Prof. Dr. Michael Harris
(Columbia University, NY)
Thursday, July 6, 2017, 17:15h
Mathematikon, Hörsaal, Im Neuenheimer Feld 205
Hilbert’s twelfth problem, also known as »Kroneckers liebster Jugendtraum« is the hope that roots of polynomial equations with ratio- nal coefficients can be expressed as special values of transcendental functions. In the original formulations of Kronecker and Hilbert, the roots were meant to generate (most or all) extensions of specific algebraic num- ber fields with abelian Galois group; the Galois action should be visible in terms of the transcendental function. This approach was developed extensively in the Shimura-Taniyama theory of complex multi plication. In the Langlands reciprocity conjectures, roots of polynomial equations are replaced by representations of Galois groups, which in general are not abelian; the role of the transcendental functions is played by auto- morphic forms, especially when they contribute to the cohomology of locally symmetric spaces. The lecture will review some recent results in this direction, with an emphasis on open problems.