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1. 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.


    1. Add remark

      Problem 1.1.

      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?


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          Problem 1.6.

          When is $\tau_F$ finite?


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              Problem 1.2.

              Find interesting classes $\mathcal F$ of fields such that for $F \in \mathcal F$, \[u(F) < \infty \Longrightarrow u(F(t)) < \infty.\]
                  Examples to try:

              \[ \mathcal F = \{F \ \ :\ \ |F^{\times}/F^{\times 2}| < \infty \} \] and \[ \mathcal F’ = \{F \ \ : \ \ F^{\times} = F^{\times 2} \}\]


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                  Problem 1.7.

                  When $F = \R$ , we have $u(F) = \infty$ but $\tau_F < \infty$. Is there an example where this holds for $F$ not formally real?

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

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                      Problem 1.3.

                      Can we compute $u_k(F)$? How about $u_3(F)$?
                          Note that forms in $I^3$ have trivial Clifford invariant.


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                          Problem 1.8.

                          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 $u(F) = 2^{k+1}$?
                            1. Remark. I think I’ve seen this somewhere before
                                •     Let \[\nu(F) = \max \{n\ |\ I^n \neq 0 \}. \]

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                                  Problem 1.4.

                                  If $\nu(F) < \infty$, is it true that $\nu(F') < \nu(F) +1$ for all $F'/F$ finite?
                                    1. Remark. $2^{\nu(F)} < u(F)$
                                        • Remark. [org.aimpl.user:c] This is related to the computation of Galois groups of quadratically closed fields.
                                            •     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 torsion index of $F$, denoted $\tau_F$ is the maximum splitting degree, taken over all even dimensional $q$.

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                                              Problem 1.5.

                                              Is $\tau_F < 2^{(\frac{u(F)}{2} -1)}$?

                                                  Cite this as: AimPL: Deformation theory and the Brauer group, available at http://aimpl.org/deformationbrauer.