On finite orbits of infinite correspondences
Manfred Buchacher
TL;DR
The paper analyzes algebraic correspondences on the projective line defined by an irreducible polynomial $p(X,Y)$, focusing on finite orbits under iteration. It establishes a field-theoretic criterion: the correspondence is finite if and only if $k(x) \cap k(y) \neq k$, and uses this to show that any infinite correspondence has only finitely many finite orbits, accompanied by a uniform bound on orbit size for finite correspondences. The key method combines algebraic geometry with valuation theory and a Fried-type argument: constructing a rational function $\Theta$ from finite orbits and showing that too many such orbits force a pole-free function on $C$, yielding a contradiction. The work concludes with open questions on determining a uniform upper bound for the size of a finite orbit in the general case and notes partial results in the literature.
Abstract
These notes collect results about algebraic correspondences and adapt them to the setting of correspondences on projective lines. The focus lies on finite orbits of algebraic correspondences. The main result is a field theoretic characterization of the (in)finiteness of the number of finite orbits.