Abstract: 

The study of congruences between modular forms is a classic and vital subject in number theory.  Hida was the first to establish the theory of p-adic modular forms with slope zero; Hida theory is a core tool in Wiles’ proof of the Iwasawa main conjecture, and also one of the main tools in proving Fermat’s Last Theorem.  Coleman extended Hida’s theory to p-adic modular forms of general finite slopes and, together with Mazur, constructed the eigencurve.  The geometry of the eigencurve is essential for understanding the congruence properties of modular forms, yet our knowledge about geometry of the eigencurve is very limited.  In a recent joint work with Nha Xuan Truong, Xiao Liang and Zhao Bin, we prove the finiteness of irreducible components of the eigencurve under a generic condition.

About the Speaker:

Ruochuan Liu is a Boya Distinguished Professor in mathematics of Peking University and a New Cornerstone Investigator.  His current research focuses on p-adic aspects of arithmetic geometry and number theory, especially p-adic Hodge theory, p-adic automophic forms and related topics.  In 2017, he received the National Natural Science Fund for Distinguished Young Scholars.  He was awarded the Tencent Xplorer prize in 2019 and the second prize of the National Award for Natural Sciences in 2020.