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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. [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.