  We investigate in detail a homomorphism which we call the 2-Selmer signature
map from the $2$-Selmer group of a number field $K$ to a nondegenerate
symmetric space, in particular proving the image is a maximal totally isotropic
subspace. Applications include precise predictions on the density of fields $K$
with given narrow class group 2-rank and with given unit group signature rank.
In addition to theoretical evidence, extensive computations for totally real
cubic and quintic fields are presented that match the predictions extremely
well. In an appendix with Richard Foote, we classify the maximal totally
isotropic subspaces of orthogonal direct sums of two nondegenerate symmetric
spaces over perfect fields of characteristic 2 and derive some consequences,
including a mass formula for such subspaces.
