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.
Links:
[1] http://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/de/node/3444
[3] http://www.mpim-bonn.mpg.de/de/node/246