% LaTeX class specification.  Any changes will not appear in the final version
\documentclass[12pt,letterpaper, reqno]{aimpl}

% The problem list should be arranged like a research paper, with
% introductory material followed be one or more main sections.
% Do not divide the sections into subsections.
% Each section could have its own introductory material.
% The main part of each section is a "problem block" which has the format
% [[see elsewhere for more documentation]]

% New macros.
% If at all possible, please do not define any new macros.  Each
% macro you introduce has the potential to cause problems with the
% long-term maintenance of the problem list.

% removed \n from join and moved it to MACROS in app.js
\newcommand{\Cat}{{\rm Cat}}
\newcommand{\A}{\mathcal A}
\newcommand{\freestar}{ \framebox[7pt]{$\star$} }

\begin{document}
\title{Deformation theory and the Brauer group}
\author{Edited by Daniel Krashen and Max Lieblich}

\urlstub{deformationbrauer}

\maketitle

This problem list begin during discussions at the AIM workshop “Deformation theory, patching, quadratic forms, and the Brauer group” in January, 2011.


\section{The $u$-invariant problem}

The problems in this section are related to the u-invariant problem on the maximal dimension of an anisotropic quadratic form.





\begin{problemblock}

\begin{problem}[.1] If $u(F) < \infty$, does there exist $B \in \N$ such that $u(F') <
  \infty$ for all $F'/F$ finite? In particular, is $B = 2u(F)$
  sufficient?
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.6] When is $\tau_F$ finite? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.2] Find interesting classes $\mathcal F$ of fields such that for $F \in
  \mathcal F$,
\[u(F) < \infty \Longrightarrow u(F(t)) < \infty.\] \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.7] When $F = \R$ , we have $u(F) = \infty$ but $\tau_F < \infty$. Is there
  an example where this holds for $F$ not formally real?
 \end{problem}




\end{problemblock}



\begin{problemblock}Define \[u_k(F) = \max \{ \dim (q)\ \ |\ \ q \in I^k \}.\]
Then $u_0 = u$.


\begin{problem}[.3] Can we compute $u_k(F)$? How about $u_3(F)$?
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.8] Find reasonable classes $\mathcal F$ of fields such that $u(F)$ is a
  power of 2 for all $F \in \mathcal F$. For these fields, assume $k$ is the
  period-index bound for $l=2$. Does this imply $u(F) = 2^{k+1}$?
 \end{problem}




\begin{remark}I think I've seen this somewhere before\end{remark}

\end{problemblock}



\begin{problemblock}Let \[\nu(F) = \max \{n\ |\ I^n \neq 0 \}. \]


\begin{problem}[.4] If $\nu(F) < \infty$, is it true that $\nu(F') < \nu(F) +1$ for all
$F'/F$ finite?  \end{problem}




\begin{remark}$2^{\nu(F)} < u(F)$\end{remark}

\begin{remark}This is related to the computation of Galois groups of
  quadratically closed fields.\end{remark}

\end{problemblock}



\begin{problemblock}For a quadratic form $q$ over $F$, we define
\[\textrm{splitting degree} = \min\{[L:F]\ :\ q_{L}\ \textrm{is a
    direct sum of hyperbolics}\}.\]
The \emph{torsion index} of $F$, denoted $\tau_F$ is the maximum
splitting degree, taken over all even dimensional $q$.


\begin{problem}[.5] Is $\tau_F < 2^{(\frac{u(F)}{2} -1)}$?
 \end{problem}




\end{problemblock}



\section{The Brauer group}





\begin{problemblock}

\begin{problem}[.1] Fix $d \geq 3$. Assume that we know:

$\operatorname{ind} (\alpha) |   \operatorname{Per}(\alpha)^{d-1}$, for all $\alpha \in \operatorname{Br}(X)$, for
  all smooth, projective $X$ over $\overline{\mathbb{F}_p}$. (Here $d
  = \dim (X)$.)

Can we use this, by a boundedness argument, to show
the same thing for $\alpha \in \operatorname{Br}(Y)$, where $Y$ smooth, projective,
$\dim(Y) = d$ over $\C$?
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.2] Is every Brauer class over $\C(s,t)$ cyclic? \end{problem}




\begin{remark}This type of question does not reduce to the prime
case.
\end{remark}

\end{problemblock}



\begin{problemblock}A variant of Problem 2.2:

\begin{problem}[.3] Higher cohomology $H^d(F, \pmb{\mu}_n^{\otimes d})$,
where $d = \dim (F)$.
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.4] When is $\operatorname{Br}(X) = \operatorname{Br'}(X)$?
 \end{problem}




\begin{remark}If $X$ is quasi-proj, then yes.
\end{remark}

\end{problemblock}



\section{Testing section}

You can use this section for testing, or we can just delete it.

Rename it to something useful if you need another section.



\begin{problemblock}
\name{Name of the problem block}
Intro to problem block

\begin{conjecture}[3.1] This is the problem statement. \end{conjecture}




\begin{remark}This is a remark.\end{remark}

\end{problemblock}



\begin{problemblock}Test introduction

\begin{problem}[3.2] the statement of a throw-away problem \end{problem}




\end{problemblock}





\begin{thebibliography}{99}

\end{thebibliography}


\end{document}
