DIAGONALS OF RATIONAL FUNCTIONS: FROM DIFFERENTIAL ALGEBRA TO EFFECTIVE ALGEBRAIC GEOMETRY

: We show that the results we had obtained on diagonals of nine and ten parameters families


Introduction
In a previous paper [1,2], using creative telescoping [3], we have obtained diagonals 1 of nine and ten parameters families of rational functions, given by (classical) modular forms expressed as pullbacked 2 F 1 hypergeometric functions [12].The natural emergence of diagonals of rational functions2 in lattice statistical mechanics is explained in [19,20].This can be seen as the reason of the frequent occurrence of modular forms, Calabi-Yau operators in lattice statistical mechanics [21][22][23][24][25][26][27].In another previous paper [17,18], dedicated to Heun functions that are diagonals of simple rational functions, or only solutions of telescopers [28,29] of simple rational functions of three variables, but most of the time four variables, we have obtained many solutions of order-three telescopers having squares of Heun functions as solutions that turn out to be squares of pullbacked 2 F 1 hypergeometric solutions corresponding to classical modular forms and even Shimura automorphic forms [30,31], strongly reminiscent of periods of extremal rational surfaces [32,33], and other foliation of K3 surfaces in elliptic curves.In other words one finds experimentally that the 2 F 1 hypergeometric functions emerging in the calculation of diagonal of rational functions, or of solutions of the telescopers of rational functions, seem to be only special 2 F 1 ([a, b], [c], x) hypergeometric functions with a selected set of parameters [a, b], [c] (see the list (B.1) in Appendix B of [17], corresponding to classical modular forms 3 , together with a finite set of parameters, like [7/24, 11/24], [5/4], corresponding to Shimura automorphic forms [30,31]), pullbacked by selected pullbacks.This last paper [17] also underlined the difference between the diagonal of a rational function Diag(R), and the solutions of the telescoper of the same rational function.
These results strongly suggested to find an algebraic geometry interpretation for all these exact results, and, more generally, suggested to provide an alternative algebraic geometry approach of the results emerging from creative telescoping 4 .This is the purpose of the present paper.In particular, we are going to show that most of these pullbacked 2 F 1 hypergeometric functions can be obtained efficiently through algebraic geometry calculations, thus providing a more intrinsic algebraic geometry interpretation of the creative telescoping calculations which are typically differential algebra calculations [28,29,34,35].
Creative telescoping [28,29,34,36] is a methodology to deal with parametrized symbolic sums and integrals that yields differential/recurrence equations for such expressions.This methodology became popular in computer algebra in the past twenty five years.By "telescoper" of a rational function, say R(x, y, z), we here refer to the output of the creative telescoping program [3], applied to the transformed rational function R = R(x/y, y/z, z)/(yz).Such a telescoper is a linear differential operator T in x and ∂ ∂x , such that T + ∂U ∂y + ∂V ∂z annihilates R, where the so-called "certificates" U, V are rational functions in x, y, z.In other words, the telescoper T represents a linear ODE that is satisfied by Diag(R).
The paper is essentially dedicated to solutions of telescopers of rational functions which are not necessarily diagonals of rational functions.These solutions correspond to periods [37] of algebraic varieties over some cycles which are not necessarily vanishing [38] cycles 5like in the case of diagonals of rational functions.The reader interested in the connection between the process of taking diagonals, calculating telescopers, and the notion of Periods, deRham cohomology (i.e.differential forms) and other Picard-Fuchs equations can read in detail the thesis of Pierre Lairez [35] (see also [41]).We just sketch some of these ideas in A.
The purpose of this paper is not to give an introduction on creative telescoping [28,29], but to provide many pedagogical (non-trivial) examples of telescopers using 6 extensively the "HolonomicFunctions" Mathematica package [3].
The paper is organised as follows.We first recall in section 2 the exact results of [1,2] for nine and ten parameters families of rational functions using creative telescoping, yielding modular forms expressed as pullbacked 2 F 1 hypergeometric functions.We show in section 3 that these exact results can be obtained, much more efficiently, by calculating the j-invariant of an elliptic curve canonically associated with the denominator of the rational functions, and we underline the fact that one can drastically generalise these results, the parameters becoming quite arbitrary rational functions.Section 4 generalises the previous calculations to denominators of the rational functions of more than three variables, corresponding to products (or foliations) of elliptic curves.In section 5 we show how modular forms expressed as pullbacked 2 F 1 hypergeometric functions occur for 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.In section (6) we sketch the situation where the denominator of the rational function is associated with algebraic varieties of low Kodaira dimension, 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 2 F 1 hypergeometric solutions, the denominator corresponding to an algebraic variety having a selected elliptic curve in the variety explaining these pullbacked 2 F 1 solutions.

Classical modular forms and diagonals of nine and ten parameters family of rational functions
In a previous paper [1,2], using creative telescoping [3], we have obtained diagonals of nine and ten parameters families of rational functions, given by (classical) modular forms expressed as pullbacked 2 F 1 hypergeometric functions.Let us recall these results.
2.1.Nine-parameters rational functions giving pullbacked 2 F 1 hypergeometric functions for their diagonals Let us recall the nine-parameters rational function in three variables x, y and z: (1) Calculating 7 the telescoper of this rational function (1), one gets an order-two linear differential operator annihilating the diagonal of the rational function (1).The diagonal of the rational function (1) can be written [1,2] as a pullbacked hypergeometric function where P 4 (x) and P 6 (x) are two polynomials of degree four and six in x, respectively.The Hauptmodul pullback in (2) has the form where 2.2.Ten-parameters rational functions giving pullbacked 2 F 1 hypergeometric functions for their diagonals.
Let us recall the ten-parameters rational function in three variables x, y and z: Calculating the telescoper of this rational function (4), one gets an order-two linear differential operator annihilating the diagonal of the rational function (4).The diagonal of the rational function ( 4) can be written [1,2] as a pullbacked hypergeometric function where P 3 (x) and P 6 (x) are two polynomials of degree three and six in x, respectively.Furthermore, the Hauptmodul pullback in ( 5) is seen to be of the form: where P 9 (x) is a polynomial of degree nine in x.Again, (5) corresponds to a classical modular form [1,2].

Deducing creative telescoping results from effective algebraic geometry
Obtaining the previous pullbacked hypergeometric results (2) and ( 5) required [1,2] an accumulation of creative telescoping calculations, and a lot of "guessing" using all the symmetries of the diagonals of these rational functions (1) and ( 4).We are looking for a more efficient and intrinsic way of obtaining these exact results.These two pullbacked hypergeometric results (2) and ( 5), are essentially "encoded" by their Hauptmodul pullbacks (3) and ( 6), or, equivalently, their corresponding j-invariants.The interesting question, which will be addressed in this paper, is whether it is possible to canonically associate elliptic curves whose j-invariants correspond precisely to these Hauptmoduls H = 1728 j .
One expects such an elliptic curve to correspond to the singular part of the rational function, namely the denominator of the rational function.Let us recall that the diagonal of a rational function in (for example) three variables is obtained through its multi-Taylor expansion [19,20] by extracting the "diagonal" terms, i.e. the powers of the product p = xyz: Consequently, it is natural to consider the algebraic curve corresponding to the intersection of the surface defined by the vanishing condition D(x, y, z) = 0 of the denominator D(x, y, z) of these rational functions (1) and ( 4), with the hyperbola p = x y z (where p is seen, here, as a constant).This amounts, for instance, to eliminating the variable z, substituting z = p x y in D(x, y, z) = 0.

Nine-parameters case:
In the case of the rational functions (1) this corresponds to the (planar) algebraic curve which can be rewritten as a (general, nine-parameters) biquadratic: Using formal calculations 8 one can easily calculate the genus of the planar algebraic curve (10), and find that it is actually an elliptic curve (genus-one).Furthermore, one can (almost instantaneously) find the exact expression of the j-invariant of this elliptic curve as a rational function of the nine parameters a, b 1 , b 2 , • • • , e in (1).One actually finds that this j-invariant is precisely the j such that the Hauptmodul H = 1728 j is the exact expression (3).In other words, the classical modular form result (2) could have been obtained, almost instantaneously, by calculating the j-invariant of an elliptic curve canonically associated with the denominator of the rational function (1).The algebraic planar curve (10) corresponds to the most general biquadratic of two variables, which depends on nine homogeneous parameters.Such general biquadratic is well-known to be an elliptic curve for generic values of the nine parameters 9 .
Thus, the nine-parameters exact result (2) can be seen as a simple consequence of the fact that the most general nine-parameters biquadratic is an elliptic curve.

Ten-parameters case:
In the case of the rational function (4), substituting z = p x y in D(x, y, z) = 0, one obtains the ten-parameters bicubic: As before, we find that this planar algebraic curve is actually an elliptic curve 10 and that the exact expression of its j-invariant is precisely the j of the Hauptmodul H = 1728/j in (6).
Thus, this ten-parameters result (5) can again be seen as a simple consequence of the fact that there exists a family of ten-parameters bicubics (see (11)) which are elliptic curves for generic values of the ten parameters.
These preliminary calculations are a strong incentive to try to replace the differential algebra calculations of the creative telescoping, by more intrinsic algebraic geometry calculations, or, at least, perform effective algebraic geometry calculations to provide an algebraic geometry interpretation of the exact results obtained from creative telescoping.

Finding creative telescoping results from j-invariant calculations.
One might think that these results are a consequence of the simplicity of the denominators of the rational functions (1) or (4), being associated with biquadratics or selected bicubics.In fact, these results are very general.Let us, for instance, consider a nineparameters family of planar algebraic curves that are not biquadratics or (selected) bicubics: One can easily calculate the genus of this planar curve and see that this genus is actually one for arbitrary values of the a n 's.Thus the planar curve ( 12) is an elliptic curve for generic values of the nine parameters a 1 , • • • , a 9 .It is straightforward to see that the algebraic surface S(x, y, z) = 0, corresponding to z • (a 1 x 4 + a 2 x 3 + a 3 x 2 + a 4 x + a 5 + a 6 x 2 y + a 7 y 2 + a 8 y) + a 9 p = 0, (13) will automatically be such that its intersection with the hyperbola p = x y z gives back the elliptic curve (12).
Using this kind of "reverse engineering" yields to consider the rational function in three variables x, y and z R(x, y, z) = which will be such that its denominator is canonically associated with an elliptic curve.Again we can immediately calculate the j-invariant of that elliptic curve.If one calculates the telescoper of this eight-parameters family of rational functions (14), one finds that this telescoper is an order-two linear differential operator with pullbacked hypergeometric solutions of the form where A(x) is an algebraic function and, where again, the pullback-Hauptmodul H = 1728/j, precisely corresponds to the j-invariant of the elliptic curve.
More generally, seeking for planar elliptic curves, one can, for given values of two integers M and N, look for planar algebraic curves defined by the set of a m, n 's which are equal to zero, apart of N homogeneous parameters a m, n being, as in (10) or (11) or (13), independent parameters.Finding such an N -parameters family of (planar) elliptic curves automatically provides an N -parameters family of rational functions such that their telescopers have a pullbacked 2 F 1 hypergeometric solution we can simply deduce from the j-invariant of that elliptic curve.
Recalling the results of section 2.2, the quite natural question to ask now is whether it is possible to find families of such (planar) elliptic curves which depend on more than ten independent parameters?Before addressing this question, let us recall the concept of birationally equivalent elliptic curves.Let us consider the monomial transformation: where M, N, P, Q are integers such that M • Q − P • N = 1, then its compositional inverse is the monomial transformation: This monomial transformation (17) is thus a birational11 transformation.A birational transformation transforms an elliptic curve, like (12), into another elliptic curve with the same j-invariant: these two elliptic curves are called birationally equivalent.In the case of the birational and monomial transformation (17), the elliptic curve ( 12) is changed into12 : With this kind of birational monomial transformation (17), we see that one can obtain families of elliptic curves (19) of arbitrary large degrees in x and y.Consequently one can find nine or ten parameters families of rational functions of arbitrary large degrees yielding pullbacked 2 F 1 hypergeometric functions.There is no constraint on the degree of the planar algebraic curves (19): the only relevant question is the question of the maximum number of (linearly) independent parameters of families of planar elliptic curves which is shown to be ten.The demonstration 13 is sketched in B.
Let us recall another important point.We have already remarked in [1,2] that once we have an exact result for a diagonal of a rational function of three variables R(x, y, z), we immediately get another exact result for the diagonal of the rational function R(x n , y n , z n ) for any positive integer n.As a result we obtain a new expression for the diagonal changing x into x n .In fact, this is also a result on the telescoper of the rational function R(x, y, z): the telescoper of the rational function R(x n , y n , z n ) is the x → x n pullback of the telescoper of the rational function R(x, y, z).Having a pullbacked 2 F 1 solution for the telescoper of the rational function R(x, y, z) (resp.the diagonal of the rational function R(x, y, z)), we will immediately deduce a pullbacked 2 F 1 solution for the telescoper of the rational function R(x n , y n , z n ) (resp. the diagonal of the rational function R(x n , y n , z n )).
Along this line, let us change in the rational function ( 1), (x, y, z) into (x 2 , y 2 , z 2 ): The diagonal of this new rational function (20) will be the pullbacked 2 F 1 exact expression (2) where we change x → x 2 .The intersection of the algebraic surface corresponding to the vanishing condition of the denominator of the new rational function (20), with the hyperbola p = x y z (i.e.z = p x y ), is nothing but the equation (10) where we have changed (x, y; p) into (x 2 , y 2 ; p 2 ) which is no longer 14 an elliptic curve but a curve of genus 9.
With that example we see that classical modular form results, or pullbacked 2 F 1 exact expressions like (2), can actually emerge from higher genus curves like (21).As far as these diagonals, or telescopers, of rational function calculations are concerned, higher genus curves like (21) must in fact be seen as "almost" elliptic curves up to an x → x n covering.
Such results for monomial transformations like (x, y, z) → (x n , y n , z n ) can, in fact, be generalised to more general (non birational 15 ) monomial transformations.This is sketched in C.

Changing the parameters into functions of the product p = x y z.
All these results for many parameters families of rational functions can be drastically generalised when one remarks that allowing any of these parameters to be a rational function of the product p = x y z also yields to the previous pullbacked 2 F 1 exact expression, like (2), where the parameter is changed into that rational function of x (see [1]).Let us consider a simple (two-parameters) illustration of this general result.Let us consider a subcase of the previous nine or ten parameters families, introducing, for example, the two parameters rational function: The diagonal of this rational function ( 22) is the pullbacked hypergeometric function: ], [1], 43200 where and Let us now consider the previous rational function (22) where the two parameters b 2 and c 3 become some rational functions of the product p = x y z, for instance: The new corresponding rational function becomes more involved but one can easily calculate the telescoper of this new rational function of three variables x, y and z, and find that it is, again, an order-two linear differential operator having the pullbacked hypergeometric solution (23) where b 2 and c 3 are, now, replaced by ( p is now x) the functions: In that case ( 22) with ( 26), one gets a diagonal which is the pullbacked hypergeometric solution where q 8 and q 20 are two polynomials with integer coefficients of degree eight and twenty in x.The exact expression ( 28) is nothing but (23) (with ( 24) and ( 25)) where b 2 and c 3 have been replaced by the rational functions (27).Similar calculations can be performed for more general rational functions (1) or ( 4), when all the (nine or ten) parameters are more involved rational functions.
When performing our creative telescoping symbolic calculations using the Holonomic-Functions package [3], such results may look quite impressive.From the algebraic geometry viewpoint, it is almost tautological 16 , if one takes for granted the result of our previous subsections 3.1 and 3.2, namely that the pullbacked hypergeometric solution of the telescoper corresponds to the Hauptmodul 1728/j, where j is the j-invariant of the elliptic curve corresponding to the intersection of the algebraic surface corresponding to the vanishing condition of the denominator, with the hyperbola p = x y z: this calculation of the j-invariant is performed for p fixed, and arbitrary (nine or ten) parameters a, b 1 , • • • .It is clearly possible to force the parameters to be functions 17 of p, the j-invariant being changed accordingly.Of course, in that case, the parameters in the rational function are the same functions but of the product p = x y z.
One thus gets pullbacked hypergeometric solutions (classical modular forms) for an (unreasonably ...) large set of rational functions in three variables, namely the families of rational functions (1) or ( 4), but where, now, the nine or ten parameters are nine, or ten, totally arbitrary rational functions (with Taylor series expansions) of the product p = x y z.
We see experimentally that changing the parameters of the rational function into functions, actually works for diagonals of rational functions, as well as for solutions of telescopers of rational functions depending on parameters.

Creative telescoping on rational functions of more than three variables associated with products or foliations of elliptic curves
Let us show that such an algebraic geometry approach to creative telescoping can be generalised to rational functions of more than three variables, when the vanishing condition of the denominator can be associated with products of elliptic curves, or more generally, algebraic varieties with foliations in elliptic curves.
• The telescoper of the rational function in the four variables x, y, z and w gives an order-three self-adjoint linear differential operator which is, thus, the symmetric square of an order-two linear differential operator.The latter has the pullbacked hypergeometric solution: ], [1], In [18] we underlined the difference between the diagonal of a rational function and solutions of the telescoper of the same rational function.In this case, the diagonal of the rational function ( 29) is zero 18 and is thus different from the pullbacked hypergeometric solution (30), which is a "Period" [37] of the algebraic variety corresponding to the denominator over some (non-vanishing 19 ) cycle.From now, we will have a similar situation in most of the following examples of this paper. 16An algebraic geometer will probably see this as a trivial remark: diagonalization is an algebraic procedure and nothing really happens to the coefficients.Therefore if one replaces the coefficients by anything else, one will find those replaced coefficients in the end result. 17The functions should be rational functions if one wants to stick with diagonals and telescopers of rational functions, but the result remains valid for algebraic functions, or even transcendental functions with reasonable Taylor series expansions at x = 0: for instance, for 2 F 1 hypergeometric functions, one gets a differentially algebraic function corresponding to the composition of 2 F 1 hypergeometric functions. 18The reason is that the integration takes place over a cycle homologically equivalent to the trivial cycle.The cycle becomes trivial after taking the limit p → 0. Integrals over non vanishing cycles usually give logarithms of p, like the second solution to the hypergeometric function 2 F 1 ([1/2, 1/2], [1], , x). 19 Diagonals of the rational functions correspond to periods over vanishing cycles [38,40].This example is a simple illustration of what we expect for products of elliptic curves, or algebraic varieties with foliations in elliptic curves.Introducing the product p = xyzw, the vanishing condition of the denominator of the rational function (29) reads the surface S(x, y, z) = 0: For fixed p and fixed y, equation ( 31) can be seen as an algebraic curve with: For fixed p and fixed y, λ can be considered as a constant, the algebraic curve (32) being an elliptic curve with an obvious Weierstrass form: The j-invariant of (32), or20 (33), is well-known and yields the Hauptmodul H: For fixed p and fixed x, equation ( 31) can be seen as an algebraic curve for: which is also an elliptic curve with an obvious Weierstrass form and the same Hauptmodul (34).This Hauptmodul is precisely the one occurring in the pullbacked hypergeometric solution (30).
More generally, the rational function of the four variables x, y, z and w where p = x y z w, and where R 1 (p) and R 2 (p) are two arbitrary rational functions (with Taylor series expansions) of the product p = x y z w, yields a telescoper which has an order-four linear differential operator which is the symmetric product21 of two order-two linear differential operators having respectively the pullbacked hypergeometric solutions (30) where x is replaced by R 1 (x) and R 2 (x).These two hypergeometric solutions thus have the two Hauptmodul pullbacks obtained by calculations similar to the ones previously performed on (31) but, now, for the Weierstrass form corresponding to the denominator (36).
A solution of the telescoper of ( 36) is thus the product of these two pullbacked hypergeometric functions.Let us give a simple illustration of this general result, with the next example.
• The telescoper of the rational function in the four variables x, y, z and w corresponding to (36) with R 1 (p) = p and R 1 (p) = 3 p, gives an order-four linear differential operator which is the symmetric product of two order-two operators having respectively the pullbacked hypergeometric solution (30) and the solution (30) where the variable x has been changed into 3 x: = (1 ], [1],

Creative telescoping on rational functions of five variables associated with products or foliations of three elliptic curves
Let us, now, introduce the rational function in five variables x, y, z, v and w where the denominator D(x, y, z, v, w) reads: where: p = x y z v w.
The telescoper of the rational function (41) of five variables gives 22 an order-eight linear differential operator which is the symmetric product of three order-two linear differential operators having respectively the pullbacked hypergeometric solution (30), the solution (30) where x has been changed into 3 x, namely (40), and the solution (30), where x has been changed into 5 x: = (1 ], [1], In other words, the order-eight telescoper of the rational function (41) has the product S = S 1 • S 2 • S 3 , of ( 30), ( 40) and ( 43) as a solution.From an algebraic geometry viewpoint, this is a consequence of the fact that, for fixed p, the algebraic variety D p = 0, where D p is given by (42), can be seen, for fixed y and z, as an elliptic curve E 1 of equation D y,z,p (v, x) = 0, for fixed x and z as an elliptic curve E 2 of equation D x,z,p (v, y) = 0, and for fixed x and y also as an elliptic curve E 3 of equation D x,y,p (v, z) = 0, the jinvariants j k , k = 1, 2, 3 of these three elliptic curves E k yielding (in terms of p), precisely, the three Hauptmoduls H k = 1728 occurring as pullbacks in the three S k 's of the solution S = S 1 • S 2 • S 3 , of the telescoper of (41).

Remark:
Other examples of rational functions of three, four, five variables where the denominators also correspond to Weierstrass (resp.Legendre) forms, are displayed in D. They provide simple illustrations of rational functions where the denominator is associated with K3 surfaces 23 , or Calabi-Yau three-folds.In these cases the algebraic varieties have simple foliations in terms of two or three families of elliptic curves, and the solutions of the corresponding telescopers can be selected 3 F 2 and 4 F 3 hypergeometric functions (see (A28) in D), naturally associated with K3 surfaces and Calabi-Yau operators [27].

Creative telescoping of rational functions in three variables associated with genus-two curves with split Jacobians
In a paper [17,18], dedicated to Heun functions that are solutions of telescopers of simple rational functions of three and four variables, we have obtained 24 an order-four telescoper of a rational function of three variables, which is the direct sum of two order-two linear differential operators, each having classical modular forms solutions which can be written as pullbacked 2 F 1 hypergeometric solutions.Unfortunately, the intersection of the algebraic surface corresponding to the denominator of the rational function with the p = x y z hyperbola, yields a genus-two algebraic curve.
Let us try to understand, in this section, how a genus-two curve can yield two classical modular forms.Let us first recall the results in section 2.2 of [18].

Periods of extremal rational surfaces
Let us recall the rational function in just three variables [18]: Its telescoper is actually an order-four linear differential operator L 4 which, not only factorizes into two order-two linear differential operators, but is actually the direct sum (LCLM) of two25 order-two linear differential operators L 4 = L 2 ⊕ M 2 .These two (non homomorphic) order-two linear differential operators have, respectively, the two pullbacked hypergeometric solutions: ], [1], and: The diagonal of ( 45) is actually the half-sum of the two series (46) and ( 47): As far as our algebraic geometry approach is concerned, the intersection of the algebraic surface corresponding to the denominator of the rational function (45) with the hyperbola p = x y z gives the planar algebraic curve (corresponding to the elimination of the z variable by the substitution z = p x y ): One easily finds that this algebraic curve is (for p fixed) a genus-two curve, and that this higher genus situation does not correspond to the "almost elliptic curves" described in subsection 3.2 namely an elliptic curve transformed by a monomial transformation.How can a "true" genus-two curve give two j-invariants, namely a telescoper with two Hauptmodul pullbacked 2 F 1 solutions?We are going to see that the answer is that the Jacobian of this genus-two curve 26 is in fact isogenous to a product E × E ′ of two elliptic curves (split Jacobian).

Split Jacobians
Let us first recall the concept of split Jacobian with a simple example.In [49], one has a crystal-clear example of a genus-two curve C such that its Jacobian J(C) is isogenous to a product of elliptic curves with j-invariants j 1 = −2 7 • 7 2 = −6272 and j 2 = −2 of j-invariant j = j 2 = −2 5 • 7 • 17 3 .We consider the following transformation 27 : This change of variable (52) actually transforms the elliptic curve (51) in u and v into the genus-two curve (50) in x and y.This provides a simple example of a genus-two curve with split Jacobian through K3 surfaces.
More generally, let us consider the Jacobian of a genus-two curve C. The Jacobian is simple if it does not contain a proper abelian subvariety, otherwise the Jacobian is reducible, or decomposable or "split".For this latter case, the only possibility for a genus-two curve is that its Jacobian is isogenous to a product E × E ′ of two elliptic curves 28 .Equivalently, there is a degree n map C → E to some elliptic curves.Classically such pairs 29 C, E arose in the reduction of hyperelliptic integrals to elliptic ones [49].The j-invariants correspond, here, to the two elliptic subfields: see [49].

Creative telescoping on rational functions in three variables associated with genus-two curves with split Jacobians: a two-parameters example.
Let us now consider the example with two parameters, a and b, given in section 4.5 page 12 of [49].Let us substitute the rational parametrisation30 u = x in the elliptic curve where This gives the genus-two curve C a, b (x, y) = 0 with: The j-invariant of the elliptic curve (54) gives the following exact expression for the Hauptmodul H = 1728 j : Let us consider the telescoper of the rational function of three variables x y/D a (x, y, z) where the denominator D a (x, y, z) is C a, b (x, y) given by ( 56), but for b = 3 + x y z: D a (x, y, z) = C a, 3 + xyz (x, y) = x 6 y 3 z 3 + x 7 y 2 z 2 + 4 x 3 y 5 z 3 + 9 x 5 y 2 z 2 + 6 x 6 yz + 3 x 4 y 2 z 2 + 36 y 4 x 2 z 2 +6 x 5 yz + 4 x 6 + 27 x 4 yz + 9 x 5 + 18 x 3 yz + 108 xy 3 z + 18 This telescoper of the rational function is an order-four linear differential operator L 4 which is actually the direct-sum, L 4 = LCLM(L 2 , M 2 ) = L 2 ⊕ M 2 , of two order-two linear differential operators, having two pullbacked hypergeometric solutions.One finds out that one of the two pullbacks precisely corresponds to the Hauptmodul H given by ( 57) for b = 3 + x.
Let us consider the a = 3 subcase 31 .For a = 3, the Hauptmodul H = 1728 j , given by (57) becomes for b = 3 + x: The telescoper of the rational function ( 59) with D a (x, y, z) given by ( 58) for a = 3, is an order-four linear differential operator which is the direct-sum of two order-two linear differential operators L 4 = LCLM(L 2 , M 2 ) = L 2 ⊕ M 2 , these two order-two linear differential operators having the pullbacked hypergeometric solutions ], [1], for L 2 , and for M 2 , where we see clearly that the Hauptmodul in ( 62) is precisely the Hauptmodul (60).The Jacobian of the genus-two curve is a split Jacobian corresponding to the product E 1 × E 2 of two elliptic curves, the j-invariant of the second elliptic curve corresponds to the Hauptmodul H = 1728 j given by ( 57) when the j-invariant of the first elliptic curve reads corresponding to the Hauptmodul 1728 (61).This second invariant is, as it should, exactly the j-invariant of the second elliptic curve E ′ , given page 48 in [53]: for c = 1, a = 3 and b = 3 + x.
5.4.Creative telescoping on rational functions of three variables associated with genus-two curves with split Jacobians: a simple example Another simpler example of a genus-two curve with pullbacked 2 F 1 solution (not product of pullbacked 2 F 1 ) of the telescoper can be given if one considers the genus-two algebraic curve C p (x, y) = 0 given in Lemma 7 of [54] (see also [55,56]) Let us introduce the rational function x y/D(x, y, z) where the denominator D(x, y, z) is given by: The telescoper of this rational function is an order-two linear differential operator which has the two hypergeometric solutions which is a Puiseux series at x = 0 and: These two hypergeometric solutions can be rewritten as 32 where the j-invariant J, in the Hauptmodul 1728 J in (69), corresponds exactly to the degree-two elliptic subfields given in the first equation of page 6 of [54].
Remark: In contrast with the previous example of subsection 5.3 where we had two j-invariants corresponding to the two order-two linear differential operators L 2 and M 2 of the direct-sum decomposition of the order-four telescoper, we have, here, just one order-two telescoper, which is enough to "encapsulate" the two j-invariants (70), since they are Galois-conjugate.

Rational functions with tri-quadratic denominator and N-quadratic denominator.
We try to find telescopers of rational functions corresponding to (factors of) linear differential operators of "small" orders, for instance order-two linear differential operators with pullbacked 2 F 1 hypergeometric functions, classical modular forms, or their modular generalisations (order-four Calabi-Yau linear differential operators [27], etc ...).As we saw in the previous sections, this corresponds to the fact that the denominator of these rational functions is associated with an elliptic curve, or products of elliptic curves, with K3 surfaces or with threefold Calabi-Yau manifolds corresponding to algebraic varieties with foliations in elliptic curves 33 .Since this paper tries to reduce the differential algebra creative telescoping calculations to effective algebraic geometry calculations 34 and structures, we want to focus on rational functions with denominators that correspond to selected algebraic varieties [45,59], beyond algebraic varieties corresponding to products of elliptic curves or foliations in elliptic curves 35 , namely algebraic varieties with an infinite number of birational automorphisms 36 .This infinite number of birational symmetries, excludes algebraic varieties of the "general type" (with finite numbers 37 of birational symmetries).For algebraic surfaces, this amounts to discarding the surfaces of the "general type" which have Kodaira dimension 2, focusing on Kodaira dimension one (elliptic surfaces), or Kodaira dimension zero (abelian surfaces, hyperelliptic surfaces, K3 surfaces, Enriques surfaces), or even Kodaira dimension −∞ (ruled surfaces, rational surfaces).
In contrast with algebraic curves where one can easily, and very efficiently, calculate the genus of the curves to discard the algebraic curves of higher genus and, in the case 32 The fact that 2 F 1 [ 1 8 , 5  8 ], [1], z can be rewritten as 2 F 1 [ 1 12 , 5  12 ], [1], H(z) where the Hauptmodul H(z) is solution of a quadratic equation is given in equation (H.14) of Appendix H of [18]. 33Even if K3 surfaces, or threefold Calabi-Yau manifolds, are not abelian varieties, the Weierstrass-Legendre forms introduced in D, amounts to saying that K3 surfaces can be "essentially viewed" (as far as creative telescoping is concerned) as foliations in two elliptic curves, and threefold Calabi-Yau manifolds as foliations in three elliptic curves. 34One has birational automorphisms in projective spaces [57,58], but since this paper is dedicated to (efficient) formal calculations we work exclusively in affine coordinates (see for instance (A41), (A42), (A43) below).For algebraic geometers an ellitic curve is a smooth complete genus 1 curve with a choice of a base point.Here our elliptic curves are, in fact, an affine piece of a genus 1 curve with no base point, but this does not really matter because the j-invariant which is all we care about in this kind of creative telescoping calculations, is determined by that much information. 35K3 surfaces, threefold Calabi-Yau manifolds, higher curves with split Jacobian corresponding to products of elliptic curves, ... 36 The best explicit illustration of this situation emerges in integrable models [45,[59][60][61] 37 There are even precise bounds for the number of automorphisms.The upper bound is 84 (g − 1) for curves of genus g and these bounds have been extensively studied in higher dimensions [62][63][64].
of genus-one, obtain the j-invariant using formal calculations 38 , it is, in practice, quite difficult to see for higher dimensional algebraic varieties, that the algebraic variety is not of the "general type", because it has an infinite number of birational symmetries.For these (low Kodaira dimension) "selected cases" we are interested in, calculating the generalisation of the j-invariant (Igusa-Shiode invariants, etc ...) is quite hard.
Along this line we want to underline that there exists a remarkable set of algebraic surfaces, namely the algebraic surfaces corresponding to tri-quadratic equations: depending on 27 = 3 3 parameters a m,n,l .More generally, one can introduce algebraic varieties corresponding to N-quadratic equations: With these tri-quadratic ( 71), or N-quadratic (72) equations, we will see, in E.1 and E.2, that we have automatically (selected) algebraic varieties that are not of the "general type" having an infinite number of birational symmetries, which is precisely our requirement for the denominator of rational functions with remarkable telescopers 39 .
Let us first, as a warm-up, consider, in the next subsection, a remarkable example of tri-quadratic (71), where the underlying foliation in elliptic curves is crystal clear.

Rational functions with tri-quadratic denominator simply corresponding to elliptic curves.
Let us first recall the tri-quadratic equation in three variables x, y and z already introduced in Appendix C of [65].This algebraic surface, symmetric in x, y and z, can be seen for z (resp.x or y) fixed, as an elliptic curve which j-invariant is independent of z yielding the corresponding Hauptmodul: This corresponds to the fact that this algebraic surface (73) can be seen as a product of two times the same elliptic curve with the Hauptmodul (74).This is a consequence of the fact that, introducing x = sn(u) 2 , y = sn(v) 2 and z = sn(u + v) 2 , and M = 1/k 2 , this algebraic surface (73) corresponds to the well-known formula for the addition on elliptic sine 40 : 38 Use with(algcurves) in Maple and the command "genus" and "j_invariant". 39Telescopers with factors of "small enough" order, possibly yielding classical modular forms, Calabi-Yau operators, ... Rational functions with denominators of the "general type" will yield telescopers of very large orders. 40See equation (C.3) in Appendix C of [65].
For M = x y z w, the LHS of the tri-quadratic equation ( 73) yields a polynomial of four variables x, y, z and w, that we denote T(x, y, z, w): T(x, y, z, w) = (76) The telescoper of the rational function in four variables x, y, z and w, x y z T(x, y, z, w) , ( 77) is an order-three (self-adjoint) linear differential operator which is the symmetric square of the order-two linear differential operator having the following pullbacked 2 F 1 hypergeometric solution: As it should the Hauptmodul in ( 78) is the same as the Hauptmodul (74).The algebraic surface ( 73) can be seen as the product of two times the same elliptic curve with the Hauptmodul (74): as expected the solution of the order-three telescoper is the square of the pullbacked 2 F 1 hypergeometric function (78) with that Hauptmodul.
More generally, we can also consider another tri-quadratic equation of three variables x, y and z and two parameters M and N: This surface, symmetric in x, y and z, can be seen for z (resp.x or y) fixed as an elliptic curve which j-invariant is, again, independent of z yielding the corresponding Hauptmodul: Let us consider the following change of variables M = m 2 and N = 8 • m 3 + p in (79).For p = x y z w, the LHS of the tri-quadratic equation ( 79) yields a polynomial in four variables x, y, z and w, that we denote T m (x, y, z, w): For z (resp.x or y) fixed the corresponding Hauptmodul (80) reads: The telescoper of the rational function in four variables x, y, z and w, x y z T m (x, y, z, w) , ( 83) is an order-three (self-adjoint) linear differential operator which is the symmetric square of an order-two linear differential operator having the following pullbacked 2 F 1 hypergeometric solution: (16 ], [1], As it should the Hauptmodul in (84) is the same as the Hauptmodul (82).The algebraic surface (79) can be seen as the product of two times the same elliptic curve with the Hauptmodul (80) (or ( 82)).As expected the solution of the order-three telescoper is the square of the pullbacked 2 F 1 hypergeometric function (84) with the Hauptmodul (82).
Remark: Let us perform some (slight) deformation of the rational function (77), changing the first −2 coefficient in (76) into a −3 coefficient.One thus considers the polynomial T(x, y, z, w): T(x, y, z, w) = (85) The telescoper of the rational function in four variables, x y z T(x, y, z, w) , is an (irreducible) linear differential operator of (only) order-four L 4 which is non-trivially homomorphic to its adjoint 41 .A priori, we cannot exclude the fact that L 4 could be homomorphic to the symmetric cube of a second-order linear differential operator, or to a symmetric product of two second-order operators.Furthermore, it could also be, in principle, that these second-order operators admit classical modular forms as solutions (pullbacks of special 2 F 1 hypergeometric functions).However, these options can both be excluded by using some results from differential Galois theory [68], specifically from [69,Prop. 7,p. 50] for the symmetric cube case, and from [69,Prop. 10,p. 69] for the symmetric product case, see also [70, §3].Indeed, if L 4 were either a symmetric cube or a symmetric product of order-two operators, then its symmetric square would contain a (direct) factor of order 3 or 1.This is ruled out by a factorization procedure which shows that the symmetric square of L 4 is (LCLM-)irreducible.
This example does not correspond to an addition formula like (75), but the polynomial T(x, y, z, w) still corresponds to a tri-quadratic.Consequently it is an algebraic variety with an infinite number of birational automorphisms, as shown in E.1.

Painlevé VI equations
Let us consider other simple examples of tri-quadratic surfaces that occur in different domains of mathematics and physics.

Telescopers of rational functions of several variables
Let us consider the rational function in four variables x, y, z, u: R(x, y, z, u) = The telescoper of this rational function of four variables is an order-two linear differential operator L 2 which has the pullbacked hypergeometric solution: The diagonal of (91) is the expansion of this pullbacked hypergeometric function (92): 1 + 648 x 2 − 72900 x 3 + 1224720 x 4 − 330674400 x 5 + 23370413220 x 6 (93) −1276733858400 x 7 + 180019474034400 x 8 − 12013427240614800 x 9 + • • • If one considers the intersection of the vanishing condition of the denominator of (91) with the hyperbola p = x y z u, eliminating for instance u = p x y z in the vanishing condition of the denominator of (91), one gets a condition, independent of x, which corresponds to a genus-one curve 11 The Hauptmodul of this elliptic curve (94) reads: which corresponds precisely to the Hauptmodul pullback in (92).

Remark :
The expansion (93) of ( 92) is not only the diagonal of the rational function R(x, y, z, u) in four variables (91), it is also the diagonal of the rational function of three variables R(x, y, z, 1).Actually, using section (3), one sees easily that eliminating x = p y z in the the vanishing condition of the denominator of R(x, y, z, 1) gives exactly the same elliptic curve (94).
Let us, now, generalize the rational function (91) of four variables x, y, z, u, introducing the rational function of The telescoper of this rational function of N + 3 variables is the same order-two telescoper as for (91), which has the pullbacked hypergeometric solution (92).Again one can verify that the diagonal of (96) is the expansion (93) of the pullbacked hypergeometric function 44(92).If one considers the intersection of the vanishing condition of the denominator of (96) with the hyperbola p = x y z u in the vanishing condition of the denominator of (96), one gets again a condition, independent of x but also of u 1 , • • • , u N , which corresponds to a genus-one curve (94): The Hauptmodul of this elliptic curve (97), or (94) reads again the Hauptmodul (95) which corresponds precisely to the Hauptmodul pullback in (92).
Other examples, corresponding to simple polynomial deformations of (91), such that their diagonal is the pullbacked 2 F 1 hypergeometric function (92) are displayed in F. This (infinite) family of rational functions correspond to a different algebraic geometry scenario: the "canonical" algebraic surface corresponding to the intersection of the vanishing condition of the denominator of the rational function with the hyperbola p = xyz, is foliated in (generically high genus) algebraic curves depending on the variable x.One sees (experimentally) that the Hauptmodul of the pullbacked 2 F 1 hypergeometric functions corresponds to the Hauptmodul of the x = 0 algebraic curve, which is an elliptic curve 45 .
In contrast with the other examples and results of this paper, we have no algebraic geometry interpretation of this experimental result yet.
We have shown that the results we had obtained on diagonals of nine and ten parameters families of rational functions in three variables, using creative telescoping yielding classical modular forms expressed as pullbacked 2 F 1 hypergeometric functions [1,2], can be obtained much more efficiently calculating the j-invariant of an elliptic curve canonically associated with the denominator of the rational functions.In the case where creative telescoping yields pullbacked 2 F 1 hypergeometric functions, we generalize this result to other families of rational functions of three, and even more than three, variables, when the denominator can be associated with products of elliptic curves or foliation in terms of elliptic curves, or when the denominator is associated with a genus-two curve with a split Jacobian corresponding to products of elliptic curves.
We have seen different scenarii.In the first cases, we have considered denominators corresponding to products of elliptic curves: in these cases the solutions of the telescoper were products of pullbacked 2 F 1 hypergeometric functions.We have also considered denominators corresponding to genus-two curves with split Jacobians isogenous to products of two elliptic curves, and in these cases the solutions of the telescoper were sums of two pullbacked 2 F 1 hypergeometric functions, sometimes one pullbacked 2 F 1 hypergeometric function being enough to describe the two Galois-conjugate j-invariants (see 5.4).We also considered denominators corresponding to algebraic varieties where the Hauptmodul pullback in the pullbacked 2 F 1 hypergeometric functions emerges from a selected (x = 0, see F.1, F.2) elliptic curve of the algebraic variety.We also encountered denominators corresponding to algebraic manifolds with an infinite set of birational automorphisms and elliptic curves foliation yielding, no longer classical modular forms represented as pullbacked 2 F 1 hypergeometric functions, but more general modular structures associated with selected linear differential operators like Calabi-Yau linear differential operators [27,66] and their generalisations.
The creative telescoping method on a rational function is an efficient way to find the periods of an algebraic variety over all possible cycles 46 .The fact that the solution of the telescoper corresponds to "periods" [37] over all possible cycles is a simple consequence of the fact that creative telescoping corresponds to purely differential algebraic manipulations on the integrand independently of the cycles, thus being blind to analytical details.In this paper, we show that the final result emerging from such differential algebra procedures (which can be cumbersome when the result depends on nine or ten parameters), can be obtained almost instantaneously from a more fundamental intrinsic pure algebraic geometry approach, calculating, for instance, the j-invariant of some canonical elliptic curve.This corresponds to a shift Analysis → Differential Algebra → Algebraic Geometry.Algebraic geometry studies of more involved algebraic varieties than product of elliptic curves, foliation in elliptic curves (Calabi-Yau manifolds, ...) is often a tedious and/or difficult task (finding Igusa-Shiode invariants, ...), and formal calculations tools are not always available or userfriendly.Ironically, for such involved algebraic varieties the creative telescoping may then become a simple and efficient tool to perform effective algebraic geometry studies.
ing 49 and more specifically the programs [3] corresponding to a fast approach to creative telescoping [43], are a powerfull way to find these linear differential operators annihilating these diagonal of rational functions in the cases emerging naturally in theoretical physics, integrable models, enumerative combinatorics, for which the order of the linear differential operators is quite large [16,23,24] and the variety D p (y, z) = 0 is (most of the time) not a smooth one.All the pedagogical (but non-trivial) examples of telescopers displayed in this paper can be viewed by an algebraic geometer as a presentation of examples of families of varieties and their Picard-Fuchs equations.
Appendix B Maximum number of parameters for families of planar elliptic curves.
We have seen, in section 3, that the previous results on diagonals of nine or ten parameters families of rational functions of three variables being pullbacked 2 F 1 hypergeometric functions (and in fact classical modular forms) can actually be seen as corresponding to the (well-known in integrable models and integrable mappings) fact that the most general biquadratic corresponding to elliptic curves is a nine-parameters family and that the most general ternary cubic corresponding to elliptic curves is a ten-parameters family.One can, for instance recall page 238 of [76], which amounts to considering the collection of all cubic curves in CP 2 with the homogeneous equation a x 3 + b x 2 y + c x y 2 + d y 3 + e x 2 z + f x z 2 + g y 2 z +h y z 2 + i z 3 + j x y z = 0, (A7) and the associated problems of passing through nine given points.One can also recall the ternary cubics in [77,78] and other problems of elliptic curves of high rank [79] (see the concept of Neron-Severi rank).
Since the rational functions of three variables we consider are essentially encoded by the denominator of these rational functions, and in the cases we have considered, the emergence of pullbacked 2 F 1 hypergeometric functions (and in fact classical modular forms) corresponds to the fact that the intersection of these denominators with the hyperbola p = x y z corresponds to elliptic curves, one sees that these rational functions are essentially classified by the possible n-parameters families P(x, y) = 0 of elliptic curves.with generic coefficients a m,n ∈ C, then the genus of the algebraic curve defined by P is determined by the support supp(P) = {(m, n) ∈ N 2 : a m,n ̸ = 0}.More precisely, the genus equals the number of interior integer lattice points inside the convex hull of supp(P) [80] (see also the discussion in [81]).For example, the support of the ten-parameters family (11) consists of the following 10 points in N 2 : (0, 0), (0, 1), (0, 2), (0, 3), (1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3,3) which form a right triangle of side length 3.Only one of these points is an interior point, namely (1,2), hence the genus is 1.
Therefore we may ask: which integer lattice polytopes exist which have exactly one interior point and what is the largest such polytope?Not surprisingly, the answer is known: there are (up to transformations like translation, rotation, shearing) exactly 16 different polytopes with a single interior point [82] (see also Figure 5, page 548 in [83]), the above-mentioned right triangle being the one with the highest total number of lattice points.This shows that there cannot be a family of elliptic curves with more than ten parameters.infinite set of birational symmetries, cannot be of the "general type" (it has Kodaira dimension less than 2).
If one considers a polynomialP(x, y) = ∑ m ∑ n a m,n • x m y n , (A8)