  This paper concerns the distribution of Selmer ranks in a family of even
Galois representations in even residual characteristic obtained by allowing
ramification at auxiliary primes. The main result is a Galois cohomological
analogue of a theorem of Friedlander, Iwaniec, Mazur and Rubin on the
distribution of Selmer ranks in a family of twists of elliptic curves. The
Selmer groups are constructed as prescribed by the Galois cohomological method
for GL(2): At each ramified place, the local Selmer condition is the tangent
space of a smooth quotient of the local deformation ring. By methods of global
class field theory, the Selmer group at the minimal level is computed
explicitly. The infinitude of primes for which the Selmer rank increases by one
is proved, and the density of such primes is shown to be 1/192. The proof
combines Wiles' formula and the global reciprocity law. The result has
implications for the algebraic structure of even deformation rings and the
distribution of their presentations in families.
