On functional equations for meromorphic functions and applications

Lower bounds for genera of fiber products

“…Thus, the functional equation (2), where X, Y are allowed to be entire or meromorphic functions, often studied in the context of Nevanlinna theory (see e.g. [4], [24], [32], [38], [64]), is also related to the low genus problem (see e. g. [7], [33], [44], [45]). Second, algebraic curves (1) with factors of genus zero or one have special Diophantine properties.…”

“…As elements of MpEq separate points of E, equality (32) implies that for every z 0 P R that is not a critical value of V the map h takes k distinct values on the set V ´1tz 0 u. Since h P U ˚MpTq, this implies in turn that the map U takes k distinct values on V ´1tz 0 u.…”