Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars

Speaker:	Prof. Preda Mihailescu，University of Gottingen
Inviter:	魏达盛
Title:	The Kummer - Vandiver Conjectures
Language:	English
Time & Venue:	2023.08.16 15:00-17:00 南楼N913室
Abstract:	We provide a complete proof of the Kummer - Vandiver Conjecture, stating that p does not divide the class number of the maximal real subfield of the p-th cyclotomic extension of Q. If K is the p-th cyclotomic field and K_infty/K its cyclotomic Z_p-extension, the negation of the conjecture implies the existence of some unramified field L/K which is abelian of degree p and unramified over K, while being galois over Q. It comes together with its cyclotomic Z_p-extension L_infty. The proof is based on an investigation of the local behavior of the real units in the fields L_n, connected to the capitulation that happens in K_n/L_n. Universal norms and a proof of the Leopoldt Conjecture for L_n are important prerequisites used.