A criterion for modules over Gorenstein local rings to have rational Poincar\'e series

We prove that modules over an Artinian Gorenstein local ring $R$ have rational Poincar\'e series sharing a common denominator if $R/\soc(R)$ is a Golod ring. If $R$ is a Gorenstein local ring with square of the maximal ideal being generated by at most two elements, we show that modules over $R$ have rational Poincar\'e series sharing a common denominator. By a result of \c Sega, it follows that $R$ satisfies the Auslander-Reiten conjecture. We provide a different proof of a result of Rossi and \c Sega concerning rationality of Poincar\'e series of modules over compressed Gorenstein local rings. We also give a new proof of the fact that modules over Gorenstein local rings of codepth at most three have rational Poincar\'e series sharing a common denominator, which is originally due to Avramov, Kustin and Miller.