Post #1434258

Gerd Faltings has been awarded the Abel Prize 2026. His work has transformed arithmetic geometry in various fundamental directions, by proving (1983) the Mordell Conjecture (1922) on solutions of diophantine equations defining curves and then the generalization in higher dimension (1989) of that conjecture by Lang. He had also revived (1984) the theory initiated by Arakelov in 1974 and showed its importance for arithmetic geometry. His proof of the Mordell-Lang conjecture uses a theory of heights which is made possible by the higher dimensional Arakelov geometry. The short presentation does not mention his proof (around 1990) of conjectures of Tate, Fontaine and others on p-adic representations associated with algebraic varieties. This work has had a tremendous impact on the theory of motives, modular forms and the Langlands program. It also does not mention that Faltings's initial proof of the Mordell conjecture appeared in a (17-page !) paper where he also proved 2 other conjectures (by Tate and Shafarevich) on abelian varieties, geometric objects that were introduced by Riemann and Jacobi (over the complex numbers) and Weil (over arbitrary fields) to illuminate the algebraic geometry of curves. https://abelprize.no/