Uploaded June 2026 | Updated September 2026, 3 weeks ago

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In his official Abel Prize lecture 2026, laureate Gerd Faltings explores the deep connections in arithmetic geometry in his lecture, "From Abel to Mordell."

Gerd Faltings revolutionized modern mathematics and number theory when he proved the legendary Mordell Conjecture (now Faltings' Theorem)—a breakthrough that originally earned him the Fields Medal in 1986. In this lecture, he traces the mathematical lineage from Niels Henrik Abel’s pioneering 19th-century work up to the geometric insights that define modern arithmetic.

Thumbnail Photo by Ilja. C. Hendel / The Abel Prize

0:00 Introduction and the Egyptians
0:54 Algebraic Varieties and Equations
2:32 Genus 0 Curves and the Circle
8:15 Elliptic Curves and Genus 1
12:40 Abelian Groups and Research History
14:36 Mordell's Conjecture Explained
15:42 Height Functions and Complexity
17:31 Diophantine Approximation (Thue-Siegel-Roth Theorem)
21:21 The 1960s: Shafarevich and Bad Reduction
24:50 Parshin's Contribution to Mordell's Conjecture
26:16 The Tate Conjecture and Abelian Varieties
29:05 Arkhelov Theory and Arithmetic Geometry
32:15 Faltings' Personal Journey and Proofs
40:45 Vojta's Approximation and Recent Developments
45:51 Q&A: Why Genus 2 and Integral Solutions?