We would like to congratulate the 2026 Abel Prize Recipient, Gerd Faltings, who is being honored “for introducing powerful tools in arithmetic geometry and resolving long-standing diophantine conjectures of Mordell and Lang.”

Solving polynomial equations over the rational numbers is a long-established, fundamental part of mathematics. Systems of such equations, which are known as diophantine equations, can be classified initially by the complex dimension of their solution set. The case of dimension zero is already non-trivial and addressed by Galois theory. Dimension one is the case of curves, which are classified topologically by their genus. The genus of a complex curve is the number of holes of the corresponding two-dimensional real Riemann surface. The diophantine problem for curves of genus zero is governed by the Hasse–Minkowski theorem. A curve of genus one gives an elliptic curve. Henri Poincaré (1854–1912) conjectured in 1901 that the group of rational points on an elliptic curve is finitely generated, and this was proved by Louis J. Mordell (1888–1972) in 1922. In the same paper, Mordell conjectured that a curve of genus two or more has only finitely many rational points. This became the central open diophantine problem for the subsequent 60 years until it was proved by Faltings in 1983. As an example, the result of Faltings establishes the finiteness of rational points on all smooth plane curves of degree four or greater, including the famous Fermat curves for . Faltings's breakthrough proof surprised the experts. Rather than employing diophantine approximation, his approach was via resolving an important case of a conjecture of John Tate (1925–2019) as well as a conjecture of Igor Shafarevich (1923–2017). 

The Abel Prize citation further states that “Gerd Faltings is a towering figure in arithmetic geometry. His ideas and results have reshaped the field, settling major long-standing conjectures, while also establishing new frameworks that have guided decades of subsequent work. His exceptional achievements unite geometric and arithmetic perspectives and exemplify the power of deep structural insight.”

Faltings had previously also received the Fields Medal in 1986 “for proving the Mordell Conjecture, using methods of arithmetic algebraic geometry.”

The Abel Prize is an international prize presented by the King of Norway to one or more outstanding mathematicians. Named after Norwegian mathematician Niels Henrik Abel (1802–1829), the Abel Prize and the Fields Medal have often been described as the "Mathematician’s Nobel Prizes". It comes with a monetary award of NOK 7,5 million (approximately US$ 840.000).
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