More details on the steps of the proof: http://www.math.columbia.edu/~calebji/Faltings_Lawrence_Venkatesh.pdf#page=7
Unsorted/Faltings theorem : for a curves \(C\) with \(g(C) \geq 2\), the number of Unsorted/rational points is finite, i.e. \({\sharp}C({\mathbf{Q}}) < \infty\).
Use of Hilbert 90 and Faltings theorem: 
Manin proves the Mordell conjecture using the Gauss-Manin connection, generalized to Areklov theory. Uses Green functions on curves. Used in Faltings theorem. Imports tools like the adjunction formula, the Hodge index theorem, Riemann-Roch.
Idea: Shaferevich conjecture types of theorems for classes of varieties. The Ax-Schanuel result established by Bakker and Tsimerman for period maps plays a key role in this paper.
- Tags: - #AG/elliptic-curves #open/conjectures - Refs: - Swinnerton-Dyer, Notes on elliptic curves. II - J. T. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. - Links: - elliptic curve - modularity - L function - motivic L function - Application: congruent number problem - Mazur’s theorem - Faltings theorem