Talk

Class field theory describes the structure of all abelian field extensions of a number field, and promises the existence of the so-called Hilbert class field.
However, its explicit construction has been a long standing open problem, and doing this with the special values of arithmetic functions is known as Hilbert's 12th problem.
We discuss some cases in which this problem has been solved, including imaginary quadratic fields through CM theory and singular moduli.
Finally, we introduce novel p-adic constructions and outline a method to compute these special values using the p-adic infinitesimal deformation theory of Eisenstein series and theta series.
 

We prove that if X, Y are derived equivalent complex K3 surfaces, then the integral Chow motives are typically not isomorphic, but are always stably isomorphic. This unfies the result of Huybrechts for isomorphism of rational Chow motives with the result of Mukai for the stable equivalence of integral Hodge structures. Under some mild assumptions, the same result holds for K3 surfaces over nonclosed fields of characteristic zero. For the proofs we rely on known results about zero-cycles on K3 surfaces and set up the machinery of Tate-stable equivalence for integral Chow motives. This is a joint work in progress with Hsueh-Yung Lin and Pavel Sechin.

Let s(n) denote the sum of proper divisors of an integer n. In honour of the Number Theory Boat Seminar, we coin the term "unsinkable numbers" for integers n whose sequence of iterates under s never eventually "sinks" to 1. We will give an overview of what is known about unsinkable numbers as well as some open conjectures.

We will then turn to a different aspect of the sum-of-proper-divisors function: the distribution of the digits of s(n). We will see that s(n) obeys Benford's law and that, asymptotically, almost every value of s(n) contains every decimal digit among both its leading and trailing k(x) digits, where k(x)→∞. We will also discuss recent work showing that values of s(n) missing a decimal digit are surprisingly rare, with prime inputs providing the main contribution, confirming a special case of a conjecture of Erdős, Granville, Pomerance, and Spiro.

This talk is based on joint work with various subsets of the following co-authors: Kübra Benli, Giulia Cesana, Cécile Dartyge, Charlotte Dombrowsky, and Paul Pollack.