Herbrand function (${\phi }_{L/K}(s)={\int }_0^s[G_0:G_t{]}^{-1}dt$)

The Herbrand function is a continuous strictly increasing piecewise linear function on the nonnegative reals. Its inverse is denoted ${\psi }_{L/K}$. It reparametrizes the lower ramification numbering to the upper ramification numbering.

The upper numbering is obtained by applying the inverse Herbrand function to the index in the lower ramification numbering. It is compatible with quotient Galois groups, whereas lower numbering is compatible with subgroups.

For a normal subgroup $H$ of a finite local Galois group, the upper ramification numbering on its quotient is obtained by projecting the upper groups. The Herbrand function is the change of variable that ensures this compatibility. In contrast, the lower ramification numbering restricts directly to subgroups.