Andreas Malmendier, Utah State University
“The special function identities from Kummer surfaces or the identity Ernst Kummer missed.”
Ernst Kummer was the first mathematicians to investigate both algebraic transformations for the Gauss? hypergeometric function, as well as nodal surfaces of degree four in P3 that we call Kummer surfaces today. For Kummer, these two were completely independent lines of his research. In this talk we will show that one can use geometry as powerful tool to find identities for special functions. In particular, we will prove that the factorization of Appell?s generalized hypergeometric series satisfying the so-called quadric property into a product of two Gauss? hypergeometric functions has a geometric origin: we first construct a generalized Kummer variety as minimal nonsingular model for a product-quotient surface with only rational double points from a pair of superelliptic curves of odd genus bigger than 3. We then show that this generalized Kummer variety is equipped with two fibrations. When periods of a holomorphic two-form over carefully crafted transcendental two-cycles on the generalized Kummer variety are evaluated using either of the two fibrations, the answer must be independent of the fibration and the aforementioned family of special function identities is obtained. This family of identities can be seen as a multivariate generalization of Clausen?s Formula.