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A remarkable theorem of Feigin and Frenkel states that, over the complexnumbers, the centre of the universal affine vertex algebra at the criticallevel is an infinite rank polynomial algebra, canonically identified with thealgebra of regular functions on the space of opers on the disk. The firstauthor and W. Wang observed that in positive characteristics, the universalaffine vertex algebra contains a large central subalgebra known as the$p$-centre. They conjectured that the centre should be generated by theFeigin--Frenkel centre and the $p$-centre, analogous to Veldkamp's theorem. Inthis paper we prove the conjecture for classical simple Lie algebras for $p$larger than the Coxeter number, and for exceptional Lie algebras in largecharacteristics.

The centre of the modular affine vertex algebra at the critical level