In mathematics, the analytic subgroup theorem is a significant result in modern transcendental number theory.
One of the central new ingredients of the proof was the theory of multiplicity estimates of group varieties developed by David Masser and Gisbert Wüstholz in special cases and established by Wüstholz in the general case which was necessary for the proof of the analytic subgroup theorem.
A direct consequence is the Tate conjecture for abelian varieties which Gerd Faltings had proved with totally different methods which has many applications in modern arithmetic geometry.
This was put into an effective form in a joint work of him with Alan Baker which marks the current state of art.
Besides the multiplicity estimates a further new ingredient was a very sophisticated use of geometry of numbers to obtain very sharp lower bounds.