Alina Ostafe, University of South Wales

Title: On some multiplicative problems for matrices

Abstract: In this talk we will discuss two multiplicative problems for matrices: in the first part we will consider various counting problems with multiplicatively dependent integer matrices, while the second part will consider a matrix analogue of the Lang problem on torsion points on plane curves. Although bearing similar multiplicative flavour, these two parts are different in methods and results, with the first one being analytic and the second algebraic. In both cases, the non-commutativity of matrices affects the methods we apply, which are very
different from those used for their scalar analogues.

More precisely, in the first part we give lower and upper bounds for the number of tuples of `multiplicatively dependent' integer matrices in a box, which is motivated by recent work by Pappalardi, Sha, Shparlinski and Stewart (2018) for the scalar case. In the second part we present some results towards a matrix analogue of Lang's problem for $2 imes 2$ matrices defined over $C$. We also pose several problems.

Québec-Vermont Number Theory
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