At the place over $t=\infty$, intrinsic regularity is regularity of the transported differential at $(0,0)$

Let $m\ge1$ be an integer and $\alpha\in\mathbb C$ with $\alpha\ne\bar\alpha$. Put $h_\alpha(t)=-t(t^m+1)(\alpha t^m+\bar\alpha)\in\mathbb C[t]$ and let $K_\alpha=\mathbb C(t)[W]/(W^2-h_\alpha(t))$, with $w$ the class of $W$, so that $w^2=h_\alpha(t)$; this is the function field of the double cover of the $t$-line in equation (7) of the note, written with $w=t(t^m+1)Y$. Let $\Omega_{K_\alpha/\mathbb C}$ be its module of Kähler differentials, with universal derivation $d$.

For the conjugate parameter, $K_{\bar\alpha}$ is built in the same way, with coordinate $s$ and root $w'$, where $w'^2=h_{\bar\alpha}(s)=-s(s^m+1)(\bar\alpha s^m+\alpha)$. Since $h_{\bar\alpha}(0)=0$, the point $(0,0)$ lies on the curve $w'^2=h_{\bar\alpha}(s)$; its local ring $\mathcal O'_{(0,0)}\subset K_{\bar\alpha}$, the localization of $\mathbb C[s][W']/(W'^2-h_{\bar\alpha})$ at the maximal ideal of $(0,0)$, is a valuation ring of $K_{\bar\alpha}$. Let $\Phi\colon K_\alpha\to K_{\bar\alpha}$ be the ring isomorphism with $\Phi(t)=1/s$ and $\Phi(w)=w'\,s^{-(m+1)}$ (the chart at $t=\infty$); it is $\mathbb C$-linear. Let $\Phi_\ast\colon\Omega_{K_\alpha/\mathbb C}\to\Omega_{K_{\bar\alpha}/\mathbb C}$ be the induced map on differentials, $\Phi_\ast(g\,dk)=\Phi(g)\,d\Phi(k)$, and let $\mathcal O_\infty=\{\xi\in K_\alpha:\ \Phi(\xi)\in\mathcal O'_{(0,0)}\}$ be the place of $K_\alpha$ over $t=\infty$.

For a valuation ring $\mathcal O$ of $K_\alpha$ (a subring with $\xi\in\mathcal O$ or $\xi^{-1}\in\mathcal O$ for every nonzero $\xi\in K_\alpha$), write $\operatorname{Reg}(\mathcal O)$ for the space of $\mathbb C$-linear combinations of differentials $a\,db$ with $a,b\in\mathcal O$; $\mathcal O_\infty$ is such a ring. A differential $\eta\in\Omega_{K_{\bar\alpha}/\mathbb C}$ is regular at $(0,0)$ if $f\in\mathcal O'_{(0,0)}$ whenever $u\in\mathcal O'_{(0,0)}$ generates the maximal ideal of $\mathcal O'_{(0,0)}$ and $\eta=f\,du$ with $f\in K_{\bar\alpha}$.

Then, for every $\omega\in\Omega_{K_\alpha/\mathbb C}$,

The right-hand side is the explicit definition of regularity at infinity used for the holomorphic differentials of the family's double cover. Together with the corresponding statement at the finite points, it identifies the intrinsically defined holomorphic differentials of $K_\alpha$ with the explicitly computed ones, in the genus computation of Lemma 4 that is used in Remark 5.

Formalization Note: the hypotheses $m>0$ and $\alpha\ne\bar\alpha$ are Fact instances; $|\alpha|=1$ is not assumed.