Lecture: Locally symmetric spaces, and Galois representations
Speaker: Peter Scholze (The University of Bonn, Germany)
Date: 25 Mar 2014, 11:30 AM
Venue: AG 66, TIFR, Mumbai
One of the most studied objects in mathematics is the modular curve, given as the locally symmetric space which is the quotient of 2-dimensional hyperbolic space by congruence subgroups of SL_2(Z). In particular, it is naturally the home of modular forms. It also has an algebraic structure as the moduli space of elliptic curves, and this algebraic structure implies that one can attach number-theoretic objects, such as Galois representations, to modular forms. The simplest generalization of the modular curve are the Bianchi manifolds, introduced in 1892 by the Italian differential geometer Luigi Bianchi, which are quotients of 3-dimensional hyperbolic space by congruence subgroups of SL_2(O_F), where F is an imaginary-quadratic field. Although these are just real manifolds, which do not admit an algebraic structure, it has been speculated already around 1970 that their singular homology, including the large torsion subgroup, knows about Galois representations. The aim of the lecture series is to first explain this conjecture and its higher-dimensional generalizations, and the recent work resolving this conjecture.