Mahler measures for a family of hyperelliptic curves

I will overview some recent progress on the Mahler measure of two-variable
polynomials corresponding to special elliptic families; these are known as
Boyd's conjectures. A machinery, which was created in our joint work with
Mat Rogers and further developed by Anton Mellit and François Brunault,
allows one to relate the Mahler measures to the $L$-values of the
underlying elliptic curves when the latter are parametrised by modular
units.
The talk will culminate with new results on Boyd's conjectures for a
hyperelliptic family obtained in joint work with Marie José Bertin.