Poincaré series of modules over compressed Gorenstein local rings

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Given positive integers e and s we consider Gorenstein Artinian local rings R of embedding dimension e whose maximal ideal m satisfies mˢ≠ 0=mˢ⁺¹. We say that R is a compressed Gorenstein local ring when it has maximal length among such rings. It is known that generic Gorenstein Artinian algebras are compressed. If s≠ 3, we prove that the Poincare series of all finitely generated modules over a compressed Gorenstein local ring are rational, sharing a common denominator. A formula for the denominator is given. When s is even this formula depends only on the integers e and s. Note that for s=3 examples of compressed Gorenstein local rings with transcendental Poincare series exist, due to Bøgvad.