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Back-ground of modular p-adic deformation theory and a brief outline

Mathematics Colloquium | Haruzo Hida | 7th Nov 2022

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Abstract : 

The deformation theory of modular forms is increasingly attracting many researchers in arithmetic geometry as it has been an important step in the proof of Fermat's last theorem by Wiles (and Taylor) and supplied an effective tool for the study of the p-adic Birch and Swinnerton Dyer conjecture in the proof by Skinner-Urban of divisibility of the characteristic power series of the Selmer group of a rational elliptic curve by its p-adic L-function under appropriate assumptions.  I try to give my back-ground motivation of creating the theory and describe an outline of the theory.


About Speaker : 

Professor Hida is a distinguished mathematician having made seminal contributions to Number Theory and Arithmetic Geometry and is currently a Distinguished Research Professor at the Department of Mathematics in University of California Los Angeles. He has been at UCLA for 35 years now, prior to which he held academic positions in Hokkaido University in Japan. Recently he was elected to the American Academy of Arts and Sciences and was also an Inaugural Fellow of the American Mathematical Society in 2012. His long list of honours include Invited Speaker at the International Congress of Mathematicians in 1986 and the Guggenheim Fellowship, 1991–1992. He initiated the theory of p-adic Hecke algebras in 1986. The research revealed many new features of modular/automorphic forms. This theory had a strong impact upon many influential number theorists – in particular, it gave a foundation of the proof of Shimura-Taniyama conjecture and Fermat’s last theorem by A. Wiles and R. Taylor. Also, it was used essentially in the proof by Skinner-Urban of the p-adic Birch and Swinnerton-Dyer conjecture. He has authored (or co-edited) 9 text-books on his research in addition to several research papers. 


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