Linear forms in logarithms provided the first unconditional partial results for the abc conjecture, by the work of Stewart-Tijdeman 1986, Stewart-Yu 1991, and Stewart-Yu 2001. An alternative approach using modular forms was developed by R. Murty and Pasten in 2011, giving effective results of similar strength.
In this talk I will recall the abc conjecture. After discussing some applications and the existing methods in the literature to approach the problem, I will introduce a new technique using modular forms. This last method gives bounds of the form d(abc)<rad(abc)^K for a uniform constant K, where d(n) is the number of divisors of n.