Effective multiplicative independence of 3singular moduli

Contact: Pieter Moree (moree @ mpim-bonn.mpg.de)

A singular modulus is the j-invariant of an elliptic curve with complex multiplication. Pila and Tsimerman  proved in 2017 that for every k there exists at most finitely many minimal multiplicatively dependent k-tuples of distinct non-zero singular moduli. The proof was non-effective, using Siegel's lower bound for the Class Number. In 2019 Riffaut obtained an effective version of this result for k=2. Moreover, he determined all the couples of 2 singular moduli with a non-trivial multiplicative combination in Q.

I will report on a joint work with Sanoli Gun and Emanuele Tron, where we extend it to k=3. We show that, if 3 distinct singular moduli have a nontrivial multiplicative combination in Q, then their discriminants do not exceed 10^{^10}.