Shigeki MATSUTANI

Title: Sigma function of y^3=x^2(x-b_1)(x-b_2)

Abstract:
Generalization of Weierstrass sigma function to a general affine curve
was started by Weierstrass himself.
Recently its study is re-evaluated and we have developed the sigma
function theory of a general affine curve from modern mathematical viewpoints.
The generalized sigma function is a very nice function which mediates between
algebraic and analytic properties of the meromorphic functions over
the algebraic curves and their Jacobian.
Recently we constructed the sigma function of an affine curve C_0 of
y^3=x^2(x-b_1)(x-b_2). Since  the sigma function for the non-degenerate
C_s y^3=x (x-s)(x-b_1)(x-b_2) has been also studied,
it is natural to consider the properties of the sigma function for the limit s to 0 of C_s.
In this talk, I give an introduction of the sigma functions with the purpose
of our study. After I show the sigma functions of C_0 and C_s (s\neq 0),
I would give some results of the limit if could.