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Mathematics
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5 May 2023

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

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and
1
Department of Mathematics, Center for Basic Education and Integrated Learning, Kanagawa Institute of Technology, 1030 Shimo-Ogino, Atsugi 243-0292, Japan
2
Electrical Engineering and Computer Science, Graduate School of Natural Science & Technology, Kanazawa University, Kakuma, Kanazawa 920-1192, Japan
3
Department of Mathematics and Statistics, Boston University, Boston, MA 02215-2411, USA
*
Author to whom correspondence should be addressed.
This article belongs to the Special Issue Partial Differential Equations and Applications
The authors wish to make the following corrections to this paper []:
  • Text Correction:
There were nine errors in the original publication [], 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 ϖ r : X P ( ( x , y ) x ) is obviously a holomorphic r-sheeted covering. When we obtain the Galois group on X, i.e., Gal ( Q ( R X ) / Q ( R P ) ) = Aut ( X / P ) = Aut ( ϖ r ) , this is denoted by G X . The ϖ r is a finite branched covering. A ramification point of ϖ r is defined as a point that is not biholomorphic. The image ϖ r of the ramification point is called the branch point of ϖ r . The number of finite ramification points is denoted by 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 ϖ x = ϖ r : X P . G x denotes the finite group action on ϖ r 1 ( x ) for x 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 P 2 ( m X 1 ) , Lemma 9, 2:
2. for a group action  ζ ^ G x , h ~ R X ( x , ζ ^ y , ζ ^ y ) h ~ R X ( x , ζ ^ y , ζ ^ y ) = h ~ R X ( x , y , y ) 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 H 0 ( X , A X ( ) ) , Lemma 16, 1:
1. for the case  x = x and a group action ζ ^ G x , h ~ X ( x , ζ ^ y , ζ ^ y ) h ~ X ( x , ζ ^ y , ζ ^ y ) = h ~ X ( x , y , y ) 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 H 0 ( X , A X ( ) ) , Proposition 13, the second paragraph:
Further, this relation is extended to the condition  Q B X \ { }  by considering the multiplicity of the action  G ϖ r ( Q ) .
The sixth correction has been made to 3. W-Normalized Abelian Differentials on X, 3.3.1. The One-Form Σ on X, Proposition 15, 1:
1. For a group action  ζ ^ G ϖ r ( P ) , Σ ( ζ ^ P , ζ ^ Q ) = Σ ( P , Q ) if ϖ r ( P ) = ϖ r ( Q ) .
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  ζ G ϖ r ( P ) , Ω ( ζ P , ζ Q ) = Ω ( P , Q ) if ϖ r ( P ) = ϖ r ( Q ) .
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  P , for the Galois action  ζ ^ G X , i.e.,  ζ ^ : X X , its associated element  ρ ζ ^ of  Sp ( 2 g , Z )  acts on  ( ω , ω )  and  ( η , η )  by
ζ ^ ( ω , ω ) = ( ω , ω ) t ρ ζ ^ , ζ ^ ( η , η ) = ( η , η ) t ρ ζ ^ ,
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  ζ ^ G X satisfies  ζ ^ = id , and  ζ ^ [ σ ( u + ~ ) / σ ( u ) ] = σ ( u + ~ ) / σ ( u ) for  ~ Γ X and  u C g , the action provides the one-dimensional representation such that
ζ ^ σ ( u ) = ρ ζ ^ σ ( u ) ,
where  ρ ζ ^ = 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.

Reference

  1. Komeda, J.; Matsutani, S.; Previato, E. Algebraic construction of the sigma function for general Weierstrass curves. Mathematics 2022, 10, 3010. [Google Scholar] [CrossRef]
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