Diagonals of rational functions, pullbacked ${}_2{F}_1$ hypergeometric functions and modular forms

We recall that diagonals of rational functions naturally occur in lattice statistical mechanics and enumerative combinatorics. We find that the diagonal of a seven parameter rational function of three variables with a numerator equal to one and a denominator which is a polynomial of degree at most two, can be expressed as a pullbacked hypergeometric function. This result can be seen as the simplest non-trivial family of diagonals of rational functions. We focus on some subcases such that the diagonals of the corresponding rational functions can be written as a pullbacked hypergeometric function with two possible rational functions pullbacks algebraically related by modular equations, thus showing explicitely that the diagonal is a modular form. We then generalize this result to nine and ten parameter families adding some selected cubic terms at the denominator of the rational function defining the diagonal. We show that each of these rational functions yields an infinite number of rational functions whose diagonals are also pullbacked hypergeometric functions and modular forms. (paper)