UConn Mathematics is honored to host a series of lectures this summer, on the theory of hypergeometric functions.

The series can be accessed** through this link (password is “hyper”).**

Video recordings can be found at this YouTube playlist.

Professor Ling Long (LSU) will be giving a series of (online) lectures on **June 29, June 30, July 1 and July 2, 2021**. Each day, Professor Long will lecture at **10am US Eastern** (lectures each day will be 90 minutes with a break in the middle). Everyone is welcome to attend! More details on how to connect to her lectures will be made available closer to the date.

**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.

Here are a few references about the material Prof. Long will be speaking about:

- Jenny Fuselier, Ling Long, Ravi Ramakrishna, Holly Swisher, and Fang-Ting Tu. Hypergeometric functions over

finite fields. Memoirs of AMS, arXiv:1510.02575 (Chapters 1-6) - Ling Long and Ravi Ramakrishna. Some supercongruences occurring in truncated hypergeometric series. Adv.

Math., 290:773–808, 2016, arXiv:1403.5232. (Section 2) - Wen-Ching Winnie Li, Ling Long, Fang-Ting Tu, A Whipple $_7F_6(1)$ formula revisited, preprint 2021 arXiv:2103.08858
- Ling Long, Fang-Ting Tu, Noriko Yui, Wadim Zudilin, Supercongruences for rigid hypergeometric Calabi–Yau threefolds, preprint 2017, arXiv:1705.01663