Similarity of quadratic forms over global fields in characteristic 2

TL;DR

This paper proves that two non-degenerate quadratic forms over a global function field of characteristic 2 are similar over the entire field if they are similar over all completions, extending known results to characteristic 2.

Contribution

It establishes an analogue of Ono's result for characteristic not 2, specifically for global function fields of characteristic 2.

Findings

Quadratic forms similar over all local completions are similar globally.

Extends classical local-global principles to characteristic 2.

Provides new insights into quadratic form theory in characteristic 2.

Abstract

Let $ K $ be a global function field of characteristic $ 2 $. For each non-trivial place $ v $ of $ K $, let $ K_{v} $ be the completion of $ K $ at $ v $. We show that if two non-degenerate quadratic forms are similar over every $ K_{v} $, then they are similar over $ K $. This provides an analogue of the version for characteristic not $ 2 $ previously obtained by T.Ono.

Equations

$$q\text{ {non-degenerate} if }\left\{\begin{array}[]{ll}\dim(\operatorname{Rad}(b_{q}))=1,\operatorname{Rad}(q)=0,&\text{ when }n\text{ is odd},\\ b_{q}\text{ is non-degenerate},&\text{ when }n\text{ is even},\\ \end{array}\right.$$

$$q\simeq\left\{\begin{array}[]{ll}\langle d\rangle\perp[a_{1},b_{1}]\perp\cdots\perp[a_{m},b_{m}],~{}d\in F^{*},~{}a_{i},b_{i}\in F,&\text{ when }n\text{ is odd},\\ \left[a_{1},b_{1}\right]\perp\cdots\perp[a_{m},b_{m}],~{}a_{i},b_{i}\in F,&\text{ when }n\text{ is even}.\\ \end{array}\right.$$

$$\operatorname{disc}(q)=\left\{\begin{array}[]{ll}dF^{*2}\in F^{*}/F^{*2},&\text{ when }n\text{ is odd},\\ \operatorname{Arf}(q)\in F/\wp F,&\text{ when }n\text{ is even}.\\ \end{array}\right.$$

$$\operatorname{clif}(q)=\left\{\begin{array}[]{ll}\left[\operatorname{C}_{0}(q)\right]\in{{}_{2}}\operatorname{Br}(F),&\text{ when }n\text{ is odd},\\ \left[\operatorname{C}(q)\right]\in{{}_{2}}\operatorname{Br}(F),&\text{ when }n\text{ is even},\\ \end{array}\right.$$

$$\operatorname{ind}(q)=\left\{\begin{array}[]{ll}\operatorname{ind}(\operatorname{C}_{0}(q)),&\text{ when }n\text{ is odd},\\ \operatorname{ind}(\operatorname{C}(q)),&\text{ when }n\text{ is even},\\ \end{array}\right.$$

$$\left\{\begin{array}[]{ll}i_{0}(f)=i_{0}(g),&\text{ when }n\text{ is odd},\\ i_{0}(f)=i_{0}(g)\text{ and }\operatorname{disc}f=\operatorname{disc}g,&\text{ when }n\text{ is even}.\\ \end{array}\right.$$

$$S=\{v\in {\Omega }_K|\mathrm{rank}(f_{v,\mathrm{an}})=\mathrm{rank}(g_{v,\mathrm{an}})=2\}.$$

$$[C(f)]=[C(f)\otimes C(g)\otimes C(g)]=[Q\otimes C(g)]=[C([a,b]\perp g)]=[C(ag)].$$

Full text

1. Introduction

Let $K$ be a global field. Let $\Omega_{K}$ be the set of non-trivial places of $K$. For each $v\in\Omega_{K}$, let $K_{v}$ be the completion of $K$ at $v$. Let $q$ be a non-degenerate quadratic form over $K$. The Hasse-Minkowski theorem establishes that $q$ is isotropic over $K$ if and only if $q\otimes_{K}K_{v}$ is isotropic over $K_{v}$ for all $v\in\Omega_{K}$. For a characteristic not $2$, see, for example, [14, 66:1]. For the characteristic $2$, see [16, Th.3.2]. As a consequence, two non-degenerate quadratic forms $f$ and $g$ over $K$ are isometric over $K$ if and only if $f\otimes_{K}K_{v}$ and $g\otimes_{K}K_{v}$ are isometric over $K_{v}$ for all $v\in\Omega_{K}$. We are also interested in the classification of quadratic forms up to similarity. Let $F$ be a field. Two quadratic forms $f$ and $g$ over $F$ are similar if there exists $a\in F^{*}$ such that $f$ and $ag$ are isometric.

Let $K$ be a global field of characteristic not $2$. Two quadratic forms $f$ and $g$ are similar over $K$ if and only if $f\otimes_{K}K_{v}$ and $g\otimes_{K}K_{v}$ are similar over $K_{v}$ for all $v\in\Omega_{K}$ by [15, Th.1]. A cohomological version of Ono’s theorem is provided in [3, Th.2.5’]. The Hasse principle does not hold for similarity of nonsymmetric bilinear forms by [4]. The Hasse principle for similarity of hermitian forms fails in general and only holds in special occasions by [11, Cons.2.11, Th.2.22, Th.3.33]. A new proof of Ono’s theorem is given by [17, Prop.8.7].

In this paper, we assume that $K$ is a global function field of characteristic $2$ and show that two non-degenerate quadratic forms $f$ and $g$ are similar over $K$ if and only if $f\otimes_{K}K_{v}$ and $g\otimes_{K}K_{v}$ are similar over $K_{v}$ for all $v\in\Omega_{K}$ (2.3). The idea of the proof is the classification of quadratic forms over $K$ or $K_{v}$ up to isometry by their rank, discriminant and Clifford invariant, see [2, Th.2.1].

Let $F$ be a field of characteristic $2$ and $F^{*}=F-\{0\}$ its multiplicative group. Let $H_{2}^{n}(F)$ be the Milne-Kato cohomology as in [13] and [8]. We say that $F$ has separable dimension $\operatorname{sd}_{2}(F)\leq 2$ if $H_{2}^{3}(L)=0$ for all finite separable extensions $L/F$ [6, p.62, just before Lem.1]. For example, if $F$ is local or global, then $\operatorname{sd}_{2}(F)\leq 2$.

Let $V$ be an $F$-vector space of dimension $n$. Let $q$ be a non-degenerate quadratic form on $V$. We denote by $q_{\operatorname{an}}$ the anisotropic part of $q$, and by $i_{0}(q)$ the Witt index of $q$. For two non-degenerate quadratic forms $q$ and $q^{\prime}$, we write $q\simeq q^{\prime}$ if they are isometric; and $q\sim q^{\prime}$ if they are similar in $F$, i.e. $q\simeq aq^{\prime}$ for some $a\in F^{*}$. The rank of $q$ is $n$. Let $b_{q}$ be the symmetric bilinear form such that $b_{q}(x,y)=q(x+y)-q(x)-q(y)$ for all $x,y\in V$. Let $\operatorname{Rad}(b_{q})=\{v\in V~{}|~{}b_{q}(v,w)=0\text{ for all }w\in V\}$ and $\operatorname{Rad}(q)=\{v\in\operatorname{Rad}(b_{q})~{}|~{}q(v)=0\}$. We call

[TABLE]

For $a,b\in F$, let $[a,b]$ denote the binary quadratic form $q$ with basis $e_{1},e_{2}$ such that $q(e_{1})=a$, $q(e_{2})=b$ and $b_{q}(e_{1},e_{2})=1$. Let $\mathbb{H}=[0,0]$ denote the hyperbolic plane over $F$. By [5, Cor.7.32], for every non-degenerate quadratic form $q$ over $F$,

[TABLE]

Let $\wp\colon F\to F$ be the Artin-Schreier map such that $\wp(x)=x^{2}+x$ for all $x\in F$. The discriminant of $q$ is defined to be

[TABLE]

To be precise, $\operatorname{Arf}(\left[a_{1},b_{1}\right]\perp\cdots\perp[a_{n},b_{n}])=a_{1}b_{1}+\cdots+a_{n}b_{n}\operatorname{mod}\wp F$.

Let $\operatorname{Br}(F)$ be Brauer group of $F$ (see, for example [7, Sec.2.4]) and let ${{}_{2}}\operatorname{Br}(F)$ be its $2$-torsion part. For a central simple $F$-algebra $A$, we denote $[A]$ its class in $\operatorname{Br}(F)$; its index $\operatorname{ind}(A)=\operatorname{ind}([A])$ is the degree of the unique central division $F$-algebra $D$ with $[D]=[A]$; its period $\operatorname{per}(A)=\operatorname{per}([A])$ is the order of $[A]$ in $\operatorname{Br}(F)$ (see, for example [7, Sec.2.8]). The Clifford invariant of $q$ is defined to be

[TABLE]

where $\operatorname{C}(q)$ is the $\mathbb{Z}/2$-graded Clifford algebra of $q$ and $\operatorname{C}_{0}(q)$ is the even part of $\operatorname{C}(q)$ (The Clifford invariant $\operatorname{clif}(q)$ is called the Witt invariant of $q$ in [9, p.242, para.(-1)]). The Clifford algebra of the binary form $[a,b]$ is $C([a,b])=\displaystyle{a,b\brack F}$, where $a,b\in F$ and $\displaystyle{a,b\brack F}$ is the quaternion $F$-algebra generated by $i,j$ such that $i^{2}=a$, $j^{2}=b$ and $ij+ji=1$. The Schur index of $q$ is defined to be

[TABLE]

2. The local-global principle

In this section, we prove the local-global principle for the similarity of non-degenerate quadratic forms over a global function field of characteristic $2$.

Lemma 2.1**.**

Let $F$ be a non-archimedean local field of characteristic $2$. Let $q$ be a non-degenerate quadratic form of rank $n$ over $F$. Let $m=\operatorname{rank}(q|_{\operatorname{Rad}(b_{q})}\perp q_{\operatorname{an}})$.

(1) Suppose $n$ is odd. We have

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$m=1$* if and only if $\operatorname{ind}(q)=1$.*

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$m=3$* if and only if $\operatorname{ind}(q)=2$.*

(2) Suppose $n$ is even. We have

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$m=0$* if and only if $\operatorname{disc}(q)$ is trivial and $\operatorname{ind}(q)=1$.*

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$m=2$* if and only if $\operatorname{disc}(q)$ is non-trivial and $\operatorname{ind}(q)\in\{1,2\}$. *

• •

$m=4$* if and only if $\operatorname{disc}(q)$ is trivial and $\operatorname{ind}(q)=2$.*

Proof.

Since $F$ is a non-archimedean local field of characteristic $2$, every quadratic form of rank $>4$ is isotropic [1, Th.1.1]. Therefore, $m\leq 4$. Since $m+2i_{0}(q)=n$, we have $m\equiv n(\operatorname{mod}2)$. Since $F$ is local, we have ${{}_{2}}\operatorname{Br}(F)\simeq\mathbb{Z}/2$ and $\operatorname{ind}(q)=\operatorname{ind}(\operatorname{clif}(q))=\operatorname{per}(\operatorname{clif}(q))\in\{1,2\}$.

(1) When $n$ is odd, $\dim(q|_{\operatorname{Rad}(b_{q})})=1$. Suppose $\operatorname{disc}(q)=dF^{*2}$.

When $m=1$, $q\simeq\langle d\rangle\perp\mathbb{H}^{(n-1)/2}$, $d\in F^{*}$. From [12, Lem.2], $\operatorname{C}_{0}(q)\simeq\operatorname{C}(d\mathbb{H}^{(n-1)/2})\simeq\operatorname{C}(\mathbb{H}^{(n-1)/2})$. Thus $\operatorname{clif}(q)$ is trivial.

When $m=3$, $q\simeq\langle d\rangle\perp[a,b]\perp\mathbb{H}^{(n-3)/2}$, $d\in F^{*}$, $a,b\in F$ and $[a,b]$ is anisotropic. From [12, Lem.2], $\operatorname{C}_{0}(q)\simeq\operatorname{C}(d[a,b]\perp d\mathbb{H}^{(n-3)/2})$. We have $d[a,b]\perp d\mathbb{H}^{(n-1)/2}\simeq[da,d^{-1}b]\perp\mathbb{H}^{(n-3)/2}$. From [12, just before Prop.5], $\operatorname{clif}(q)$ is the Brauer class of $\displaystyle{da,d^{-1}b\brack F}$. The quaternion algebra $\displaystyle{da,d^{-1}b\brack F}$ is not split, otherwise the two quadratic forms $\langle d\rangle\perp[a,b]\perp\mathbb{H}^{(n-3)/2}$ and $\langle d\rangle\perp\mathbb{H}^{(n-1)/2}$ would have the same rank $n$, the same discriminant $dF^{*2}\in F^{*}/F^{*2}$ and the same Clifford invariant $0\in{{}_{2}}\operatorname{Br}(F)$. Thus, by [2, Th.2.1], $\langle d\rangle\perp[a,b]\perp\mathbb{H}^{(n-3)/2}\simeq\langle d\rangle\perp\mathbb{H}^{(n-1)/2}$, a contradiction to the uniqueness of the Witt decomposition [5, Th.8.5].

(2) When $m=0$, $q\simeq\mathbb{H}^{n/2}$ is a hyperbolic space, which has trivial Arf invariant and split Clifford algebra.

When $m=2$, $q\simeq[a,b]\perp\mathbb{H}^{(n-2)/2}$, $a,b\in F$ and $[a,b]$ is anisotropic. Therefore, $\operatorname{Arf}(q)=ab$ is nontrival. By [12, just before Prop.5], $\operatorname{clif}(q)$ is the Brauer class of $\displaystyle{a,b\brack F}$. Thus $\operatorname{ind}(q)\in\{1,2\}$.

When $m=4$, $q\simeq[a_{1},b_{1}]\perp[a_{2},b_{2}]\perp\mathbb{H}^{(n-4)/2}$, $a_{1},b_{1},a_{2},b_{2}\in F$ and $[a_{1},b_{1}]\perp[a_{2},b_{2}]$ is anisotropic. We have $\operatorname{Arf}(q)=a_{1}b_{1}+a_{2}b_{2}\operatorname{mod}\wp F$. From [12, just before Prop.5], $\operatorname{clif}(q)$ is the Brauer class of $\displaystyle{a_{1},b_{1}\brack F}\otimes{a_{2},b_{2}\brack F}$. Similar to the case $m=3$, $\operatorname{ind}(q)\neq 1$ and hence $\operatorname{ind}(q)=2$. The biquaternion algebra $\displaystyle{a_{1},b_{1}\brack F}\otimes{a_{2},b_{2}\brack F}$ has Albert form $\varphi=[1,a_{1}b_{1}+a_{2}b_{2}]\perp[a_{1},b_{1}]\perp[a_{2},b_{2}]$. By [10, (16.5)], $\operatorname{ind}(q)=2$ if and only if $i_{0}(\varphi)=1$. Thus, $\operatorname{disc}(q)=\operatorname{Arf}(q)=a_{1}b_{1}+a_{2}b_{2}$ is trivial. ∎

Lemma 2.2**.**

Let $F$ be a non-archimedean local field of characteristic $2$. Let $V$ be an $F$-vector space of dimension $n$. Let $f$ and $g$ be two non-degenerate quadratic forms on $V$. The two forms $f\simeq g$ if and only if

[TABLE]

Proof.

If $f\sim g$, then $f\simeq ag$ for some $a\in F^{*}$. Therefore, $i_{0}(f)=i_{0}(ag)=i_{0}(g)$. When $n$ is even, $\operatorname{Arf}(f)=\operatorname{Arf}(ag)=\operatorname{Arf}(g)$ since $a[b,c]\simeq[ab,a^{-1}c]$ for all $b,c\in F$.

Conversely, we suppose $i_{0}(f)=i_{0}(g)$. Therefore, $n-2i_{0}(f)=n-2i_{0}(g)$ which we denote by $m$.

Case 1: $n$ is odd. Suppose $\operatorname{disc}f=d_{1}F^{*2}$ and $\operatorname{disc}g=d_{2}F^{*2}$ for $d_{1},d_{2}\in F^{*}$.

When $m=1$, since $\langle d_{1}\rangle\sim\langle d_{2}\rangle$, $f\simeq\langle d_{1}\rangle\perp\mathbb{H}^{(n-1)/2}\sim\langle d_{2}\rangle\perp\mathbb{H}^{(n-1)/2}\simeq g$.

When $m=3$, $\operatorname{ind}(f)=\operatorname{ind}(g)=2$ by 2.1(1). Since $F$ is a local field, ${{}_{2}}\operatorname{Br}(F)\simeq\mathbb{Z}/2$. Also, $\operatorname{per}(\operatorname{clif}(f))=2=\operatorname{per}(\operatorname{clif}(g))$. Therefore, $\operatorname{clif}(f)=\operatorname{clif}(g)$. By [5, Prop.11.4], $\operatorname{clif}(d_{1}d_{2}^{-1}g)=[\operatorname{C}_{0}(d_{1}d_{2}^{-1}g)]=[\operatorname{C}_{0}(g)]=\operatorname{clif}(g)$. The two forms $f$ and $d_{1}d_{2}^{-1}g$ have the same rank $n$, the same discriminant $d_{1}F^{*2}$ and the same Clifford invariant $\operatorname{C}(g)$. Now, by [2, Th.2.1], $f\simeq d_{1}d_{2}^{-1}g$. Thus, $f\sim g$.

Case 2: $n$ is even. When $m=0$, $f\simeq\mathbb{H}^{n/2}\simeq g$ and hence $f\sim g$.

When $m=2$, suppose $f\simeq[a_{1},b_{1}]\perp\mathbb{H}^{(n-2)/2}$, $a_{1},b_{1}\in F$ and $[a_{1},b_{1}]$ is anisotropic; $g\simeq[a_{2},b_{2}]\perp\mathbb{H}^{(n-2)/2}$, $a_{2},b_{2}\in F$ and $[a_{2},b_{2}]$ is anisotropic. We may assume that $a_{1}\neq 0$ and $a_{2}\neq 0$. If $\operatorname{Arf}(f)=\operatorname{Arf}(g)$, i.e. $a_{1}b_{1}=a_{2}b_{2}\operatorname{mod}\wp F$, then $[a_{1},b_{1}]\simeq a_{1}[1,a_{1}b_{1}]\sim a_{2}[1,a_{2}b_{2}]\simeq[a_{2},b_{2}]$ by [5, Ex.7.6]. Thus, $f\simeq[a_{1},b_{1}]\perp\mathbb{H}^{(n-1)/2}\sim[a_{2},b_{2}]\perp\mathbb{H}^{(n-1)/2}\simeq g$ and hence $f\sim g$.

When $m=4$, $\operatorname{disc}(f)=\operatorname{disc}(g)$ is trivial, and we obtain $\operatorname{ind}(f)=\operatorname{ind}(g)=2$ by 2.1(2). Again, since $F$ is a local field, ${{}_{2}}\operatorname{Br}(F)\simeq\mathbb{Z}/2$. Also, $\operatorname{per}(\operatorname{clif}(f))=2=\operatorname{per}(\operatorname{clif}(g))$. Therefore, $\operatorname{clif}(f)=\operatorname{clif}(g)$. Now, by [2, Th.2.1], $f\simeq g$ and hence $f\sim g$. ∎

Theorem 2.3**.**

Let $K$ be a global function field of characteristic $2$. Let $V$ be a $K$-vector space of dimension $n$. Let $f$ and $g$ be non-degenerate quadratic forms on $V$. Let $\Omega_{K}$ be the set of all non-trivial places of $K$. For each $v\in\Omega_{K}$, let $K_{v}$ be the completion of $K$ at $v$ and $f_{v},g_{v}$ the scalar extensions of $f,g$ to $K_{v}$. If $f_{v}\sim g_{v}$ in $K_{v}$ for all $v\in\Omega_{K}$, then $f\sim g$ in $K$.

Proof.

Since $K$ is a global function field, every non-trivial place $v\in\Omega_{K}$ is non-archimedean. We are free to use 2.1 and 2.2 for all $K_{v}$.

Case 1: $n$ is odd. Suppose $\operatorname{disc}(f)=d_{1}K^{*2}$ and $\operatorname{disc}(g)=d_{2}K^{*2}$ for $d_{1},d_{2}\in K^{*}$. Let $a=d_{1}d_{2}^{-1}$. Since $f_{v}\sim g_{v}$ in $K_{v}$, by 2.2, we have $i_{0}(f_{v})=i_{0}(g_{v})=i_{0}(ag_{v})$. Also $\operatorname{disc}(f_{v})=d_{1}K_{v}^{*2}=ad_{2}K_{v}^{*2}=\operatorname{disc}(ag_{v})$ for all $v$. By 2.1(1), $\operatorname{ind}(f_{v})=\operatorname{ind}(ag_{v})\in\{1,2\}$. Since $K_{v}$ is a local field, ${{}_{2}}\operatorname{Br}(K_{v})\simeq\mathbb{Z}/2$. Also, $\operatorname{per}(\operatorname{clif}(f_{v}))=\operatorname{ind}(f_{v})=\operatorname{ind}(ag_{v})=\operatorname{per}(\operatorname{clif}(ag_{v}))$. Therefore, $\operatorname{clif}(f_{v})=\operatorname{clif}(ag_{v})$. Now, by [2, Th.2.1], $f_{v}\simeq ag_{v}$ in $K_{v}$ for all $v$. By the Hasse-Minkowski theorem, $f\simeq ag$. Thus, $f\sim g$.

Case 2: $n$ is even. Since $f_{v}\sim g_{v}$ in $K_{v}$ for all $v$, by 2.2, we have $i_{0}(f_{v})=i_{0}(g_{v})$ and $\operatorname{disc}(f_{v})=\operatorname{disc}(g_{v})$. Let

[TABLE]

If $v\in\Omega_{K}-S$, i.e. $\operatorname{rank}(f_{v,\operatorname{an}})=\operatorname{rank}(g_{v,\operatorname{an}})\in\{0,4\}$, then by 2.1(2), $\operatorname{ind}(f_{v})=\operatorname{ind}(g_{v})\in\{1,2\}$. Since $K_{v}$ is a local field, ${{}_{2}}\operatorname{Br}(K_{v})\simeq\mathbb{Z}/2$. Also, $\operatorname{per}(\operatorname{clif}(f_{v}))=\operatorname{ind}(f_{v})=\operatorname{ind}(g_{v})=\operatorname{per}(\operatorname{clif}(g_{v}))$. Therefore, $\operatorname{clif}(f_{v})=\operatorname{clif}(g_{v})$. By [2, Th.2.1], $f_{v}\simeq g_{v}$ for all $v\in\Omega_{K}-S$.

Subcase 2a: $\operatorname{clif}(f_{v})=\operatorname{clif}(g_{v})$ for all $v\in S$. By [2, Th.2.1], $f_{v}\simeq g_{v}$ for all $v\in S$. We have $f_{v}\simeq g_{v}$ for all $v\in\Omega_{K}$. By the Hasse-Minkowski theorem, $f\simeq g$. Hence $f\sim g$.

Subcase 2b: $\operatorname{ind}(\operatorname{C}(f_{s})\otimes\operatorname{C}(g_{s}))=2$ for some $s\in S$. Therefore, $\operatorname{ind}(\operatorname{C}(f_{s})\otimes\operatorname{C}(g_{s}))=2$. Since $K_{v}$ is local, we have ${{}_{2}}\operatorname{Br}(K_{v})\simeq\mathbb{Z}/2$ and $\operatorname{ind}(\operatorname{C}(f_{v})\otimes\operatorname{C}(g_{v}))=\operatorname{per}(\operatorname{C}(f_{v})\otimes\operatorname{C}(g_{v}))\in\{1,2\}$ for all $v\in\Omega_{K}$. By the Albert-Brauer-Hasse-Noether theorem, $\operatorname{ind}(\operatorname{C}(f)\otimes\operatorname{C}(g))=\operatorname{lcm}\limits_{v\in\Omega_{K}}\operatorname{ind}(\operatorname{C}(f_{v})\otimes\operatorname{C}(g_{v}))=2$. Suppose $Q$ is a quaternion division $K$-algebra such that $[Q]=[\operatorname{C}(f)\otimes\operatorname{C}(g)]$ in $\operatorname{Br}(K)$.

Now we take $d\in\wp^{-1}(\operatorname{disc}(g))$ and show that $Q\otimes K(d)$ is split.

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For all $v\in\Omega_{K}-S$, we have $f_{v}\simeq g_{v}$ from subcase 2a. Then $[Q\otimes K_{v}]=[\operatorname{C}(f_{v})\otimes\operatorname{C}(g_{v})]=2[\operatorname{C}(g_{v})]=0$ in $\operatorname{Br}(K_{v})$. Therefore, $[Q\otimes K(d)_{w}]=0$ for all places $w\in\Omega_{K(d)}$ lying over $v$.

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For all $v\in S$ and for all places $w\in\Omega_{K(d)}$ lying over $v$, $K(d)_{w}$ is an extension of $K_{v}$. We have $\operatorname{rank}(f_{K(d)_{w},\operatorname{an}})\leq 2$ and $\operatorname{rank}(g_{K(d)_{w},\operatorname{an}})\leq 2$. Since $f_{v}\sim g_{v}$, $\operatorname{disc}(f_{v})=\operatorname{disc}(g_{v})$ by 2.2. Therefore, $\operatorname{disc}(f_{K(d)_{w}})=\operatorname{disc}(g_{K(d)_{w}})$. We obtain $f_{K(d)_{w}}\simeq g_{K(d)_{w}}$ and thus $[Q\otimes K(d)_{w}]=[\operatorname{C}(f_{K(d)_{w}})\otimes\operatorname{C}(g_{K(d)_{w}})]=0$ in $\operatorname{Br}(K(d)_{w})$.

We obtain that $Q\otimes K(d)_{w}$ is split for all $w\in\Omega_{K(d)}$. By the Albert-Brauer-Hasse-Noether theorem, $Q\otimes K(d)$ is split.

It follows that $Q=(K(d),\sigma,a)$ is a cyclic algebra where $\sigma$ is the generator of $\operatorname{Gal}(K(d)/K)$ given by $\sigma(d)=d+1$ and $Q=K(d)\oplus cK(d)$ for some $c\in K^{*}$ with $c^{2}=a$ and $dc=cd+c$. Let $b=a^{-1}\wp(d)$ and consider the binary quadratic form $[a,b]$, we have $\operatorname{disc}([a,b])=\operatorname{disc}(g)$ in $K/\wp K$ and $\operatorname{C}([a,b])=\displaystyle{a,b\brack K}\simeq Q$.

Next, we show that $f$ and $ag$ have the same discriminant. Since $n$ is even, $\operatorname{disc}(g_{v})=\operatorname{disc}(ag_{v})$, by 2.2, $\operatorname{disc}(f_{v})=\operatorname{disc}(g_{v})=\operatorname{disc}(ag_{v})$ for all $v\in\Omega_{K}$. Suppose $q_{1}=[1,\operatorname{disc}(f)]$ and $q_{2}=[1,\operatorname{disc}(ag)]$. We have $(q_{1})_{v}\simeq(q_{2})_{v}$ for all $v\in\Omega_{K}$. By the Hasse-Minkowski theorem, $q_{1}\simeq q_{2}$ and hence $\operatorname{disc}(f)=\operatorname{disc}(q_{1})=\operatorname{disc}(q_{2})=\operatorname{disc}(ag)$.

Now, we show that $f$ and $ag$ have the same Clifford invariant. Since $\operatorname{disc}([a,b])=\operatorname{disc}(g)$, $\operatorname{disc}([a,b]\perp g)$ is trivial. By [12, Prop.5], $\operatorname{C}([a,b]\perp g)\simeq M_{2}(\operatorname{C}(ag))$ and hence $[\operatorname{C}([a,b]\perp g)]=[\operatorname{C}(ag)]$ in $\operatorname{Br}(K)$. Since the discriminant module of $g$ is trivial (see [9, Ch.III, (4.2.4) Prop.]), we also have $[\operatorname{C}([a,b]\perp g)]=[\operatorname{C}([a,b])\otimes\operatorname{C}(g)]=[Q\otimes\operatorname{C}(g)]$ in $\operatorname{Br}(K)$ by [9, Ch.IV, (8.1.1) Prop. 2)]. We obtain

[TABLE]

Hence $\operatorname{clif}(f)=\operatorname{clif}(ag)$.

Summarizing subcase 2b, $f$ and $ag$ have the same rank, the same discriminant and the same Clifford invariant; by Theorem 2.1 in [2], $f\simeq ag$. Hence, $f\sim g$. ∎

3. Acknowledgements

The author is supported by National Natural Science Foundation of China (No.11701352) and Shantou University Scientific Research Foundation for Talents (No.130-760188). The author thanks Yong Hu and Peng Sun for helpful discussions.