Algebraic Construction of the Sigma Function for General Weierstrass Curves

Jiryo Komeda, Shigeki Matsutani () and Emma Previato
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Jiryo Komeda: Department of Mathematics, Center for Basic Education and Integrated Learning, Kanagawa Institute of Technology, 1030 Shimo-Ogino, Atsugi 243-0292, Japan
Shigeki Matsutani: Electrical Engineering and Computer Science, Graduate School of Natural Science & Technology, Kanazawa University, Kakuma, Kanazawa 920-1192, Japan
Emma Previato: Department of Mathematics and Statistics, Boston University, Boston, MA 02215-2411, USA

Mathematics, 2022, vol. 10, issue 16, 1-31

Abstract: The Weierstrass curve X is a smooth algebraic curve determined by the Weierstrass canonical form, y r + A 1 ( x ) y r − 1 + A 2 ( x ) y r − 2 + ⋯ + A r − 1 ( x ) y + A r ( x ) = 0 , where r is a positive integer, and each A j is a polynomial in x with a certain degree. It is known that every compact Riemann surface has a Weierstrass curve X , which is birational to the surface. The form provides the projection ϖ r : X → P as a covering space. Let R X : = H 0 ( X , O X ( ∗ ∞ ) ) and R P : = H 0 ( P , O P ( ∗ ∞ ) ) . Recently, we obtained the explicit description of the complementary module R X c of R P -module R X , which leads to explicit expressions of the holomorphic form except ∞ , H 0 ( P , A P ( ∗ ∞ ) ) and the trace operator p X such that p X ( P , Q ) = δ P , Q for ϖ r ( P ) = ϖ r ( Q ) for P , Q ∈ X \ { ∞ } . In terms of these, we express the fundamental two-form of the second kind Ω and its connection to the sigma function for X .

Keywords: Weierstrass canonical form; fundamental two-form of the second kind; sigma function; plane and space curves with the higher genera (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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