  We investigate certain arithmetic properties of field theories. In
particular, we study the vacuum structure of supersymmetric gauge theories as
algebraic varieties over number fields of finite characteristic. Parallel to
the Plethystic Programme of counting the spectrum of operators from the
syzygies of the complex geometry, we construct, based on the zeros of the
vacuum moduli space over finite fields, the local and global Hasse-Weil zeta
functions, as well as develop the associated Dirichlet expansions. We find
curious dualities wherein the geometrical properties and asymptotic behaviour
of one gauge theory is governed by the number theoretic nature of another.
