Gabriel A. Dill (Universität Bonn): The Modular Support Problem

Herr Gabriel A. Dill (Universität Bonn)

The Modular Support Problem

Abstract:
In 1988, Erdös asked: let a and b be positive integers such that for all n, the set of primes dividing aⁿ − 1 is equal to the set of primes dividing bⁿ − 1. Is a = b? Corrales and Schoof answered this question in the affirmative and showed more generally that, if every prime dividing aⁿ − 1 also divides bⁿ − 1, then b is a power of a.

In joint work that was inspired by the analogy between roots of unity and singular moduli, i.e. j-invariants of elliptic curves with complex multiplication, Francesco Campagna and I have studied this so-called support problem with the Hilbert class polynomials H_D(T ) instead of the polynomials Tⁿ − 1.

In this talk, I will introduce singular moduli, Hilbert class polynomials, and the analogy with roots
of unity and present the results we obtained.

Before this lecture, from 4:45 p.m., there will be coffee and tea again in room 1404. Everyone is warmly invited to attend.