The centre of the modular affine vertex algebra

The Feigin--Frenkel theorem states that, over the complex numbers, the centre of the universal affine vertex algebra at the critical level is an infinite rank polynomial algebra. The first author and W.~Wang observed that in positive characteristics, the universal affine vertex algebra contains a large central subalgebra known as the $p$-centre. They conjectured that at the critical level the centre should be generated by the Feigin--Frenkel centre and the $p$-centre. In this paper we prove the conjecture for classical simple Lie algebras for $p$ larger than the Coxeter number, and for exceptional Lie algebras in large characteristics. Finally, we give an example which shows that at non-critical level the center is larger than the $p$-centre.