  Single-scale Feynman diagrams yield integrals that are periods, namely
projective integrals of rational functions of Schwinger parameters. Algebraic
geometry may therefore inform us of the types of number to which these
integrals evaluate. We give examples at 3, 4 and 6 loops of massive Feynman
diagrams that evaluate to Dirichlet $L$-series of modular forms and examples at
6, 7 and 8 loops of counterterms that evaluate to multiple zeta values or
polylogarithms of the sixth root of unity. At 8 loops and beyond, algebraic
geometry informs us that polylogs are insufficient for the evaluation of terms
in the beta-function of $\phi^4$ theory. Here, modular forms appear as
obstructions to polylogarithmic evaluation.
