in $F$, which means that $(\alpha^{\mathbb{d}})_{\mathbb{d}\in\mathbb{2}^{n}}$ is a linearly independent family in $F$ viewed as an $F^{2}$-vector space. Then the forms $\varphi_{\mathbb{d}}$ for $\mathbb{d}\in\mathbb{2}^{n}$ are anisotropic and have no common $1$-fold factor.

The main result from which Theorems A and B derive is the following:

Proposition 3**.**

Let $F=k(x_{1},\ldots,x_{n})$ be the field of rational functions in $n$ indeterminates over an arbitrary field $k$ of characteristic zero, for some $n\geq 2$. Let $\varphi_{\mathbb{d}}$ for $\mathbb{d}\in\mathbb{2}^{n}$ be the Pfister forms defined as in Notation 1 with the sequence $x_{1}$, …, $x_{n}$ for $\alpha_{1}$, …, $\alpha_{n}$. The forms $\varphi_{\mathbb{d}}$ do not have a common $1$-fold factor.

Proof.

A theorem of Chevalley (see [5, Theorem 3.1.1]) shows that the $2$-adic valuation on $\mathbb{Q}$ extends to a valuation $v_{0}$ on $k$. Let $\overline{k}$ be the residue field of this valuation, which has characteristic $2$. The valuation $v_{0}$ has a Gauss extension to a valuation $v$ on $F$ such that $v(x_{i})=0$ for $i=1$, …, $n$ and $\overline{x_{1}}$, …, $\overline{x_{n}}$ are algebraically independent over $\overline{k}$; see [5, Cor. 2.2.2]. The residue field of $v$ is thus $\overline{F}=\overline{k}(\overline{x_{1}},\ldots,\overline{x_{n}})$, a field of rational functions in $n$ indeterminates over $\overline{k}$. Since the coefficients of the forms $\{\varphi_{\mathbb{d}}:\mathbb{d}\in\mathbb{2}^{n}\}$ are all of value [math], they have residue forms $\{\overline{\varphi}_{\mathbb{d}}:\mathbb{d}\in\mathbb{2}^{n}\}$, where the coefficients of $\overline{\varphi}_{\mathbb{d}}$ are the residues of the coefficients of $\varphi_{\mathbb{d}}$. The forms $\overline{\varphi}_{\mathbb{d}}$ are bilinear Pfister forms as defined in Notation 1, with the $2$-independent sequence $\overline{x_{1}}$, …, $\overline{x_{n}}$ for $\alpha_{1}$, …, $\alpha_{n}$.

For $\mathbb{d}\in\mathbb{2}^{n}$, let $\mathbb{t}_{\mathbb{d}}=(t_{1,\mathbb{d}},\ldots,t_{2^{n}-1,\mathbb{d}})$ be a $(2^{n}-1)$-tuple of indeterminates. Suppose the bilinear forms $\varphi_{\mathbb{d}}$ have a common factor ${\langle\!\langle{\alpha}\rangle\!\rangle}$. Then the pure subforms $\varphi^{\prime}_{\mathbb{d}}$ defined by the equation $\varphi_{\mathbb{d}}={\langle{1}\rangle}\perp\varphi^{\prime}_{\mathbb{d}}$ all represent $-\alpha$, hence the system of equations

[TABLE]

has a solution. We may therefore find nontrivial solutions to the system of equations

[TABLE]

Since these equations are homogeneous, upon scaling we may find solutions $(\mathbb{u}_{\mathbb{d}})_{\mathbb{d}\in\mathbb{2}^{n}}$ such that

[TABLE]

Taking residues, we obtain

[TABLE]

Since at least one $\overline{u_{i,\mathbb{d}}}$ is nonzero and the forms $\overline{\varphi}^{\prime}_{\mathbb{d}}$ are anisotropic, it follows that these forms all represent some $\beta\in\overline{F}^{\times}$, hence the forms $\overline{\varphi}_{\mathbb{d}}$ have a common factor ${\langle\!\langle{\beta}\rangle\!\rangle}$ by [4, Lemma 6.11]. This yields a contradiction to Proposition 2. ∎

Theorem A readily follows from Proposition 3 with $n=2$ and $k=\mathbb{C}$, because the forms $\varphi_{\mathbb{0}}$, $\varphi_{(0,1)}$, $\varphi_{(1,0)}$, and $\varphi_{(1,1)}$ are the norm forms of the quaternion algebras $(x_{1},x_{2})$, $(x_{1},x_{2}+1)$, $(x_{2},x_{1}+1)$ and $(x_{2},x_{1}x_{2}+1)$ respectively.

Proof of Theorem B.

The field $F=\mathbb{C}(x_{1},\ldots,x_{n})$ is a $C_{n}$-field, hence $F(t)$ is a $C_{n+1}$-field, see [4, Cor. 97.6]. In particular, u\bigl{(}F(t)\bigr{)}=2^{n+1}, and it follows from [1, Cor. 5.4] that $I^{n}F$ is $3$-linked. Apply Proposition 3 with $k=\mathbb{C}$ to obtain a set of $n$-fold Pfister forms of cardinality $2^{n}$ that do not have a common $1$-fold factor, hence are not linked. ∎