Marston Morse Lectures

One of the most studied objects in mathematics is the modular curve, which is the quotient of hyperbolic 2-space by the action of \(\mathrm{SL}_2(\mathbb Z)\). It is naturally the home of modular forms, but it also admits an algebraic structure. The interplay of these structures lies at the heart of the Langlands correspondence for \(\mathrm{GL}_2/\mathbb Q\), connecting modular forms with Galois representations. The natural generalizations of the modular curve to higher dimension are the arithmetic locally symmetric spaces. Some of them, namely Shimura varieties, admit an algebraic structure; however, others don't, and this has blocked progress for a long time. The first example is the case of Bianchi manifolds, which are quotients of hyperbolic 3-space by the action of \(\mathrm{SL}_2(O)\), where \(O\) is the ring of integers in an imaginary-quadratic field. These manifolds have a lot of torsion in their singular homology, which has long been expected to carry arithmetic information. I will start by explaining the conjectures relating these objects with Galois representations, and then explain the recent progress made on these conjectures. Ultimately, this relies on new results on the \(p\)-adic geometry of Shimura varieties and the theory of perfectoid spaces, of which I will try to give some impression.