---
abstract: 'Let $K$ be a complete discretely valued field with residue field $\kappa$. If char$(K) = 0$, char$(\kappa) = 2$ and $[\kappa : \kappa^2] = d$, we prove that there exists an integer $N$ depending on $d$ such that the $u$-invariant of any function field in one variable over $K$ is bounded by $N$. The method of proof is via introducing the notion of uniform boundedness for the $p$-torsion of the Brauer group of a field and relating the uniform boundedness of the 2-torsion of the Brauer group to finiteness of the $u$-invariant. We prove that the 2-torsion of the Brauer group of function fields in one variable over $K$ are uniformly bounded.'
address: |
    Department of Mathematics & Computer Science\
    Emory University\
    400 Dowman Drive NE\
    Atlanta, GA 30322, USA
author:
- 'R. Parimala'
- 'V. Suresh'
title: ' On the $u$-invariant of function fields of curves over complete discretely valued fields'
---

Let $K$ be a complete discretely valued field with residue field $\kappa$ and $F$ a function field in one variable over $K$. Suppose char$(\kappa) \neq 2$. A bound for the $u$-invariant of $F$ in terms of the $u$-invariant of function fields in one variable over $\kappa$ were obtained by Harbater-Hartmann-Krashen [@hhk:patching_qf_csa] using patching techniques. This recovers the $u$-invariant of function fields of non-dyadic $p$-adic curves ([@ps:uinv_8]). Leep ([@leep]), using results of Heath-Brown ([@heath-brown]), proved that the $u$-invariant of function fields of all $p$-adic curves (including dyadic curves) is 8. An alternate proof for function fields of dyadic curves is given in ([@ps:period_index]). In fact more generally we proved that if char$(K) = 0$, char$(\kappa) =2$ and $\kappa$ is perfect, then $u(F) \leq 8$. If $[\kappa : \kappa^2]$ is infinite it is easy to construct anisotropic quadratic forms over $K$ and hence over $F$ of arbitrarily large dimension. The question remained open whether the $u$-invariant of $F$ is finite if char$(\kappa) = 2$ and $[\kappa : \kappa^2]$ is finite. The aim of this article is to give an affirmative answer to this question. More precisely we prove the following (\[theorem:u\_invariant\])

Let $K$ be complete discretely valued field with residue field $\kappa$ and $F$ a function field of a curve over $K$. Suppose that char$(K)  = 0$, char$(\kappa) = 2$ and $[\kappa : \kappa^2]$ is finite. Then there exists an integer $M$ which depends only on $[\kappa :
\kappa^2]$ such that for any finite extension $F$ of $K(t)$, $u(F) \leq M$.

It was conjectured in ([@ps:period_index]) that $u(F)$ is at most $8[\kappa: \kappa^2]$. The bound we give for the $u(F)$ is not effective and far from the conjectural bound.

Let $L$ be a field of characteristic not equal to 2 with $H^M(L,
\mu_2^{\otimes M}) = 0$ for some $M \geq 1$. Suppose that there exists an integer $N$ such that for all finite extensions $E$ of $L$ and for any $\alpha \in H^n(E, \mu_2^{\otimes n}), n \geq 2$, there exists an extension $E'$ of $E$ of degree at most $N$ such that $\alpha  \otimes_E E' = 0$. Then a theorem of Krashen (\[cor:krashen2\]) asserts that the $u$-invariant of $L$ is finite. Our aim is to prove that if $K$ is a complete discretely valued field with residue field $\kappa$ of characteristic 2 and $[\kappa : \kappa^2]$ finite and $F$ a function field in one variable over $K$, then such an integer $N$ exists for $F$, thereby proving the finiteness of the $u$-invariant of $F$.

We introduce the notion of uniform boundedness for the $\ell$-torsion of the Brauer group $\operatorname{\mathrm{Br}}(L)$ of $L$, where $L$ is any field. We say that the Brauer group of $L$ is [*uniformly $\ell$-bounded*]{} if there exists an integer $N$ such that for any finite extension $E$ of $L$ and for any set of finitely many elements $\alpha_1, \cdots , \alpha_n \in
{{}_\ell\mathrm{Br}}(E)$, there is a finite extension $E'$ of $E$ of degree at most $N$ such that $\alpha_i \otimes_E E' = 0$ for $1 \leq i \leq n$. Using the result of Krashen (\[cor:krashen2\]), we show that if $L$ is a field of characteristic not equal to 2 with $H^M(L, \mu_2^{\otimes M}) = 0$ for some $M \geq 1$ and with the Brauer group of $L$ is $2$-uniformly bounded, then $u(L)$ is finite (\[cor:uniformbound\_uinvariant\]). It looks plausible that there are fields $L$ of finite $u$-invariant with $L$ not uniformly 2-bounded.

The main result of the paper is to prove the uniform $p$-boundedness for the Brauer group of any function field $F$ in one variable over a complete discretely valued field $K$ with residue field $\kappa$, where char$(\kappa) = p$, char$(F) \neq p$ and $[\kappa :
\kappa^p]$ is finite. We also prove the uniform $\ell$-boundedness for the Brauer group of any function field in one variable over a complete discretely valued field with residue field $\kappa$ and $\ell \neq $char$(\kappa)$ under the assumption that the Brauer groups of $\kappa$ and $\kappa(t)$ are uniformly $\ell$-bounded. This result for function fields of $p$-adic curves ($p \neq \ell$) is due to Saltman ([@saltman:jrms]). To prove our theorems we use the patching techniques of Harbater-Hartmann-Krashen and results of ([@ps:period_index]).

We thank D. Harbater for his very useful comments on the text.

 Galois cohomology, Symbol length, $u$-invariant and Uniform bound
==================================================================

In this section we recall the recent results of Krashen and Saltman connecting the symbol length and effective index in Galois cohomology with the $u$-invariant of a field.

Let $K$ be a field and $\ell$ a prime not equal to the characteristic of $K$. Let $\mu_\ell$ denote the Galois module of $\ell^{\rm th}$ roots of unity and $H^n(K, \mu_\ell^{\otimes m})$ denote the $n^{\rm
  th}$ Galois cohomology group with values in $\mu_\ell^{\otimes m}$. We have $H^1(K, \mu_\ell)  \simeq K^*/K^{*\ell}$. For $a \in K^*$, let $(a) \in H^1(K, \mu_\ell)$ denote the image of $aK^{*\ell}$. Let $a_1, \cdots , a_n \in K^*$. The cup product $(a_1) \cdot (a_2) \cdots (a_n) \in H^n(K, \mu_\ell^{\otimes n})$ is called a [*symbol*]{} in $H^n(K, \mu_\ell^{\otimes n})$. A theorem of Voevodsky ([@voevodsky]) asserts that every element in $H^n(K,
\mu_\ell^{\otimes n})$ is a sum of symbols.

Let $\alpha \in H^n(K,
\mu_\ell^{\otimes n})$. The [*symbol length*]{} of $\alpha$, denoted by $\lambda(\alpha)$, is defined as the smallest $m$ such that $\alpha $ is a sum of $m$ symbols in $H^n(K, \mu_\ell^{\otimes n})$. For any $\alpha \in H^n(K, \mu_\ell^{\otimes m})$, the [ *effective index*]{} of $\alpha$, denoted by $eind(\alpha)$, is defined to be the minimum of the degrees of finite field extensions $E$ of $K$ with $\alpha_E = 0$, where $\alpha_E$ is the image of $\alpha$ in $H^n(E, \mu_\ell^{\otimes m})$. Since $H^2(K, \mu_\ell) \simeq
{{}_\ell\mathrm{Br}}(K)$, for $\alpha \in {{}_\ell\mathrm{Br}}(K)$, eind$(\alpha)$ is equal to the index of a central simple algebra $A$ over $K$ representing $\alpha$. The following lemma asserts that this definition of effective index coincides with the definition in ([@krashen:symbol_length]).

\[lemma:sep\_ext\] Let $K$ be a field and $\ell$ a prime not equal to char$(K)$. Let $\alpha \in H^n(K, \mu_\ell^{\otimes
    m})$. Suppose that exists an extension $L$ of $K$ of degree at most $N$ with $\alpha \otimes_K L = 0$. Then there exists a separable field extension $E$ of $K$ of degree at most $N$ such that $\alpha
  \otimes_K E = 0$.

Let $L$ be an extension of $K$ of degree at most $N$ with $\alpha \otimes_KL  = 0$. Let $E$ be the separable closure of $K$ in $L$. Let $p $ be the characteristic of $(K)$. Suppose $p > 0$. Then $L/E$ is of degree $p^r$ for some $r \geq 0$. Since $\ell \neq
  p$, the restriction map $H^n(E, \mu_\ell^{\otimes m}) \to H^n(L,
  \mu_\ell^{\otimes m})$ is injective ([@serre:gc Cor. on p.12]). Hence $\alpha \otimes_K E = 0$.

We know recall a theorem of Krashen ([@krashen:symbol_length 4.2], cf. [@saltman:symbol_length]).

\[theorem:krashen1\] Let $K$ be a field and $\ell$ a prime not equal to the characteristic of $K$. Let $n \geq 1$. Suppose that there exists an integer $N$ such that for every finite extension $L$ of $K$ and for every element $\beta \in H^d(L, \mu_\ell^{\otimes d})$, $ 1 
  \leq d \leq n-1$, eind$(\beta) \leq N$. Then for any $\alpha \in H^n(K,
  \mu_\ell^{\otimes n})$, $\lambda(\alpha)$ is bounded in term of eind$(\alpha)$, $N$ and $n$.

The following is a consequence of the above theorem.

\[cor:krashen1\] Let $K$ be a field and $\ell$ a prime not equal to the characteristic of $K$. Let $n\geq 1$. Suppose that there exists an integer $N$ such that for all finite extensions $L$ of $K$ and for all $\alpha \in H^d(L, \mu_\ell^{\otimes d})$, $1\leq d \leq
  n$, eind$(\alpha) \leq N$. Then there exists an integer $N'$ which depends only $N$ and $n$ such that $\lambda(\alpha) \leq N'$ for all finite extensions $L$ of $K$ and $\alpha
  \in H^n(K, \mu_\ell^{\otimes n})$.

Let $K$ be a field of characteristic not equal to 2. The [*$u$-invariant*]{} of $K$ is defined to be the supremum of dimensions of anisotropic quadratic forms over $K$. The following theorem is a consequence of a theorem of Orlov, Vishik, and Voevodsky ([@ovv]) on the Milnor conjecture (cf. [@kahn], [@ps:symbol_length]).

\[theorem:symbol\_length\_u\_invariant\] Let $K$ be a field of characteristic not equal to 2. Suppose that there exist integer $M\geq 1 $ and $N$ such that $H^M(K,
\mu_2) =0$ and $\lambda(\alpha) \leq N$ for all $\alpha \in H^d(K,
\mu_2^{\otimes d})$, $ 1 \leq d < M$. Then the $u$-invariant is bounded by a function of $M$ and $N$.

The following follows from (\[theorem:symbol\_length\_u\_invariant\]) and (\[cor:krashen1\]) (cf. [@krashen:symbol_length 5.5]).

\[cor:krashen2\] Let $K$ be a field of characteristic not equal to 2. Suppose that there exist integers $N \geq 1$ and $M \geq
1$ such that for all finite extensions $L$ of $K$, $H^{M}(L,
\mu_2) = 0$ and for all $ n \geq 1$ and $\alpha \in H^n(L, \mu_2)$, eind$(\alpha) \leq N$. Then there exists an integer $N'$, which depends only on $N$ and $M$, such that $u(K)  \leq N'$.

Let $K$ be a field and $\ell$ a prime not equal to the characteristic of $K$. We say that $K$ is $(n , \ell)$-[*uniformly bounded* ]{} if there exists an integer $N$ such that for any finite extension $L$ of $K$ and $\alpha_1, \cdots , \alpha_m
\in H^n(L, \mu_\ell^{\otimes   n})$ there is an extension $E$ of $L$ with $[E : L] \leq N$ and $\alpha_{i} \otimes_L E = 0$ for $1 \leq i \leq m$. Such an $N$ is called an $(n, \ell)$-[*uniform bound*]{} for $K$. We note that if $N$ is an $(n, \ell)$-uniform boud for $K$, then $N$ is also an $(N, \ell)$-uniform bound for any finite extension $L$ of $K$.

In view of a theorem of Voevodsky ([@voevodsky]) on the Bloch-Kato conjecture, every element in $H^n(K, \mu_\ell^{\otimes n})$ is a sum of symbols. In particular $N$ is an $(n, \ell)$-uniform bound of $K$ if and only if for given symbols $\alpha_1, \cdots , \alpha_m \in H^n(K, \mu_\ell^{\otimes
  n})$ there is an extension $L$ of $K$ with $[L : K] \leq N$ and $\alpha_{i_L} = 0$ for $1 \leq i \leq m$.

\[lemma:uniform\_bound\_d\_d+1\] Let $K$ be a field and $\ell$ a prime not equal to the characteristic of $K$. If $N$ is an $(n,\ell)$-uniform bound for $K $, then $N$ is also a $(d,
\ell)$-uniform bound for $K $ for all $d \geq n$.

Suppose $N$ is an $(n, \ell)$-uniform bound for $ K$. It is enough to prove the lemma for $d = n+1$. Let $L$ be a finite extension of $K$ and $\alpha_1, \cdots , \alpha_m
\in H^{n+1} (L, \mu_\ell^{\otimes (n+1)})$ be symbols. Then $\alpha_i =
\beta_i \cdot (a_i)$ for some symbols $\beta_i \in  H^{n}(L,
\mu_{\ell}^{\otimes n})$ and $a_i \in L^*$, $1 \leq i \leq m$. Since $N$ is an $(n, \ell)$-uniform bound for $K$, there exists a field extension $E$ of $L$ with $[E : L] \leq N$ and $\beta_ {i_E} = 0$ for $1 \leq i \leq
  m$. Then clearly $\alpha_{i_E} = 0$ for $1 \leq i \leq m$. Thus $N$ is also an $(n+1, \ell)$-uniform bound for $K$.

\[cor:symbol\_length\] Let $K$ be a field and $\ell$ a prime not equal to the characteristic of $K$. Suppose that $N$ is a $(2, \ell)$-uniform bound for $K$. Then for every $n \geq 2$, there exists an integer $N_n$, which depends only on $N$ and $n$ such that $\lambda(\alpha) \leq N_n$ for all $\alpha \in H^n(K,
\mu_\ell^{\otimes n})$.

Since $N$ is $(2, \ell)$-uniform bound for $K$, by ( \[lemma:uniform\_bound\_d\_d+1\]), $N$ is an $(n, \ell)$-uniform bound for $K$ for all $n \geq 2$. Let $\alpha \in H^n(K, \mu_\ell^{\otimes
   n})$. Then, by (\[theorem:krashen1\]), $\lambda(\alpha)$ is bounded in terms of eind$(\alpha)$, $N$ and $n$. Since eind$(\alpha) \leq N$, $\lambda(\alpha)$ is bounded in terms of $N$ and $n$.

\[cor:uniformbound\_uinvariant\] Let $K$ be a field of characteristic not equal to 2. Suppose that there exists an integer $M$ such that for all finite extensions $L$ of $K$, $H^M(K,
  \mu_2)  = 0$ and $N$ is a $(2,2)$-uniform bound for $K$. Then there exists $N'$ which depends only on $N$ and $M$ such that for any finite extension $L$ of $K$, $u(L) \leq N'$.

Since the conditions on $K$ are also satisfied by any finite extensions of $K$, it is enough to find an $N'$ which depends only on $N$ and $M$ such that $u(K) \leq N'$.

Since $N$ is a $(2,2)$-uniform bound for $K$, by (\[cor:symbol\_length\]) there exist integers $N_n$ for $1 \leq n
  < M$, which depends only on $N$ and $n$ such that for all $\alpha \in H^n(K, \mu_\ell^{\otimes n})$, $\lambda(\alpha)
  \leq N$. Let $N'$ be the maximum of $N_n$ for $ 1 \leq n < M$. Thus corollary follows from (\[theorem:krashen1\]).

Let $R$ be a discrete valuation ring with field of fractions $K$ and residue field $\kappa$. Let $\ell$ be a prime not equal to char$(\kappa)$. Then there is a residue homomorphism $\partial : H^n(K, \mu_\ell^{\otimes m} ) \to H^{n-1}(\kappa,
\mu_\ell^{\otimes (m-1)})$ with kernel $H^n_{{\mathrm{\acute{e}t}}}(R,\mu_\ell^{\otimes
  m})$.

Let $A$ be an integral domain with field of fractions $K$. Let $\ell$ be a prime which is a unit in $A$. We have the natural map $H^n_{\mathrm{\acute{e}t}}(A, \mu_\ell^{\otimes m}) \buildrel{\iota}\over{\to} H^n(K,
\mu_\ell^{\otimes m})$. An element $\alpha$ of $H^n(K, \mu_\ell^{\otimes m})$ is said to be [ *unramified*]{} on $A$ if $\alpha$ is in the image of $\iota$. Suppose that $A$ is a regular ring. For each height one prime ideal $P$ of $A$, we have the residue homomorphism $\partial_P : H^n(K, \mu_\ell^{\otimes m}) \to
H^{n-1}(\kappa(P), \mu_\ell^{\otimes (m-1)})$, where $\kappa(P)$ is the residue field at $P$. We have the following

([@auslander-goldman 7.4]) \[auslander-goldman\] Let $A$ be a regular two dimensional integral domain with field of fractions $K$ and $\ell$ a prime which is a unit in $A$. The sequence $$0 \to H^2_{\mathrm{\acute{e}t}}(A, \mu_\ell) \to H^2(K, \mu_\ell) \to \oplus_{P \in
  Spec(A)^{(1)}} H^1(\kappa(P), {\mathbb Z}/ \ell{\mathbb Z})$$ is exact, where Spec$(A)^{(1)}$ is the set of height one prime ideals of $A$.

We now recall a few notation from ([@hhk:patching_qf_csa]). Let $R$ be a complete discrete valuation ring with field of fractions $K$ and residue field $\kappa$. Let $F$ be the function field of a curve over $K$. Let ${{\mathscr{X}}}$ be a regular proper model of $F$ over $R$ and $X$ its reduced special fiber. For any codimension one point $\eta$ of ${{\mathscr{X}}}$, let $F_\eta$ be the completion of $F$ at the discrete valuation of $F$ given by $\eta$ and $\kappa(\eta)$ the residue field at $\eta$. For a closed point $P$ of ${{\mathscr{X}}}$, let $F_P$ be the field of fractions of the completion of the local ring at $P$ and $\kappa(P)$ the residue field at $P$. Let $U$ be an open subset of $X$. Let $R_U$ be the ring of all those functions in $F$ which are regular on $U$. Then $R
\subset R_U$. Let $t$ be a parameter in $R$. Let $\hat{R}_U$ be the completion of $R_U$ at the ideal $(t)$. Let $F_U$ be the field of fractions of $\hat{R}_U$.

Let $A$ be a regular integral domain with field of fractions $F$. For a maximal ideal $m$ of $A$, let $\hat{A}_m$ denote the completion of the local ring $A_m$ and $F_m$ the field of fractions of $\hat{A}_m$.

Uniform bound - bad characteristic case 
========================================

Let $K$ be a complete discretely valued field with residue field $\kappa$. Let $p = char(\kappa)$. In this section we show that there is a $(2, p)$-uniform bound for $K(t)$ which depends only on $[\kappa : \kappa^p]$.

First we recall the following two results from ([@ps:period_index]).

\[theorem:ps1\] [@ps:period_index 2.4] Let $R$ be a complete discrete valuation ring with field of fractions $K$ and residue field $\kappa$. Suppose that char$(K) = 0$, char$(\kappa)
  = p >0$ and $[\kappa : \kappa^p] = p^d$. Let $\pi \in R$ be a parameter and $u_1,\cdots ,u_d \in R^*$ units such that $\kappa =
  \kappa^p(\overline{u}_1, \cdots , \overline{u}_d)$, where for any $u \in R$, $\overline{u}$ denotes the image of $u$ in $\kappa$. Suppose that $K$ contains a primitive $p^{\rm th}$ root of unity. Then any $\alpha \in H^2(K, \mu_p) $ splits over $K(\sqrt[p]{\pi}, \sqrt[p^2]{u_1}, \cdots ,  \sqrt[p^2]{u_{d-1}}, \sqrt[p]{u_d})$. In particular if $d > 0$, $p^{2d}$ is a $(2, p)$-uniform bound for $K$ and if $d = 0$, $p$ is a $(2, p)$-uniform bound for $K$.

\[prop:ps2\] ([@ps:period_index 3.5]) Let $A$ be a complete regular local ring of dimension 2 with field of fractions $F$ and residue field $\kappa$. Suppose that char$(F) =
  0$, char$(\kappa) = p > 0$ and $[\kappa : \kappa^p] = p^d$. Let $\pi, \delta \in A$ and $u_1, \cdots , u_d \in A^*$ such that the maximal ideal $m$ of $A$ is generated by $\pi$ and $\delta$, and $\kappa = \kappa^p(\overline{u}_1, \cdots ,
\overline{u}_d)$, where for any $u \in A$, $\overline{u}$ denotes the image of $u$ in $\kappa$. Suppose that $F$ contains a primitive $p^{\rm th}$ root of unity. Then any $\alpha \in H^2(F, \mu_p)$ which is unramified on $A$ except possibly at $(\pi)$ and $(\delta)$ splits over $F(\sqrt[p]{\pi},  \sqrt[p]{\delta}, \sqrt[p^2]{u_1},
  \cdots ,   \sqrt[p^2]{u_d})$.

\[lemma:choice\_of\_f\_g\] Let ${{\mathscr{X}}}$ be a regular integral two dimensional scheme and $F$ its function field. Suppose $C$ and $E$ are regular curves on ${{\mathscr{X}}}$ with normal crossings. Let ${{\mathscr{P}}}$ be a finite set of closed points of ${{\mathscr{X}}}$ with ${{\mathscr{P}}}\subset C \cup E$. Then there exist $f, g \in F^*$ such that the maximal ideal at $P$ is generated by $f$ and $g$ for each $P \in {{\mathscr{P}}}$ and $f$ (resp. $g$) defines $C$ (resp. $E$) at each $P \in {{\mathscr{P}}}\cap C$ (resp. $P \in {{\mathscr{P}}}\cap E$).

Let $R$ be the semi-local ring at all $P \in
  {{\mathscr{P}}}$. Since $R$ is a unique factorisation domain, there exist $f_1, g_1 \in R$ such that div$_{Spec(R)}(f_1) = C\mid _ {Spec(R)}$ and div$_{Spec(R)}(g_1) = E\mid _ {Spec(R)}$. For each $P \in {{\mathscr{P}}}$, let $m_P$ be the maximal ideal of the local ring $R_P$ at $P$. Then for each $P \in C\cap E \cap {{\mathscr{P}}}$, $m_P = (f_1, g_1)$. Let $P \in {{\mathscr{P}}}$. Suppose $P \not\in C$. Then $P \in E$. Since $E$ is regular on ${{\mathscr{X}}}$, by the choice of $g_1$, there exists $\theta_P \in m_P$ such that $m_P = (\theta_P, g_1)$. By the Chinese remainder theorem, there exists $\pi_P \in m_P$ such that $\pi_P
 \not\in m_Q$ for all $Q \in {{\mathscr{P}}}$, $Q \neq P$ and $\pi_p = \theta_P$ module $m_P^2$. Then $m_P = (\pi_P, g_1)$. Similarly for each $P \not\in E$, choose $\delta_P 
 \in m_P$ such that $m_P = (f_1, \delta_P)$ and $\delta_P \not\in m_Q$ for all $Q \in {{\mathscr{P}}}$, $Q \neq P$. Let $$f_2  =   \prod_{P \in {{\mathscr{P}}}\setminus C} \pi_P, ~~f = f_1f_2$$ and $$g_2 =  \prod_{P \in {{\mathscr{P}}}\setminus E} \delta_P, ~~ g = g_1g_2.$$ Then $f_2$ and $g_2$ are units at all $P \in C\cap E$. We claim that $f$ and $g$ have the required properties. Let $P \in {{\mathscr{P}}}$. Suppose $P \in C \cap E$. Then by the choices $f_2$ and $g_2$, they are units at $P$ and $m_P = (f_1, g_1)$. In particular $m_P  = (f, g)$ and $f, g$ define $C$ and $E$ respectively at $P$. Suppose that $P \not\in C$. Then $f_1$ and $g_2$ are units at $P$ and $f_2 = \pi_P u_P$ for some unit $u_P$ at $P$. Since $m_P = (\pi_P, g_1)$, we have $m_P = (f, g)$. Since $g_1$ defines $E$ at $P$ and $g_2$ is a unit at $P$, $g$ defines $E$ at $P$. Similarly if $P \not\in E$, then $m_P = (f, g)$ and $f$ defines $C$ at $P$.

\[theorem:bad\_char\] Let $K$ be a complete discretely valued field with residue field $\kappa$. Suppose that char$(K) = 0$ and char$(\kappa) = p > 0$ and $[\kappa : \kappa^p] = p^d$. Assume that $K$ contains a primitive $p^{th}$ root of unity. Then $p^{4d+4}$ is a $(2, p)$-uniform bound for $K(t)$.

Let $F$ be a finite extension of $K(t)$. Let $\alpha_1, \cdots , \alpha_m \in H^2(F,\mu_p)$. Let ${{{\mathscr{X}}}}$ be a regular proper model of $F$ over the ring of integers $R$ of $K$ such that the support of ram$(\alpha_i)$ for all $i$ and the special fiber is contained in $C \cup E$, where $C$ and $E$ are regular curves on ${{\mathscr{X}}}$ having only normal crossings. Let $\eta$ be the generic point of an irreducible component $X_\eta$ of the special fiber of ${{\mathscr{X}}}$. Then $\kappa(\eta)$ is a function field in one variable over $\kappa$. Since $[\kappa : \kappa^p] = p^d$, $[\kappa(\eta) : \kappa(\eta)^p] =
d+1$ ([@bourbaki:algebra; @ll A.V.135, Corollary 3]). Let $\pi_\eta$ be a parameter at $\eta$ and $u_{\eta, 1},  \cdots ,  u_{\eta, d+1} \in F^*$ be lifts of a $p$-basis of $\kappa(\eta)$. Then, by (\[theorem:ps1\]), $\alpha \otimes F_\eta(\sqrt[p^2]{u_{\eta, 1}}, 
\sqrt[p^2]{u_{\eta, d}}, \sqrt[p]{u_{\eta, d+1}}, \sqrt[p]{\pi_\eta}) = 0$ for all $i$. Let $f \in F^*$ be chosen such that $\nu_\eta(f) = 1$ for all $\eta$. By the Chinese remainder theorem, choose $u_1,  \cdots , u_d \in F^*$ units at each $\eta$ such that $\overline{u}_j = 
\overline{u}_{\eta, j} \in
\kappa(\eta)$. Then $\alpha_i \otimes F_\eta(\sqrt[p]{f}, 
 \sqrt[p^2]{u_1}, \cdots , \sqrt[p^2]{u_{d}}, \sqrt[p]{u_{d+1}}) = 0$ for all $i$. By ([@hhk:lgp_torsors 5.8], [@kmrt 1.17]), there exists a non-empty open set $U_\eta$ of the component $X_\eta$ of the special fiber, such that $\alpha \otimes F_{U_\eta}(\sqrt[p]{f}, 
 \sqrt[p^2]{u_1}, \cdots , \sqrt[p^2]{u_{d}}, \sqrt[p]{u_{d+1}}) 
 = 0$ for all $i$.

Let ${{{\mathscr{P}}}}$ be the finite set of closed points of ${{{\mathscr{X}}}}$ which are not in $U_\eta$ for any $\eta$. Let $A$ be the semi-local ring at the points of ${{\mathscr{P}}}$. For $P \in {{\mathscr{P}}}$, let $A_P$ be the local ring at $P$. Since the ramifications of $\alpha_i$ for all $i$ are in normal crossings, for each $P  \in {{{\mathscr{P}}}}$, the maximal ideal $m_p$ at $P$ is $(\pi_P, \delta_P)$ for some $\pi_P$ and $\delta_P$ such that $\alpha_i$ is unramified on $A_P$ except possibly at $(\pi_P)$ and $(\delta_P)$. Since the residue field $\kappa(P)$ at $P$ is a finite extension of $\kappa$ and $[\kappa : \kappa^p] = p^d$, $[\kappa(P) :
\kappa(P)^p] = p^d$ ([@bourbaki:algebra; @ll A.V.135, Corollary 3]). Let $v_{P, 1}, \cdots , v_{P, d} \in A_P^*$ be lifts of a $p$-basis of $\kappa(P)$. By the Chinese remainder theorem, choose $h_1, \cdots , h_d \in A^*$ such that $h_i = v_{P, i}$ modulo the maximal ideal at $P$ for all $P
\in {{\mathscr{P}}}$ and $1\leq i
\leq d$. By (\[lemma:choice\_of\_f\_g\]), there exist $g_1, g_2 \in F^*$ such that for any point $P \in {{\mathscr{P}}}$, we have $m_P = 
(g_1, g_2)$ and $g_1$ defines $C$ at all $P \in {{\mathscr{P}}}\cap C$ and $g_2$ defines $E$ at all $P \in {{\mathscr{P}}}\cap E$. In particular, each $\alpha_i$ is unramified on $A_P$ except possibly at $(g_1)$ and $(g_2)$. Then, by (\[prop:ps2\]), $\alpha_i \otimes F_P(\sqrt[p^2]{h_1},
\cdots , \sqrt[p^2]{h_d}, \sqrt[p]{g_1},   \sqrt[p]{g_2}) = 0$ for all $i$.

Let $L = F(\sqrt[p]{f}, 
 \sqrt[p^2]{u_1}, \cdots , \sqrt[p^2]{u_{d}}, \sqrt[p]{u_{d+1}}, \sqrt[p^2]{h_1}, \cdots , 
 \sqrt[p^2]{h_d}, \sqrt[p]{g_1},  \sqrt[p]{g_2})$. We claim that $\alpha_i \otimes L = 0$ for all $i$. Let ${{{\mathscr{Y}}}}$ be a regular proper model of $L$ and $Y$ its special fiber. Let $y $ be a point of $Y$. If $y$ lies over a generic point of a component of the special fiber $X$ of ${{{\mathscr{X}}}}$, then by the choice of $f, u_1, \cdots , u_d$, $\alpha_i  \otimes L_y = 0$. Suppose the image of $y$ lies over a closed point $P$ of $X$. Suppose $P \in U_\eta$ for some $\eta$. Then $F_{U_\eta} \subset F_P$ and hence once again by the choice of $f, u_1, \cdots , u_d$, $\alpha_i \otimes L_y = 0$ for all $i$. Suppose $P \not\in U_\eta$ for all $\eta$. Then $P \in {{{\mathscr{P}}}}$. Since $F_P \subset L_y$, by the choices of $h_1, \cdots , h_d, g_1, g_2$, $\alpha_i \otimes L_y = 0$ for all $i$.

Since $[L : F] \leq p^{4d + 4}$, the theorem follows.

\[cor:bad\_char\_2\] Let $K$ be a complete discretely valued field with residue field $\kappa$. Suppose that char$(K) = 0$ and char$(\kappa) = p > 0$ and $[\kappa : \kappa^p] = p^d$. Let $\zeta$ be a primitive $p^{th}$ root of unity. Then $[K(\zeta) : K]p^{4d+4}$ is a $(2, p)$-uniform bound for $K(t)$.

\[cor:bad\_char\_n\] Let $K' = K(\zeta)$. Then $K'$ is a complete discretely valued field with residue field $\kappa$. Let $F$ be a finite extension of $K(t)$. Let $\alpha_1, \cdots , \alpha_m \in H^2(F, \mu_p^{\otimes 2})$. Since $F'= F(\zeta)$ is also a function field over $K'$, by (\[theorem:bad\_char\]), there exists an extension $L$ of $F'$ of degree at most $p^{4d+4}$ such that $\alpha_i \otimes L  = 0$ for all $i$. Since $[L : F] = [L :
  F'][F' : F]$, the corollary follows.

The above corollary and (\[lemma:uniform\_bound\_d\_d+1\]) give the following

Let $K$ be a complete discretely valued field with residue field $\kappa$. Suppose that char$(K) = 0$ and char$(\kappa) = p > 0$ and $[\kappa : \kappa^p] = p^d$. Let $\zeta$ be a primitive $p^{th}$ root of unity. Then $[K(\zeta) : K]p^{4d+4}$ is an $(n, p)$-uniform bound for $K(t)$ for all $n \geq 2$.

Uniform bound - good characteristic case 
=========================================

Let $F$ be the function field of a $p$-adic curve. In ([@saltman:jrms]), Saltman proved that if $\ell$ is a prime not equal to $p$ and $\alpha_1, \cdots , \alpha_m \in H^2(F, \mu_\ell)$, then there exists an extension $L$ of $F$ such that $[L : F] \leq \ell^2$ and $\alpha_{i_L} = 0 $ for all $i$, i.e., $F$ is $(2,
\ell)$-uniformly bounded.

Let $K$ be a complete discretely valued field with residue field $\kappa$. Let $\ell$ be a prime not equal to char$(\kappa)$. In this section we show that $K(t)$ is $(2,\ell)$-uniformly bounded under some conditions on $\kappa$.

\[theorem:good\_char\_dvr\] Let $K$ be a complete discretely valued field with residue field $\kappa$. Let $\ell$ be a prime not equal to char$(\kappa)$ and $n \geq 1$. If $N$ is an $(n, \ell)$-uniform bound for $\kappa$, then $\ell N$ is an $(n, \ell)$-uniform bound for $K$.

Let $L$ be a finite extension of $K$. Then $L$ is a complete discretely valued field with residue $\kappa'$ a finite extension of $\kappa$. Let $R$ be the valuation ring of $R$ and $\pi \in R$ be a parameter. Let $\alpha_1, \cdots, \alpha_m \in H^n(L,
  \mu_\ell^{\otimes n})$. Let $S$ be the integral closure of $R$ in $L(\sqrt[\ell]{\pi})$. Then $S$ is also a complete discrete valuation ring with residue field $\kappa'$. Since $S/R$ is ramified, $\alpha_i \otimes
  L(\sqrt[\ell]{\pi})$ is unramified at $S$ for each $i$. Hence there exists $\beta_i \in H^n_{{\mathrm{\acute{e}t}}}(S, \mu_\ell^{\otimes n})$ such that $\beta_i
  \otimes_SK(\sqrt[\ell]{\pi}) = \alpha_i$. Since $N$ is an $(n,
  \ell)$-uniform bound for $\kappa$, there exists an extension $L_0$ of $\kappa$ of degree at most $N$ such that $\beta_i \otimes_\kappa L_0 =
  0$ for all $i$. Let $L$ be the extension of $K$ of degree equal to $[L_0 : \kappa]$ with residue field $L_0$. Let $T$ be the integral closure of $R$ in $L(\sqrt[\ell]{\pi})$. Then $T$ is a complete discrete valued ring with residue field $L_0$ and $S \subset T$. Since $\beta_i \otimes_\kappa L_0 = 0$, $\beta_i
  \otimes_S T = 0$ for all $i$. In particular $\alpha_i \otimes_K
  L(\sqrt[\ell]{\pi}) = 0$ for all $i$. Since the degree of $L$ over $K$ is equal to the degree of $L_0$ over $\kappa$ and $[L_0 :
  \kappa] \leq N$, we have $[L(\sqrt[\ell]{\pi}) : K] \leq \ell N$.

\[lemma:good\_char\_dim2\] Let $A$ be a regular local ring with residue field $\kappa$ and maximal ideal $m = (\pi,\delta)$. Let $F$ be the field of fractions of $A$, $\ell$ a prime not equal to char$(\kappa)$. Let $B$ (resp. $B'$) be the integral closure of $A$ in $F(\sqrt[\ell]{\pi}, \sqrt[\ell]{\delta})$ (resp. $F(\sqrt[\ell]{\pi})$). Let $\alpha \in H^2(F, \mu_\ell^{\otimes 2})$. If $\alpha$ is unramified on $A$ except possibly at $(\pi)$ and $(\delta)$ (resp. except possibly at $(\pi)$), then $\alpha \otimes_F F(\sqrt[\ell]{\pi},
\sqrt[\ell]{\delta})$ is unramified on $B$ (resp. $\alpha \otimes_F
F(\sqrt[\ell]{\pi})$ is unramified on $B'$).

By ([@ps:period_index 3.3]), $B$ is a regular local ring of dimension 2 with residue field $\kappa$. Let $P$ be a height one prime ideal of $B$ and $Q = P \cap A$. Then $Q$ is a height one prime ideal of $A$. If $Q \neq (\pi)$ and $(\delta)$, then $\alpha$ is unramified at $Q$ and hence $\alpha \otimes F(\sqrt[\ell]{\pi}, \sqrt[\ell]{\delta})$ is unramified at $P$. Suppose $Q = (\pi)$ or $(\delta)$. Then $Q$ is ramified in $B$ and hence $\alpha \otimes F(\sqrt[\ell]{\pi}, \sqrt[\ell]{\delta})$ is unramified at $P$. Since $B$ is a regular local ring of dimension 2 and $\alpha \otimes F(\sqrt[\ell]{\pi}, \sqrt[\ell]{\delta})$ is unramified at every height one prime ideal of $B$, $\alpha \otimes F(\sqrt[\ell]{\pi}, \sqrt[\ell]{\delta})$ is unramified on $B$ (cf. \[auslander-goldman\]). The other case follows similarly.

\[theorem:good\_char\_killing\_ramification\] Let $R$ be a complete discrete valuation ring with field of fractions $K$ and residue field $\kappa$. Let $F$ be the function field of a curve over $K$. Let $\ell$ be a prime not equal to char$(\kappa)$ and $\alpha_1, \cdots ,
  \alpha_r \in H^n(K, \mu_\ell^{\otimes m})$. Then there exist $f, g,  h \in F^*$ such that each $\alpha_i \otimes F(\sqrt[\ell]{f}, \sqrt[\ell]{g},
  \sqrt[\ell]{h})$ is unramified at all codimension one points of any regular proper model of $F(\sqrt[\ell]{f}, \sqrt[\ell]{g},
  \sqrt[\ell]{h})$.

Let ${{\mathscr{X}}}$ be a regular proper model of $F$ over $R$ such that the union of the support of ramification locus of $\alpha_i$, for $1
  \leq i \leq r$, is contained in the union of regular curves $C$ and $E$ with $C \cup E$ having only normal crossings. Let $f \in F^*$ be such that $${\rm div}_{{\mathscr{X}}}(f) = C + E + F$$ for some divisor $F$ on ${{\mathscr{X}}}$ which does not contain any irreducible component of $C \cup E$ and does not pass through any point of $C
\cap E$. Let $g \in F^*$ be such that $${\rm div}_{{\mathscr{X}}}(g) = C  + G$$ for some divisor $G$ on ${{\mathscr{X}}}$ which does not contain any irreducible component of $C \cup E \cup F$ and does not pass through any point of $C
\cap E$, $C \cap F$ and $E \cap F$. Let $h \in F^*$ be such that $${\rm div}_{{\mathscr{X}}}(h) =  E + H$$ for some divisor $H$ on ${{\mathscr{X}}}$ which does not contain any irreducible component of $C \cup E \cup F \cup G$ and does not pass through any point of $C
\cap E$, $C\cap F$, $C \cap G$, $E \cap F$, $E \cap G$ and $F \cap
G$.

Let $L = F(\sqrt[\ell]{f}, \sqrt[\ell]{g}, \sqrt[\ell]{h})$ and ${{\mathscr{Y}}}$ be a regular proper model of $L$ over $R$. We claim that $\alpha_i
\otimes_F L$ is unramified on ${{\mathscr{Y}}}$. Let $y \in {{\mathscr{Y}}}$ be a codimension one point of ${{\mathscr{Y}}}$. Then $y$ lies over a point $x$ of ${{\mathscr{X}}}$. If $x$ is not on $C$ or $E$, then each $\alpha_i$ is unramified at $x$ and hence $\alpha
\otimes_F L$ is unramified at $y$. Assume that $x \in C\cup E$.

Suppose that $x$ is a codimension one point of ${{\mathscr{X}}}$. Since $x$ is on $C$ or $E$, by the choice of $f$, $f$ is a parameter at $x$ and hence $L/K$ is unramified at $x$. In particular $\alpha_i \otimes_F L $ is unramified at $y$.

Suppose that $x$ is a closed point of ${{\mathscr{X}}}$. Suppose $x \in C$ and $x
\not\in E$. Let $A_x$ be the local ring of ${{\mathscr{X}}}$ at $x$ and $S_y$ be the local ring of ${{\mathscr{Y}}}$ at $y$. Suppose $x \not\in F$. Then the maximal ideal $m_x$ at $x$ is $(f, \delta_x)$ for some $\delta_x \in m_x$ and each $\alpha_i$ is unramified on the local ring at $x$ except possibly at $(f)$. Thus, by (\[lemma:good\_char\_dim2\]), each $\alpha_i \otimes_F
F(\sqrt[\ell]{f})$ is unramified at the integral closure of $A$ in $F[\sqrt[\ell]{f}]$. Since the integral closure of $A$ in $F[\sqrt[\ell]{f}]$ is contained in $S_y$, each $\alpha_i\otimes_F \otimes
  L$ is unramified at $S_y$. Suppose $x \in F$. If $x \not\in G$, the as above each $\alpha_i \otimes_F L$ is unramified at $S_y$. If $x
  \in F\cap G$, then by the choice $h$, $x \not\in H$ and hence as above, each $\alpha_i \otimes_F L$ is unramified at $S_y$. Similarly if $x \in E$ and $x \not\in C$, then each $\alpha_i
  \otimes_F L$ is unramified at $S_y$.

Suppose $x \in C \cap E$. Then $x \not\in G$ and $x \not\in H$. In particular, $m_x = (g, h)$ and each $\alpha_i$ is unramified on $A_x$ except possibly at $(g)$ and $(h)$. As above each $\alpha_i \otimes_F
L$ is unramified at $S_y$.

\[lemma:crt\] Let $A$ be a semi-local regular domain with field of fractions $F$. For each maximal ideal $m$ of $A$, let $\mathfrak{s}(m)$ be a separable finite extension of the residue field $\kappa(m)$ at $m$ of degree $N_m$. Let $N$ be a common multiple of $N_m$, $m$ varying over all maximal ideals of $A$. Then there exists an extension $E$ of $F$ of degree at most $N$ such that for each maximal ideal $m$ of $A$ and for each maximal ideal $m'$ of the integral closure $B_m$ of $A_m$ in $E$, $B_m/m'$ contains a field isomorphic to $\mathfrak{s}(m)$.

Since $\mathfrak{s}(m)$ is a finite separable extension of $\kappa(m)$, there exists $\theta_m
   \in \mathfrak{s}(m) $ such that $\mathfrak{s}(m) = \kappa(m)(\theta_m)$. Let $f_m(X) \in \kappa(m)[X]$ be the minimal polynomial of $\theta_m$ over $\kappa(m)$. Then the degree of $f_m(X)$ is $N_m$. Let $f(X) \in A[X]$ be a monic polynomial of degree $N$ such that $f(X) = f_m(X)^{N/deg(f_m)}$ modulo $m$ for each maximal ideal $m$ of $A$. Let $g(X)$ be any monic irreducible factor of $f(X)$ over $A$. Let $E = F[X]/(g(X))$. We claim that $E$ has the required property.

Let $m$ be a maximal ideal of $A$. By the choice of $f(X)$ and $g(X)$, we have $g(X) = f_m(X)^{r_m}$ modulo $m$ for some $r_m \geq
1$. Let $B_m$ be the integral closure of $A_m$ in $E$. Since $g(X)$ is monic, $A_m[X]/(g(X))$ is isomorphic to a subring of $B_m$. Let $g_m(X) \in A_m[X]$ be a monic polynomial with $g_m(X) = f_m(X)$ modulo $m$. Since $f(X) = f_m(X)^{r_m} = g_m(X)^{r_m}$ modulo $m$, the ideal $\tilde{m}$ of $ A_m[X]/(g(X))$ generated by $m$ and $g_m(X)$ is a maximal ideal with $(A_m[X]/(g(X)))/\tilde{m} \simeq \mathfrak{s}(m)$. Since $B_m$ is integral over a subring isomorphic to $A_m[X]/(g(X))$, for every maximal ideal $m'$ of $B_m$, $B_m/m'$ contains a subfield isomorphic to $(A_m[X]/(g(X)))/\tilde{m} \simeq \mathfrak{s}(m)$.

\[cor:crt\] Let $A$ be a semi-local regular domain with field of fractions $F$. Let $\ell$ be a prime. Suppose that $\ell$ is a unit in $A$. Let $\beta \in H^2_{\mathrm{\acute{e}t}}(A, \mu_\ell)$. Suppose that for every maximal ideal $m$ of $A$, there exists a finite separable extension $\mathfrak{s}(m)$ of $\kappa(m)$ of degree $N_m$ such that $\beta \otimes_A \mathfrak{s}(m) = 0$. Let $N$ be a common multiple of $N_m$, where $m$ varies over maximal ideals of $A$. Let $E$ be the field constructed in (\[lemma:crt\]). Then for any maximal ideal $m$ of $A$, $\beta \otimes_A (E \otimes_F F_m) = 0$.

Let $B$ be the integral closure of $A$ in $E$. Let $m$ be a maximal ideal of $A$. and $\hat{A}_m$ the completion of $A$ at $m$. Then, $B \otimes_A \hat{A}_m$ is complete and by the choice of $E$, $B \otimes_A \hat{A}_m$ modulo its radical is isomorphic to a product of fields with each factor containing a field isomorphic to $\mathfrak{s}(m)$. Since $\beta \otimes \mathfrak{s}(m) = 0$, it follows that $\beta
\otimes B \otimes  \hat{A}_m = 0$. Since $E$ is the field of fractions of $B$, $\beta \otimes_A (E
\otimes_F F_m) = 0$.

\[theorem:good\_char\] Let $K$ be complete discretely valued field with residue field $\kappa$. Let $\ell$ be a prime not equal to char$(\kappa)$. Suppose that $N_1$ is a $(2, \ell)$-uniform bound for $\kappa(t)$ and $N_2$ is a $(2, \ell)$-uniform bound for $\kappa$. Then $\ell^3 (N_1!)(N_2!)$ is a $(2, \ell)$-uniform bound for $K(t)$.

Let $F$ be a finite extension of $K(t)$. Let $\alpha_1, \cdots , \alpha_m \in H^2(F, \mu_\ell^{\otimes 2})$. Then, by (\[theorem:good\_char\_killing\_ramification\]), there exist $f, g, h \in F^*$ such that $\alpha_i \otimes 
F(\sqrt[\ell]{f}, \sqrt[\ell]{g}, \sqrt[\ell]{h})$ is unramified at every codimension one point of any regular proper model of $F(\sqrt[\ell]{f}, \sqrt[\ell]{g}, \sqrt[\ell]{h})$ over $R$. Let $L  = F(\sqrt[\ell]{f}, \sqrt[\ell]{g}, \sqrt[\ell]{h})$. Let ${{{\mathscr{Y}}}}$ be a regular proper model of $L$ over $R$ and $Y$ the reduced special fiber of ${{\mathscr{Y}}}$.

Let $\eta$ be the generic point of an irreducible component $Y_\eta$ of $Y $. Let $A_\eta$ be the local ring at $\eta$. Since each $\alpha_i \otimes_F L$ is unramified at $\eta$, there exists $\beta_i \in H^2_{\mathrm{\acute{e}t}}(A_\eta, \mu_\ell^{\otimes 2})$ such that $\beta_i 
\otimes_{A_\eta} F_\eta = \alpha_i$ for $1 \leq i \leq m$. Since $\kappa(\eta)$ is a finite extension of $\kappa(t)$ and $N_1$ is a $(2, \ell)$-uniform bund for $\kappa(t)$, there exists a finite extension $\mathfrak{s}(\eta)$ of degree at most $N_1$ such that $\beta_i \otimes_{A_\eta} \mathfrak{s}(\eta) = 0$ for all $i$. By (\[lemma:sep\_ext\]), we assume that $\mathfrak{s}(\eta)$ is separable over $\kappa(\eta)$. Let $A$ be the semi-local ring at the generic points of all irreducible components of the special fiber $Y$ of ${{\mathscr{Y}}}$. Then $A$ is a semi-local regular ring with field of fractions $L$. By (\[lemma:crt\]), there exists a field extension $L_1$ of $L$ of degree at most $N_1!$ such that for every maximal ideal $m'$ of the integral closure $B_m$ of $A_m$ in $L$, $B_m/m'$ contains a subfield isomorphic to $\mathfrak{s}(\eta)$. Hence, by (\[cor:crt\]), $\alpha_i \otimes (L_1 \otimes_{L} L_{\eta}) = \beta_i  \otimes (L_1
\otimes_{L} L_{\eta}) = 0$ for all $i$. By ([@hhk:lgp_torsors 5.8], [@kmrt 1.17]), there exists a non-empty open set $U_\eta$ of the component $Y_\eta$ of the special fiber $Y$, such that $\alpha_i
\otimes L_1  \otimes L_{{U_\eta}}   = 0$ for all $i$.

Let ${{\mathscr{P}}}$ be the finite set of closed points of ${{{\mathscr{Y}}}}$ which are not in $U_\eta$ for any $\eta$. Let $A_{{\mathscr{P}}}$ be the regular semi-local ring at the closed points of ${{\mathscr{P}}}$. Since each $\alpha_i $ is unramified on ${{\mathscr{Y}}}$, there exists $\beta_i \in H^2_{\mathrm{\acute{e}t}}(A_{{\mathscr{P}}}, \mu_\ell^{\otimes 2})$ such that $\beta_i \otimes L = \alpha_i \otimes L$. Let $P \in {{\mathscr{P}}}$. Since the residue field $\kappa(P)$ at $P$ is a finite extension of $\kappa$, by the assumption on $\kappa$, there exists an extension $\mathfrak{s}(P)$ of $\kappa(P)$ of degree at most $N_2$ such that $\beta_i \otimes \mathfrak{s}(P) = 0$ for all $i$. Once again, by (\[lemma:sep\_ext\]), we assume that each $\mathfrak{s}(P)$ is a separable extension of $\kappa(P)$. Let $L_2$ be as in (\[lemma:crt\]). Then, as above, by (\[cor:crt\]), $\alpha_i \otimes (L_2 \otimes L_{P}) = 0$ for all $i$.

Let $L = LL_1L_2$. Then as in (\[theorem:bad\_char\]), $L \otimes \alpha_i = 0$ for all $i$. Since $[L : F] \leq \ell^3(N_1!)(N_2!)$, the theorem follows.

\[cor:good\_char\_2\] Let $K$, $\kappa$, $\ell$, $N_1$ and $N_2$ be as in (\[theorem:good\_char\]). Let $\zeta$ be a primitive $\ell^{\rm th}$ root of unity. Then $[K(\zeta), K]\ell^3(N_1!)(N_2!)$ is a $(2, \ell)$-uniform bound of $K(t)$.

The above corollary and (\[lemma:uniform\_bound\_d\_d+1\]) gives the following

\[cor:good\_char\_n\] Let $K$, $\kappa$, $\ell$, $N_1$ and $N_2$ be as in (\[theorem:good\_char\]). Let $\zeta$ be a primitive $\ell^{\rm th}$ root of unity. Then $[K(\zeta), K]\ell^3(N_1!)(N_2!)$ is an $(n, \ell)$-uniform bound of $K(t)$ for all $n \geq 2$.

 Symbol length and $u$-invariant
================================

\[theorem:symbol\_length1\] Let $K$ be a complete discretely valued field with residue field $\kappa$. Let $\ell$ be a prime not equal to char$(\kappa)$. Suppose that there exist integers $N_1$ and $N_2$ such that $\kappa(t)$ is $(2, \ell)$-uniformly bounded by $N_1$and $\kappa$ is $(2, \ell)$-uniformly bounded by $N_2$. Let $n \geq
  2$. Then there exists an integer $M_n$ which depends only on $N_1$, $N_2$ and $n$ such that for every finite extension $F$ of $K(t)$ and for all $\alpha
  \in H^n(F, \mu_\ell^{\otimes n})$, $\lambda(\alpha) \leq M_n$.

By (\[cor:good\_char\_2\]), $K(t)$ is $(2,
  \ell)$-uniformly bounded by $N = (\ell-1)\ell^3(N_1!)(N_2!)$. Hence any finite extension $F$ of $K(t)$ is also $(2, \ell)$-uniformly bounded by $N$. The theorem follows from (\[cor:symbol\_length\]).

\[theorem:symbol\_length2\] Let $K$ be a complete discretely valued field with residue field $\kappa$. Let $p = $ char$(\kappa)$. Suppose that char$(K) = 0$, $p > 0$ and $[\kappa : \kappa^d] = p^d$. Then there exists an integer $M$, which depends only on $d$ such that for any finite extension $F$ of $K(t)$ and for all $\alpha \in H^n(F, \mu_p^{\otimes n})$, $n
  \geq 1$, $\lambda(\alpha) \leq M$.

By (\[cor:bad\_char\_2\]), $K(t)$ is $(2,
  p)$-uniformly bounded by $(p-1)p^{4d + 4}$. Let $F$ be a finite extension of $K(t)$. Then $F$ is also $(2, p)$-uniformly bounded by $(p-1)p^{4d+4}$. Let $n \geq 1$. By (\[cor:symbol\_length\]), there exists an integer $N_n$, which depends only on $d$ and $n$ such that for all $\alpha \in H^n(F,   \mu_p^{\otimes n})$, $\lambda(\alpha) \leq N_n$. Since the $p$-cohomological dimension of $K$ is at most $d + 2$ ([@g-o]) and $F$ is a function field in one variable over $K$, the $p$-cohomological dimension of $F$ is $d +
  3$. Hence $H^n(F, \mu_p^{\otimes n}) = 0$ for all $n \geq
  d+4$. Let $N$ be the maximun of $N_n$ for $2 \leq n \leq d+4$. Then $\lambda(\alpha) \leq  N$ for all $\alpha \in H^n(F,
  \mu_p^{\otimes n})$ and $n \geq 2$.

\[theorem:u\_invariant\] Let $K$ be complete discretely valued field with residue field $\kappa$ and $F$ a function field of a curve over $K$. Suppose that char$(K)  = 0$, char$(\kappa) = 2$ and $[\kappa : \kappa^2]$ is finite. Then there exists an integer $M$ which depends only on $[\kappa :
\kappa^2]$ such that for any finite extension $F$ of $K(t)$, $u(F) \leq M$.

The theorem follows from (\[theorem:symbol\_length2\]) and (\[cor:uniformbound\_uinvariant\]).

We end with the following

Let $L$ be a field of characteristic not equal to 2 with $u(L)$ finite. Is the Brauer group of $L$ uniformly 2-bounded?

[10]{}

A. A. Albert, *Structure of Algebras*, Amer. Math. Soc. Colloq. Publ., Vol. 24, Amer. Math. Soc., Providence, RI, 1961, revised printing.

Asher Auel, Eric Brussel, Skip Garibaldi, Uzi Vishne, *Open problems on central simple algebras*, Transformation Groups [**16**]{} (2011), 219-264.

\] N. Bourbaki, *Algebra II*, Springer-Verlag, New York, 1988.

M. Cipolla, *Remarks on the lifting of algebras over henselian pairs*, Mathematische Zeitschrift, [**152**]{} (1977), 253–257.

Auslander Maurice and Goldman Oscar, *The Brauer group of a commutative ring*, Trans. Amer. Math. Soc. [**97**]{} (1960) 367–409.

O. Gabber and F. Orgogozo, *Sur la $p$-dimension des corps*, Invent. Math. [**174**]{} (2008), 47-80.

D. Harbater, J. Hartmann and D. Krashen, *Applications of patching to quadratic forms and centrals simple algebras*, Invent. Math. [**178**]{} (2009), 231-263.

D. Harbater, J. Hartmann and D. Krashen, *Local-global principles for torsors over arithmetic curves*, arxive:1108.3323v2

Heath Brown, D.R., *Zeros of systems of p-adic quadratic forms*, Compos. Math. [**146**]{} (2010), 271-287.

Kahn, B., *On "horizontal” invariants attached to quadratic forms*, Algebra and number theory, 21–33, Hindustan Book Agency, Delhi, 2005.

M.-A. Knus, A.S. Merkurjev, M. Rost and J.-P. Tignol, *The Book of Involutions*, A.M.S, Providence RI, 1998.

Krashen, D., *Period and index, symbol lengths, and generic splittings in galois cohomology*, preprint 2013.

Leep, D.B., *The u-invariant of p-adic function fields*, J. Reine Angew. Math., to appear

Milne,J.S., *Étale cohomology*, Princeton Univ. Press, Princeton, N.J., 1980,

Orlov, D., Vishik, A., Voevodsky, V., *An exact sequence for $K^M_\ast/2$ with applications to quadratic forms*, Ann. of Math. [**165**]{} (2007), no. 1, 1–13.

Parimala, R. and Suresh, V., *On the length of a quadratic form*, Algebra and number theory, 147–157, Hindustan Book Agency, Delhi, 2005.

Parimala, R. and Suresh, V., *The $u$-invariant of the function fields of $p$-adic curves*, Annals of Mathematics [**172**]{} (2010), 1391-1405.

Parimala, R. and Suresh, V., *Period-index and $u$-invariant questions for function fields over complete discretely valued fields*, to appear in Invent. Math.

Saltman, D.J., *Division Algebras over $p$-adic curves*, J. Ramanujan Math. Soc. [**12**]{} (1997), 25-47.

Saltman, D.J., *Finite $u$ Invariant and Bounds on Cohomology Symbol Lengths*, preprint, 2011.

Serre, J.-P., *Galois Cohomology*, Springer monographs in Mathematics, 1996.

Voevodsky, V., *On motivic cohomology with ${\mathbb Z}/l$-coefficients*, Ann. of Math. (2) 174 (2011), no. 1, 401–438.
