Diagonals of Rational Functions: From Differential Algebra to Effective Algebraic Geometry

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