= Frobenius-corrected logarithm
{c}
{title2=$\mathcal L_f(t)=\frac1p\log\frac{f(t)^p}{f([\pi]t)}$}
Assume $K_\mathfrak p=\mathbb Q_p$, $p>2$, $f(t)\in\mathbb Z_p[[t]]^\times$, and $[\pi]t\equiv t^p\pmod p$. Then the displayed ratio lies in $1+p\mathbb Z_p[[t]]$, so its logarithm divided by $p$ is integral. A formal-group isomorphism over the completed unramified coefficient ring transfers it to an integral <Amice transform>. A norm relation makes the corresponding <p-adic measure> supported on units. Its $k$th moment has the Euler correction $1-\pi^k/p$.
Back to article page
