Problems on Abelian Functions at the Time of Poincaré and Some at Present by Jun-ichi Igusa
نویسنده
چکیده
1. Abelian functions by Poincaré. 1-1. If the variable x and a general solution y of a linear differential equation with polynomial coefficients are algebraically dependent, the periods of abelian integrals of the first kind associated with the curve f(x, y) = 0 satisfy certain relations. In his earliest works on abelian functions Poincaré examined such relations in some special cases. He also used a similar relation in a joint paper with Picard of 1883 on a "theorem of Riemann". Poincaré later developed a general theory of reducible integrals. This theory played some role in almost all of his works on abelian functions. We shall start by recalling the theorem of Riemann: There are three related theorems concerning a complex torus. If ƒ is a meromorphic function on C, an element a of C such that f(z + a) = f(z) for a variable z in C* is called a period off; the set of all periods of ƒ forms a closed subgroup of C, called the period group of ƒ. Let A denote a lattice in C*, i.e., a discrete subgroup of C with compact quotient; then a meromorphic function ƒ on C whose period group contains A, i.e., a meromorphic function on the complex torus T = C*/A considered as a function on C, is called an abelian function relative to A and a holomorphic function ® on C with the property
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