Correction: Komeda et al. Algebraic Construction of the Sigma Function for General Weierstrass Curves. Mathematics 2022, 10, 3010

The authors wish to make the following corrections to this paper [1]:

• Text Correction:

There were nine errors in the original publication [1], related to the incorrect use of the mathematical terminology Galois covering; this should be referred to as holomorphic r-sheeted covering.

The first correction has been made to 2. Weierstrass Canonical Form and Weierstrass Curves (W-curves), 2.5.1 Galois Covering, the second paragraph:

Following the above description, we consider the W-curve X. The covering ${\varpi }_r:X\to \mathbb{P}$$((x,y_{•})\mapsto x)$ is obviously a holomorphic r-sheeted covering. When we obtain the Galois group on X, i.e., $\mathrm{Gal}(\mathcal{Q}\left(R_X\right)/\mathcal{Q}\left(R_{\mathbb{P}}\right))=\mathrm{Aut}(X/\mathbb{P})=\mathrm{Aut}\left({\varpi }_r\right)$, this is denoted by $G_X$. The ${\varpi }_r$ is a finite branched covering. A ramification point of ${\varpi }_r$ is defined as a point that is not biholomorphic. The image ${\varpi }_r$ of the ramification point is called the branch point of ${\varpi }_r$. The number of finite ramification points is denoted by ${\ell }_{\mathcal{B}}$.

The second correction has been made to 2. Weierstrass Canonical Form and Weierstrass Curves (W-curves), 2.5.1. Galois Covering, the third paragraph:

We basically focus on the holomorphic r-sheeted covering ${\varpi }_x={\varpi }_r:X\to \mathbb{P}$. $G_x$ denotes the finite group action on ${\varpi }_r^{-1}\left(x\right)$ for $x\in \mathbb{P}$, referred to as group action at x in this paper.

The third correction has been made to 2. Weierstrass Canonical Form and Weierstrass Curves (W-curves), 2.5.3. Embedding of X into ${\mathbb{P}}^{2(m_X-1)}$, Lemma 9, 2:

2. for a group action $\widehat{\zeta }\in G_x$, ${\displaystyle \frac{{\tilde{h}}_{R_X}(x,\widehat{\zeta }{y}_{•},\widehat{\zeta }{y}_{•}^{\prime })}{{\tilde{h}}_{R_X}(x,\widehat{\zeta }{y}_{•},\widehat{\zeta }{y}_{•})}=\frac{{\tilde{h}}_{R_X}(x,y_{•},y_{•}^{\prime })}{{\tilde{h}}_{R_X}(x,y_{•},y_{•})}}$.

The fourth correction has been made to 3. W-Normalized Abelian Differentials on X, 3.1. W-Normalized Abelian Differentials ${\mathbf{H}}^0(X,{\mathcal{A}}_X(\ast \infty ))$, Lemma 16, 1:

1. for the case $x^{\prime }=x$ and a group action $\widehat{\zeta }\in G_x$, ${\displaystyle \frac{{\tilde{h}}_X(x,\widehat{\zeta }{y}_{•},\widehat{\zeta }{y}_{•}^{\prime })}{{\tilde{h}}_X(x,\widehat{\zeta }{y}_{•},\widehat{\zeta }{y}_{•})}=\frac{{\tilde{h}}_X(x,y_{•},y_{•}^{\prime })}{{\tilde{h}}_X(x,y_{•},y_{•})}}$.

The fifth correction has been made to 3. W-Normalized Abelian Differentials on X, 3.1. W-Normalized Abelian Differentials ${\mathbf{H}}^0(X,{\mathcal{A}}_X(\ast \infty ))$, Proposition 13, the second paragraph:

Further, this relation is extended to the condition $Q\in {\mathcal{B}}_X\backslash \{\infty \}$ by considering the multiplicity of the action $G_{{\varpi }_r\left(Q\right)}$.

The sixth correction has been made to 3. W-Normalized Abelian Differentials on X, 3.3.1. The One-Form $\Sigma$ on X, Proposition 15, 1:

1. For a group action $\widehat{\zeta }\in G_{{\varpi }_r\left(P\right)}$, $\Sigma (\widehat{\zeta }P,\widehat{\zeta }Q)=\Sigma (P,Q)$ if ${\varpi }_r\left(P\right)={\varpi }_r\left(Q\right)$.

The seventh correction has been made to 3. W-Normalized Abelian Differentials on X, 3.3.5. W-Normalized Differentials of the Second Kind, Theorem 3, 2 (b):

(b) for any $\zeta \in G_{{\varpi }_r\left(P\right)}$, $\Omega (\zeta P,\zeta Q)=\Omega (P,Q)$ if ${\varpi }_r\left(P\right)={\varpi }_r\left(Q\right)$.

The eighth correction has been made to 3. W-Normalized Abelian Differentials on X, 3.3.5. W-Normalized Differentials of the Second Kind, Lemma 30:

Lemma 30. If X is the Galois covering on $\mathbb{P}$, for the Galois action $\widehat{\zeta }\in G_X$, i.e., $\widehat{\zeta }:X\to X$, its associated element ${\rho }_{\widehat{\zeta }}$of $\mathrm{Sp}(2g,\mathbb{Z})$ acts on $({\omega }^{\prime },{\omega }^{″})$ and $({\eta }^{\prime },{\eta }^{″})$ by

$$\widehat{\zeta }({\omega }^{\prime },{\omega }^{″})=({\omega }^{\prime },{\omega }^{″}{)}^t{\rho }_{\widehat{\zeta }},\widehat{\zeta }({\eta }^{\prime },{\eta }^{″})=({\eta }^{\prime },{\eta }^{″}{)}^t{\rho }_{\widehat{\zeta }},$$

and the generalized Legendre relation (35) is invariant for the action.

The ninth correction has been made to 4. Sigma Function for W-curves, 4.3. Sigma Function and W-curves, Theorem 4, 7:

7. If $\widehat{\zeta }\in G_X$ satisfies ${\widehat{\zeta }}^{\ell }=\mathrm{id}$, and $\widehat{\zeta }[\sigma (u+\tilde{\ell })/\sigma (u)]=\sigma (u+\tilde{\ell })/\sigma (u)$ for $\tilde{\ell }\in {\Gamma }_X$ and $u\in {\mathbb{C}}^g$, the action provides the one-dimensional representation such that

$$\widehat{\zeta }\sigma \left(u\right)={\rho }_{\widehat{\zeta }}\sigma \left(u\right),$$

where ${\rho }_{\widehat{\zeta }}^{\ell }=1$.

The authors state that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated.