The authors wish to make the following corrections to this paper [
1]:
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 is obviously a holomorphic r-sheeted covering. When we obtain the Galois group on X, i.e., , this is denoted by . The is a finite branched covering. A ramification point of is defined as a point that is not biholomorphic. The image of the ramification point is called the branch point of . The number of finite ramification points is denoted by .
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 . denotes the finite group action on for , 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 , Lemma 9, 2:
2. for a group action , .
The fourth correction has been made to 3. W-Normalized Abelian Differentials on X, 3.1. W-Normalized Abelian Differentials , Lemma 16, 1:
1. for the case and a group action , .
The fifth correction has been made to 3. W-Normalized Abelian Differentials on X, 3.1. W-Normalized Abelian Differentials , Proposition 13, the second paragraph:
Further, this relation is extended to the condition by considering the multiplicity of the action .
The sixth correction has been made to 3. W-Normalized Abelian Differentials on X, 3.3.1. The One-Form on X, Proposition 15, 1:
1. For a group action , if .
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 , if .
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 , for the Galois action , i.e., , its associated element of acts on and byand 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 satisfies , and for and , the action provides the one-dimensional representation such thatwhere .
The authors state that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated.