Westlake Online Math Forum第十七期 | Loïc Merel: About the unit 1^1.2^2.3^3.4^4....((N-1)/2)^((N-1)/2) modulo a prime number N

2024-03-20 09:26:19


地点:E4-233 & ZOOM

ZOOM ID:950 0888 0530

PASSCODE: 486951

主讲人: Loïc Merel, Institut de Mathématiques de Jussieu-Paris Rive Gauche

主讲人介绍: After studying at École Normale Supérieure in Paris, Loïc Merel obtained his PhD degree from Sorbonne Université in 1993 (then UPMC-Paris 6). He has been professor at Université Paris Cité since 1997. He has been director of the large joint department of both these universities from 2016 to 2020. He works on Number theory, in particular the arithmetic of automorphic forms. His best known work is the proof of the Uniform Boundedness Conjecture for elliptic curves.

讲座主题:About the unit modulo a prime number N


This is mostly joint work with E. Lecouturier.

Let N be a prime number >3. In his great work from 1977, Modular curves and the Eisenstein Ideal, Mazur has defined, from the theory of modular forms, a unit u in Z/N. In 1993, this unit was determined and turned out to be, up to a 6th root of unity,\prod_{k=1}^{(N-1)/2}k^k. We will discuss subsequent developments.

1. The unit is connected, in a non-elementary way, to various objects in elementary algebraic number theory (class groups, Hasse polynomial, Mersenne primes).

2. A more advanced application concerns the structure of Hecke algebras, as originally proposed by Mazur.

3. The unit plays an important role in the developments around the more sophisticated conjecture of Harris and Venkatesh.