GL2ROrbitClosure enhancements

[videlec]

• starting from a guess (absolute) tangent space $V \subset H^1(M; \mathbb{R})$ there are two purely algebraic ingredients that could be used to make it bigger
  • the field of definition must be totally real. So one could pick the largest totally real subfield of the field of definition of $V$ and use it as a starting field. (question: (efficient) computation of this field?)
  • the final $V_\infty$ must satisfy that for any field automorphism of $\overline{\mathbb{Q}}$ either $V_\infty = V_\infty^\sigma$ or $V_\infty$ and $V_\infty^\sigma$ are disjoint. So if we find a $\sigma$ such that $V \cap V^\sigma$ is a non-trivial proper subspace of $V$ then we can start with $V' = V + V^\sigma$.
• if we deal with unfoldings, one can use the $\pi_1$ of configuration spaces acting on cohomology (see also Use symmetries in GL(2,R)-orbit closure computation #35).

[videlec]

Note: to compute the largest totally real sub-field of a given number field, Aurel said that it is almost in PARI/GP. What is there is about CM fields but could be modified to handle totally real. A reasonable option seems to enumerate all sub-fields and pick the maximal degree totally real one.