Computations of Sha for maximal tori of \(\mathrm{GSp}\) split by a Galois CM-field when all decomposition groups are cyclic.

Note that this serves as a bound for regular Tate-Shafarevich group.

The result only depends on the \(2\)-Sylow subgroups of the Galois group, so we restrict the description to \(2\)-groups, and the Tate-Shafarevich group is trivial when the \(2\)-Sylow subgroups have order \(4\) or less.

Also note that we know how to compute the Tate-Shafarevich group exactly when the Galois groups is abelian.

Below are complete computations are done for nonabelian Galois groups up to order \(128\). The page for groups of order \(256\) only contains the first \(29631\) groups.

• order 8.
• order 16.
• order 32.
• order 64.
• order 128.
• order 256 part 1.
• order 256 part 2.

Computations of \(|H^1(\mathbb{Q}, \mathbf{X}^\star(\mathbf{T}))|\) and Sha_C for maximal tori of \(\mathrm{GSp}\) split by a CM-field of degree 8: Here.

• modules/glattice.py.
• modules/glattice_morphism.py.
• modules/glattice_homspace.py.
• schemes/groups_chemes/tori.py.