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We recall that diagonals of rational functions naturally occur in latticestatistical mechanics and enumerative combinatorics. We find that aseven-parameter rational function of three variables with a numerator equal toone (reciprocal of a polynomial of degree two at most) can be expressed as apullbacked 2F1 hypergeometric function. This result can be seen as the simplestnon-trivial family of diagonals of rational functions. We focus on somesubcases such that the diagonals of the corresponding rational functions can bewritten as a pullbacked 2F1 hypergeometric function with two possible rationalfunctions pullbacks algebraically related by modular equations, thus showingexplicitely that the diagonal is a modular form. We then generalise this resultto eight, nine and ten parameters families adding some selected cubic terms atthe denominator of the rational function defining the diagonal. We finally showthat each of these previous rational functions yields an infinite number ofrational functions whose diagonals are also pullbacked 2F1 hypergeometricfunctions and modular forms.

Diagonals of rational functions, pullbacked 2F1 hypergeometric functions and modular forms (unabrigded version)