  The terms in the quantum scattering in scalar field theory models is
parameterized by the invariants $\prod s_{ij}^{n_{ij}}$. The $s_{ij}$ are
kinematic two-particle invariants, and the $n_{ij}$ are integers. The
coefficients of these terms are computed via integrating all Feynman diagrams,
or within the derivative expansion by solving the iteration equations. The
latter has been provided recently; the functions which are prefactors of the
individual terms $\prod s_{ij}^{n_{ij}}$ can be interpreted as terms in the
expansions of L-series, which may be specified by collections of their zeroes.
Once finding the appropriate elliptic curve coefficients, these quantum field
solutions provide an algorithm to determining all of the mod p zeros to the
algebraic curves. The latter is presumably determined by 'experimental'
computer modeling or by the appropriate determination of the quantum
prefactors.
