  We prove high-frequency modulational instability of small-amplitude Stokes
waves in deep water under longitudinal perturbations, providing the first isola
of unstable eigenvalues branching off from $\mathtt{i}\frac34$. Unlike the
finite depth case this is a degenerate problem and the real part of the
unstable eigenvalues has a much smaller size than in finite depth. By a
symplectic version of Kato theory we reduce to search the eigenvalues of a
$2\times 2$ Hamiltonian and reversible matrix which has eigenvalues with
non-zero real part if and only if a certain analytic function is not
identically zero. In deep water we prove that the Taylor coefficients up to
order three of this function vanish, but not the fourth-order one.
