p -adic vertex operator algebras.

We postulate axioms for a chiral half of a nonarchimedean 2-dimensional bosonic conformal field theory, that is, a vertex operator algebra in which a p -adic Banach space replaces the traditional Hilbert space. We study some consequences of our axioms leading to the construction of various examples, including p -adic commutative Banach rings and p -adic versions of the Virasoro, Heisenberg, and the Moonshine module vertex operator algebras. Serre p -adic modular forms occur naturally in some of these examples as limits of classical 1-point functions.