Abstract.We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations of the renormalization group. We first illustrate covariance properties on order-two linear differential operators associated with identities relating the same 2 F 1 hypergeometric function with different rational pullbacks. These rational transformations are solutions of a differentially algebraic equation that already emerged in a paper by Casale on the Galoisian envelopes. We provide two new and more general results of the previous covariance by rational functions: a new Heun function example and a higher genus 2 F 1 hypergeometric function example. We then focus on identities relating the same 2 F 1 hypergeometric function with two different algebraic pullback transformations: such remarkable identities correspond to modular forms, the algebraic transformations being solution of another differentially algebraic Schwarzian equation that also emerged in Casale's paper. Further, we show that the first differentially algebraic equation can be seen as a subcase of the last Schwarzian differential condition, the restriction corresponding to a factorization condition of some associated order-two linear differential operator. Finally, we also explore generalizations of these results, for instance, to 3 F 2 , hypergeometric functions, and show that one just reduces to the previous 2 F 1 cases through a Clausen identity. The question of the reduction of these Schwarzian conditions to modular correspondences remains an open question. In a 2 F 1 hypergeometric framework the Schwarzian condition encapsulates all the modular forms and modular equations of the theory of elliptic curves, but these two conditions are actually richer than elliptic curves or 2 F 1 hypergeometric functions, as can be seen on the Heun and higher genus example. This work is a strong incentive to develop more differentially algebraic symmetry analysis in physics.

We provide a set of diagonals of simple rational functions of four variables that are seen to be squares of Heun functions. Each time, these Heun functions, obtained by creative telescoping, turn out to be pullbacked 2 F 1 hypergeometric functions and in fact classical modular forms. We even obtained Heun functions that are automorphic forms associated with Shimura curves as solutions of telescopers of rational functions.

We recall that diagonals of rational functions naturally occur in lattice statistical mechanics and enumerative combinatorics. We find that a sevenparameter rational function of three variables with a numerator equal to one (reciprocal of a polynomial of degree two at most) can be expressed as a pullbacked 2 F 1 hypergeometric function. This result can be seen as the simplest nontrivial 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 2 F 1 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 generalise this result to eight, nine and ten parameters families adding some selected cubic terms at the denominator of the rational function defining the diagonal. We finally show that each of these previous rational functions yields an infinite number of rational functions whose diagonals are also pullbacked 2 F 1 hypergeometric functions and modular forms.

We show that the results we had previously obtained on diagonals of 9- and 10-parameter families of rational functions in three variables x, y, and z, using creative telescoping, yielding modular forms expressed as pullbacked 2F1 hypergeometric functions, can be obtained much more efficiently by calculating the j-invariant of an elliptic curve canonically associated with the denominator of the rational functions. These results can be drastically generalized by changing the parameters into arbitrary rational functions of the product p=xyz. In other cases where creative telescoping yields pullbacked 2F1 hypergeometric functions, we extend this algebraic geometry approach to other families of rational functions in three or more variables. In particular, we generalize this approach to rational functions in more than three variables when the denominator can be associated to an algebraic variety corresponding to products of elliptic curves, or foliations in elliptic curves. We also extend this approach to rational functions in three variables when the denominator is associated with a genus-two curve such that its Jacobian is a split Jacobian, corresponding to the product of two elliptic curves. We sketch the situation where the denominator of the rational function is associated with algebraic varieties that are not of the general type, having an infinite set of birational automorphisms. We finally provide some examples of rational functions in more than three variables, where the telescopers have pullbacked 2F1 hypergeometric solutions, because the denominator corresponds to an algebraic variety that has a selected elliptic curve.

We show that the unresolved examples of Christol's conjecture 3 F 2 ([2/9, 5/9, 8/9], [2/3, 1], x) and 3 F 2 ([1/9, 4/9, 7/9], [1/3, 1], x), are indeed diagonals of rational functions.We also show that other 3 F 2 and 4 F 3 unresolved examples of Christol's conjecture are diagonals of rational functions. Finally we give two arguments that show that it is likely that the 3 F 2 ([1/9, 4/9, 5/9], [1/3, 1], 27 · x) function is a diagonal of a rational function.