Title: Hypergeometric Functions, Character Sums and Applications
Speaker: Prof. Ling Long, Louisiana State University
Abstract: Hypergeometric functions form a class of special functions satisfying a lot of symmetries. They are closely related to the arithmetic of one-parameter families of algebraic varieties, such as the Legendre curves. p-adic hypergeometric functions were investigated by Dwork leading to his fundamental unit root theory. Hypergeometric functions over finite fields were initiated by Katz and Greene, further developments were made in recent years by many authors. In the proposed mini course, we will explore hypergeometric functions over different settings. Applications of the combined perspectives include obtaining character sum identities, computing special L-values of modular forms, finding and proving supercongruences (influenced by important conjectures of Beukers, Van Hamme, Rodriguez-Villegas, Roberts and Rodriguez-Villegas, constructing decomposable Galois representations, and computing arithmetic invariance of certain abelian varieties.

For more info: https://alozano.clas.uconn.edu/hyperg...