  This work lies at an intersection of three subjects: quantum field theory,
algebraic geometry and number theory, in a situation where dialogue between
practitioners has revealed rich structure. It contains a theorem and 7
conjectures, tested deeply by 3 optimized algorithms, on relations between
Feynman integrals and L-series defined by products, over the primes, of data
determined by moments of Kloosterman sums in finite fields. There is an
extended introduction, for readers who may not be familiar with all three of
these subjects. Notable new results include conjectural evaluations of
non-critical L-series of modular forms of weights 3, 4 and 6, by determinants
of Feynman integrals, an evaluation for the weight 5 problem, at a critical
integer, and formulas for determinants of arbitrary size, tested up to 30
loops. It is shown that the functional equation for Kloosterman moments
determines much but not all of the structure of the L-series. In particular,
for problems with odd numbers of Bessel functions, it misses a crucial feature
captured in this work by novel and intensively tested conjectures. For the
9-Bessel problem, these lead to an astounding compression of data at the
primes.
