• “Faltings’ work has revolutionized arithmetic geometry. He proved some of the most important conjectures formulated in this area throughout the 20th century,” says Daniel Macías (ICMAT-CSIC).
• His proof of the Mordell conjecture was fundamental for later advances in the field.
• The Abel Prize recognizes extraordinary contributions over an entire career in mathematics and is considered the equivalent of the Nobel Prize in this discipline.

Faltings’ work has revolutionized arithmetic geometry. Image: Peter Badge_Typos1_The Abel Prize.

The German mathematician Gerd Faltings has been awarded the Abel Prize, the most prestigious award in mathematics, recognizing an entire career. The jury highlighted his contributions to arithmetic geometry, a field at the intersection of the two oldest branches of mathematics: number theory and geometry.

Faltings, emeritus director of the Max Planck Institute for Mathematics, is the first German to receive the prize from the Norwegian Academy of Science. At just 29 years old, he became, almost overnight, a celebrity within the mathematical community after resolving an important conjecture proposed 60 years earlier. For that work, he received the Fields Medal in 1986.

“Faltings’ work has revolutionized arithmetic geometry, an area that studies problems arising from number theory using the language and tools of geometry,” says Daniel Macías, a researcher at the Spanish National Research Council (CSIC) at the Institute of Mathematical Sciences (ICMAT) and an expert in the field.

“He proved some of the most important conjectures formulated in this area throughout the 20th century,” he adds. His central result, the so-called Faltings’ theorem, answers a simple question to state: how many rational solutions can a Diophantine equation have? These are polynomial equations, that is, combinations of additions, subtractions, and multiplications of variables raised to integer exponents, whose solutions are initially sought only among integers.

One of the great questions of the 20th century

Various ancient civilizations, from the Babylonians to the Mayans, already knew solutions to these equations, such as x² + y² = z², which correspond to the sides of a right triangle. There are infinitely many such triples—the so-called Pythagorean triples. However, as equations become more complex, solutions cease to be infinite and become scarce, sometimes even nonexistent.

Precisely, one of the great questions of the 20th century was to understand in which cases an equation of this type has many, few, or no solutions. If rational solutions (ratios of integers) are also allowed, it is possible to use more powerful geometric and arithmetic tools to analyze the equation. “Quadratic equations (with exponent 2) are relatively simple to study. Cubic ones are much more complicated, but there are examples that have infinitely many rational solutions,” explains Macías.

From that point on, things change. Equations of so-called genus greater than 1 appear, marking the point at which the behavior of Diophantine equations changes radically. In the simplest cases (genus 0 or 1), equations can have infinitely many rational solutions. However, when the genus is greater than 1, the geometric complexity of the curve imposes a very strong arithmetic rigidity.

Mordell’s question

In 1922, the British mathematician Louis Mordell focused on this type of Diophantine equations and conjectured that the set of their rational solutions is finite. Faltings was the one who, 60 years later, proved that his intuition was correct. “This result has had very important consequences for the study of Diophantine equations, including Fermat equations,” says Macías. “In fact, Andrew Wiles proved Fermat’s Last Theorem after Faltings’ theorem. And Faltings was one of the people involved in checking Wiles’ proof,” he adds.

Faltings’ proof surprised experts, as it used a completely unexpected approach. In 1989, another mathematician, Paul Vojta, found another proof of the Mordell conjecture, following more traditional lines. In 1991, Faltings adapted this approach to prove a broad generalization of the Mordell conjecture, the so-called Mordell–Lang conjecture. “Throughout his career, Faltings has proved very important conjectures about algebraic varieties that have had a major impact on the field,” concludes Macías.

Leader of German mathematics

Faltings is emeritus director of the Max Planck Institute for Mathematics in Bonn. Peter Badge_Typos1_The Abel Prize.

Gerd Faltings (1954, Gelsenkirchen, Germany) grew up in an academic environment, with both parents holding doctorates in science. Initially interested in physics, he eventually chose mathematics for its absolute notion of truth. After earning his PhD at the University of Münster (Germany) under the supervision of Hans Joachim Nastold, he worked at Harvard University (USA) and Princeton University (USA).

In 1994, he returned to Germany as a scientific member of the Max Planck Institute for Mathematics in Bonn. A year later, he became director of the institution, a position he held until 2023, when he became emeritus director.

Throughout his career, Faltings has received numerous international awards and distinctions, including the Fields Medal, the Gottfried Wilhelm Leibniz Prize, the Georg Cantor Medal, and the Shaw Prize in Mathematical Sciences.

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