Invariants of the diagonal CpCp-action on V3V3

Let CpCp denote the cyclic group of order p   where p⩾3p⩾3 is prime. We denote by V3V3 the indecomposable three-dimensional representation of CpCp over a field F of characteristic p  . We compute a set of generators, in fact a SAGBI basis, for the ring of invariants FCp[V3⊕V3]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 CpCp are generated by rational invariants, norms and transfers.