Invariants of the diagonal Cp-action on V3

Let Cp denote the cyclic group of order p where p 3 is prime. We denote by V3 the indecomposable three-dimensional representation of Cp over a field F of characteristic p. We compute a set of generators, in fact a SAGBI basis, for the ring of invariants F[V3 V3]Cp. Our main result confirms the conjecture of Shank [R.J. Shank, Classical covariants and modular invariants, in: H.E.A. Campbell, D.L. Wehlau (Eds.), Invariant Theory in All Characteristics, CRM Proc. Lecture Notes, vol. 35, Amer. Math. Soc., 2004, pp. 241–249], for this example, that all modular rings of invariants of Cp are generated by rational invariants, norms and transfers.