Westlake Math Colloquium第四十六期 | Ivan Fesenko: Higher adelic structures

2023-11-16 10:37:54



主讲人Ivan Fesenko

讲座主题:Higher adelic structures


For global fields the ring of adeles and its relation to field play most fundamental roles in simplified presentations of class field theory and in the Langlands program. In dimension 1 arithmetic, algebra, geometry and analysis are peacefully coexistent, since codimension 1 is the same as dimension 0 here.

This is not the case in dimension 2, i.e. for arithmetic schemes fibered over a 1d base. Geometry gets divorced from analysis and number theory, since the former corresponds to geometric adelic structure on the scheme, which corresponds to 1-cocycles (or their Arakelov extension) while the latter corresponds to analytic adelic structure on the scheme, which corresponds to 0-cycles and 0+1 is not 2. However, not everything is lost as there is a non-trivial interaction between the multiplicative groups of the two adelic structures which comes from explicit higher class field theory.

This talk will present these two adelic structures and their applications as an adelic insight on major open problems remaining unsolved for 50 or more years.