Lattice structure of modular vertex algebras

In this paper we study the integral form of the lattice vertex algebra $V_L$. Based on the lattice structure, we introduce and study the associated modular vertex algebras $V_p$ and the quotient algebra $overline{V}_p$ as well as their irreducible modules over $mathbb Z_p$. We show that divided powers of general vertex operators preserve the integral lattice spanned by Schur functions indexed by partition-valued functions. We also show that the Garland operators, counterparts of divided powers of Heisenberg elements in affine Lie algebras, also preserve the integral form. These construe analogs of the Kostant $mathbb Z$-forms for the enveloping algebras of simple Lie algebras and the algebraic affine Lie groups in the situation of the lattice vertex algebras. We also prove that the irreducible modules (modulo Heisenberg generators of degree divisible by $p$) remain irreducible for the modular vertex algebra $overline{V}_p$.