Decidability of theories of modules over tubular algebras

We show that the common theory of all modules over a tubular algebra (over a recursive algebraically closed field) is decidable. This result supports a long standing conjecture of Mike Prest which says that a finite-dimensional algebra (over a suitably recursive field) is tame if and only if its common theory of modules is decidable (Prest, Model theory and modules (Cambridge University Press, Cambridge, 1988)). Moreover, as a corollary, we are able to confirm this conjecture for the class of concealed canonical algebras over algebraically closed fields. Tubular algebras are the first examples of non-domestic algebras which have been shown to have decidable theory of modules. We also correct results in Harland and Prest (Proc. Lond. Math. Soc. (3) 110 (2015) 695–720), in particular, Corollary 8.8 of that paper.