1.1. The Dannie Heineman Prize.

The Göttingen Academy of Sciences and Humanities and the Minna James Heineman Foundation has awarded the Dannie Heineman Prize biennially since 1961. It is awarded to young researchers in natural sciences or mathematics for excellent recently published publications in a new research field of current interest. The prize was endowed by Dannie Heineman (1872-1962), who was born in America, trained as an engineer, but worked in Belgium for many years. For 50 years he was the managing director of Sofina, one of the most important public utility engineering management and holding companies in the world.

1.2. Faltings awarded the Dannie Heineman Prize.

The 1983 Dannie Heineman Prize was awarded to Gerd Faltings, Wuppertal, for his outstanding work in mathematics, particularly for his contributions to Diophantine approximation. The award recognised his doctoral and early postdoctoral research on bounds for approximations of algebraic numbers which was establishing him as a rising talent.

2.1. The Fields Medal.

John Charles Fields' will established the Fields Medal. The International Congress of Mathematicians at Zurich in 1932 adopted his proposal, and the Fields Medal was first awarded at the next congress, held at Oslo in 1936. Fields wished that the awards should recognise both existing mathematical work and also the promise of future achievement. To fit with these wishes Fields Medals may only be awarded to mathematicians under the age of 40.

2.2. The Field Medal Committee for the 1986 awards.

The Fields Medals Committee, consisting of P Deligne, J Glimm, L Hörmander, K Ito, J Milnor, J Moser (Chairman), S Novikov, and C S Seshadri arrived at its decisions in early 1986.

2.2. Gerd Faltings awarded the Fields Medal.

Gerd Faltings was awarded the Fields Medal in 1986. Using methods of arithmetic algebraic geometry, he received Medal primarily for his proof of the Mordell Conjecture.

Barry Mazur gave a talk at the International Congress of Mathematicians at Berkeley in 1986 describing some of the mathematical contributions of Gerd Faltings leading to the Fields Medal award. He began as follows:-

One of the recent great moments in mathematics was when Gerd Faltings revealed the circle of ideas which led him to a proof of the conjecture of Mordell.

The conjecture, marvellous in the simplicity of its statement, had stood as a goad and an elusive temptation for over half a century: it is even older than the Fields Medal! In modern language it takes the following form:

If $K$ is any number field and $X$ is any curve of genus > 1 defined over $K$, then $X$ has only a finite number of f-rational points.

To get a feeling for our level of ignorance in the face of such questions, consider that, before Faltings, there was not a single curve $X$ (of genus > 1) for which we knew this statement to be true for all number fields $K$ over which $X$ is defined!

Having described Gerd Faltings's approach to the conjecture of Mordell, Mazur ended his lecture saying:-

... but his other mathematical contributions, whether they be concerned with moduli spaces of abelian varieties, the Riemann-Roch theorem for arithmetic surfaces, or p-adic Hodge theory, all immediately impress one as the work of a marvellously original mind from which we may expect similarly wonderful things in the future.

3.1. The Gottfried Wilhelm Leibniz Prize.

The Gottfried Wilhelm Leibniz Prize, awarded by the Deutsche Forschungsgemeinschaft, is the most important research award in Germany. The Leibniz Programme, established in 1985, aims to honour outstanding scientists and academics, expand their research opportunities, and help them employ particularly qualified early career researchers. A maximum of €2.5 million is provided per award. Prizewinners are first chosen from a slate of nominations put forward by third parties; the Joint Committee selects the actual prizewinners based on a recommendation from the Selection Committee for the Leibniz Programme.

3.2. Gerd Faltings awarded the Gottfried Wilhelm Leibniz Prize.

Gerd Faltings was awarded the Gottfried Wilhelm Leibniz Prize for his work in arithmetic geometry, most notably for proving the Mordell conjecture.

4.1. The Karl Georg Christian von Staudt Prize.

The Karl Georg Christian von Staudt Prize is presented every three to six years by the Otto and Edith Haupt Foundation and the Friedrich-Alexander University of Erlangen-Nürnberg. The prize is awarded to one or more mathematicians working permanently (not just temporarily) at a German university or research institution. The prize honours "outstanding, pioneering and published research results in the field of theoretical mathematics". The award is named after Karl Georg Christian von Staudt, who held the chair of mathematics at the Friedrich-Alexander University of Erlangen-Nürnberg from 1835 to 1867. In 1986 the Otto and Edith Haupt Foundation made a bequest to support mathematical excellence at the University of Erlangen. Otto Haupt was one of Staudt's successors to the chair in Erlangen serving from 1921 to 1953.

4.2. Gerd Faltings awarded the Karl Georg Christian von Staudt Prize.

Gerd Faltings was awarded the von Staudt Prize in 2008 for his outstanding achievements in the field of theoretical mathematics, for the proofs of numerous conjectures in arithmetic geometry and for his research on cohomology and the theory of vector bundles on curves

5.1. The Heinz Gumin Prize.

The Heinz Gumin Prize for Mathematics is granted every three to four years by the Carl Friedrich von Siemens Foundation. The prize bears the name of the mathematician and computer scientist Heinz Gumin (1928-2008), who served as Chairman of the Board of the Carl Friedrich von Siemens Foundation for over 24 years, from 1984 to 2008. With a prize of €50,000, the Heinz Gumin Prize for Mathematics of the Carl Friedrich von Siemens Foundation is the most prestigious mathematics prize in Germany. The Carl Friedrich von Siemens Foundation is an independent institution dedicated to the advancement of science. Since 1960, it has conducted a comprehensive scientific program at its headquarters in Nymphenburg, awarded fellowships to outstanding researchers, and in recent years has provided university libraries in Germany with more than €16 million for the acquisition of urgently needed scientific literature.

5.2. Gerd Faltings awarded the Heinz Gumin Prize.

The Heinz Gumin Prize for Mathematics, first awarded by the Carl Friedrich von Siemens Foundation in 2010, was bestowed upon Professor Faltings, as stated in the award certificate,

"for his groundbreaking methods and results in arithmetic geometry, which have had a lasting impact on the fields of number theory and geometry. His scientific work has brought great international renown to mathematical research in Germany."

The Heinz Gumin Prize for Mathematics of the Carl Friedrich von Siemens Foundation was presented to Gerd Faltings on 19 November 2010, during a ceremony at the foundation's headquarters in Nymphenburg. The laudatory speech was given by mathematician Michael Rapoport [an extract is given below]. Friedrich Hirzebruch from the Max Planck Institute for Mathematics in Bonn spoke about the prize's namesake.

5.3. Michael Rapoport gives the laudatory speech for Faltings Heinz Gumin Prize.

I have been given the task of delivering the laudatory speech for the recipient of the first Heinz Gumin Prize of the Carl Friedrich von Siemens Foundation, Professor Dr Gerd Faltings of the Max Planck Institute for Mathematics in Bonn. This is certainly an honour. At the same time, I consider it a difficult task, given the very diverse audience gathered here, ranging from close colleagues to professional mathematicians and even laypeople whose last contact with mathematics was during their school days (and which they may only recall with horror). My aim is to:

1. explain why Faltings was an excellent choice as the first recipient of the Heinz Gumin Prize,

2. introduce Faltings as a person,

3. try to convey some of the fascination of Faltings's scientific field.

It is clear that I cannot truly achieve the third goal in particular; but if I could elicit from you the phrase (a saying of my 98-year-old mother) that "I don't understand it, but I find it burning with interest," I would be satisfied. Therefore, contrary to the general advice of my colleagues, I dared to project slides with mathematical formulas onto the wall: I wanted to try to truly involve you in the topics discussed in Faltings' mathematics.

Gerd Faltings was born in 1954. His mathematical talent was already evident as a schoolboy, which he demonstrated by winning the German Federal Mathematics Competition twice. Faltings studied mathematics and physics in Münster from 1972 to 1978. He graduated with a diploma and received his doctorate in the same year (both under J Nastold). He then spent a year abroad at Harvard University and followed this with a research assistantship in Münster from 1979 to 1982, where he completed his habilitation in 1981. During this time, Faltings published 13 papers on commutative algebra, particularly on finiteness theorems concerning local cohomology groups. In 1982, at the age of 27, Faltings was appointed to the University of Wuppertal. That same year brought his major breakthrough. First, he made fundamental contributions to Arakelov geometry, especially Faltings' index theorem and the Faltings-Riemann-Roch theorem. Second, he proved the three major finiteness theorems of arithmetic algebraic geometry: the Mordell conjecture (finiteness of the set of rational points on a smooth projective curve of genus ≥ 2 over a number field), the Shafarevich conjecture (finiteness of the set of smooth projective curves of a given genus over a number field with a given set of bad reduction points), and the Tate conjecture (on homomorphisms of abelian varieties). More precisely, Faltings, partly following a programme by L Szpiro, proves the Shafarevich conjecture in a more rigorous form, from which (according to A Parshin) the Mordell conjecture and (according to J Tate) the Tate conjecture follow. Faltings' work is only 17 pages long; however, this brevity is also a result of the subsequently infamous telegraphic style of Faltings' work, which makes it so difficult for the reader. I give a diagram which explains "the Mordell." It deals with solutions of polynomial equations, more precisely with the investigation of the geometric structure of their solution set. This is the subject of algebraic geometry. In arithmetic algebraic geometry, the focus is specifically on equations with integer coefficients and their integer solutions. In recent decades, methods from topology, especially cohomological methods, have been made available for these investigations. Faltings' entire body of work after his beginnings in commutative algebra is dedicated to this area of ​​problem-solving, but with a mathematical and methodological breadth that is unparalleled. The "Mordell" is the spectacular beginning, but by no means the end. The fact that his later results are more difficult to explain to the average person does not mean that they are of lesser importance.

Faltings made the leap from the relatively narrow world of commutative algebra to arithmetic algebraic geometry by participating in various advanced training courses at the Mathematical Research Institute in Oberwolfach. It was also in Oberwolfach that he presented his very recent proof of the Tate conjecture at a conference on commutative algebra (although of the 30 conference participants, only 5 attended his presentation, which, in fact, did not address any topic in commutative algebra).

Here I'd like to make a comment for those of you who aren't mathematicians. You may have seen the film "Good Will Hunting," in which a highly gifted boy, who, due to mental health issues, only manages to become a janitor (but at least he's at MIT), solves a major mathematical problem without any formal training. A similar theme appears in other Hollywood films. But believe me: although Faltings was very young when he proved this theorem, he was already a highly educated and thoroughly competent mathematician and a brilliant technician. In fact, I consider it impossible for someone without rigorous academic training to prove such a significant theorem.

In 1984, he married Angelika Faltings, née Tschimmel, and they had two daughters, Christina (born 1985) and Ulrike (born 1988). In 1985, Faltings accepted a professorship at Princeton University, where he remained until 1994. During this time, Faltings initially focused on two sets of problems. Inspired by his proof of the Tate conjecture, he published a construction of arithmetic compactifications of the Siegel moduli space of abelian varieties. In doing so, he significantly expanded upon earlier work on the corresponding construction over $\mathbb{C}$ by D Mumford (with A Ash, M Rapoport, and Y Tai) and on the arithmetic compactification of the Hilbert moduli space of abelian varieties, which I had presented in my thesis. The other set of problems that continues to occupy Faltings is the Padian Hodge theory, i.e., the connection between Padian etal cohomology and crystalline cohomology. This connection - called the "mysterious functor" by A Grothendieck - was first described by J-M Fontaine had predicted this, and Fontaine and W Messing had found an approach to this problem that subsequently led to success with the work of K Kato and T Tsuji. Faltings developed a highly original, completely different method that still possesses great potential that remains largely untapped even today.

In 1986, Faltings received the Fields Medal at the International Congress of Mathematicians in Berkeley. At the age of 32, he was well below the age limit of 40 for Fields Medalists.

Another area of ​​interest for Faltings subsequently became Diophantine approximation. Inspired by the work of P Vojta, Faltings proved the Lang conjecture on abelian varieties over number fields (the finiteness of the set of rational points of a subvariety of an abelian variety that does not contain a translation of a subgroup of positive dimension). This theorem also provides a new proof of Mordell's conjecture. The main tool of the proof is Faltings' product theorem, which replaces Roth's lemma from classical Diophantine approximation in arithmetic algebraic geometry. Faltings and G Wüstholz subsequently used this tool to provide a new proof of Schmidt's subspace theorem. The following diagram illustrates the elegance and impact of Faltings' conceptualisation in these works. The definition sounds banal, and understanding it requires only knowledge of linear algebra. Although an analogue had existed for some time in the theory of vector bundles, it was Faltings who recognised the full implications of this conceptualisation in other theories as well.

In 1994, Faltings returned to Germany and became a director at the Max Planck Institute in Bonn. He actively participates in the mathematical life of Bonn, attends the weekly colloquium, and regularly gives a lecture course on arithmetic algebraic geometry at the University of Bonn each semester. Faltings received the Leibniz Prize in 1986 and the von Staudt Prize in 2008. To this day, Faltings continues to make important contributions to algebraic geometry.

[Note. We omit the more technical description of Faltings' contributions.]

A report on Faltings would be incomplete without mentioning his strong sense of duty: for instance, he served as editor-in-chief of Inventiones for twelve years, a task he took very seriously, often writing the referee's report himself. But Faltings also has interests outside his profession: he's an avid gardener, his house has been nominated for an architectural award, and his wine cellar contains some fine bottles. As for his sense of humour, it's sometimes as complex as his scientific work. Some of you may wonder how it feels for Faltings's colleagues to have such an outstanding mathematician in their field, and whether they might feel intimidated or even frustrated by it. But science is a collaborative endeavour and doesn't simply produce winners and losers in a competition. Furthermore, we are immensely proud that a Faltings is interested in our field. And finally, Faltings has consistently provided new impetus to our field, removed obstacles, and thus ensured its continued progress. And nothing is more frustrating in science than stagnation.

And so I would like to conclude with this exclamation: We are delighted to have you with us, Gerd - and congratulations on today's award!

6.1. The King Faisal International Prize.

The King Faisal International Prize is awarded to "scientists and scholars whose research results in significant advances in specific areas that benefit humanity." It consists of a certificate, hand-written in Arabic calligraphy summarising the laureate's work, a commemorative 24 carat gold medal, uniquely cast for each prize, and a cash award of 750,000 Saudi riyal (150,000 Euro). Prizes are awarded in five categories: Service to Islam; Islamic Studies; Arabic Language and Literature; Medicine; and Science. The Science Prize is announced to be for: Physics; Mathematics; Chemistry; Biology.

6.2. Inception of the King Faisal Prize.

The following is taken from the website of the King Faisal Prize in April 2026.

The idea of establishing the King Faisal Prize came out of the belief in the importance of knowledge and science in opening broader horizons in all fields. It also confirms the cause for which the Foundation was established, which is to spread goodness and hope everywhere, and acknowledge and reward the efforts of scholars and intellectuals in the development and welfare of mankind. From the day it was first established in 1979, the King Faisal Prize has been awarded to 275 laureates, from 43 countries. Many of the Prize's objectives have been reached, and the Foundation has assumed an outstanding leadership in its efforts to honour some of the world's scholars and scientists has been well acknowledged. In this regard, 24 among the Prize recipients in medicine and sciences were subsequently awarded the Nobel Prize in their respective fields.

6.3. Gerd Faltings awarded the King Faisal Prize.

Gerd Faltings, Director at the Max Planck Institute for Mathematics in Bonn and Professor at the University of Bonn, was awarded the 2014 King Faisal International Prize for Science for his groundbreaking contributions to algebraic geometry and number theory. This was announced by the president of the King Faisal Foundation, Prince Khaled Al-Faisal, on 14 January 2014.

Previous Mathematics winners of the Prize were: Michael Atiyah (1987); Dennis P Sullivan (1994); Andrew J Wiles (1998); Peter W Shor (2002); Yuri I Manin (2002); Simon Kirwan Donaldson (2006); Mudumbai S Narasimhan (2006); Terence Chi-Shen Tao (2010); Enrico Bombieri (2010).

The prize committee stated that Professor Faltings "work combines ingenuity, vision and technical power. He has introduced stunning new tools and techniques which are now constantly used in modern mathematics. His deep insights into the p-adic cohomology of algebraic varieties have been crucial to modern developments in number theory. His work on moduli spaces of abelian varieties has had great influence on arithmetic algebraic geometry. He has introduced new geometric ideas and techniques in the theory of Diophantine approximation, leading to his proof of Lang's conjecture on rational points of abelian varieties and to a far-reaching generalisation of the subspace theorem. Professor Faltings has also made important contributions to the theory of vector bundles on algebraic curves with his proof of the Verlinde formula."

6.4. The Laureate's Biography.

Gerd Faltings studied mathematics and physics from 1972 to 1978 at the Westphalian Wilhelm University of Muenster. He received his diploma and Ph.D. in 1978, then was appointed as a visiting scientist at Harvard University from 1978 - 1979. Between 1979 and 1982 he worked as a Scientific Assistant at the University of Muenster where he received his habilitation in 1981. Between 1982 and 1984, he held a professorship in Pure Mathematics at the University of Wuppertal and was the youngest professor of mathematics in Germany. Between 1985 and 1994, he was appointed as a professor of Mathematics at Princeton University. He then came back to Germany in 1994 as a Scientific Member of the Max Planck Institute for Mathematics in Bonn and became its Director in 1995.

Professor Faltings has made seminal contributions to mathematics, particularly to algebraic geometry, number theory and arithmetic. At the age of 27, he made a breakthrough which revolutionised the Arakelov theory by proving his index theorem and the Faltings-Riemann-Roch theorem. During the following two years, he proved three major arithmetic finiteness theorems: the Mordell Conjecture, the Tate Conjecture and the Shafarevich Conjecture, all of which have become attached to his name. He gained fame through his proof of the Mordell conjecture, a problem about Diophantine equations that date back to the Greek era. He introduced new geometric ideas and techniques to the theory of Diophantine approximation, which have led to his proof of Lang's conjecture on rational points of abelian varieties and to a far-reaching generalisation of the subspace theorem. He has also made important contributions to the theory of vector bundles on algebraic curves with his proof of the Verlinde formula.

Professor Faltings has authored numerous publications in leading mathematical journals and is Associate Editor of Compositio Mathematica and Editorial Board Member of the Journal of Algebraic Geometry. His accomplishments in mathematics have been recognised by numerous awards and honours, including the Dannie Heineman Prize of the Göttingen Academy of Sciences (1983); Fields Medal of the International Mathematical Union, which he received the medal for proving the Mordell conjecture (1986), a proof that led him to interesting research on the toroidal compactification of the moduli space of Abelian varieties and on the relationship between p-adic estate and crystalline cohomology. Professor Faltings is a member of the National Academy of Sciences Leopoldina, and the North Rhine Westphalian Academy of Sciences and Arts.

6.5. The Laureate's Reply.

Speech of Professor Gerd Faltings 2014 Winner of the King Faisal International Prize For Science

36th KFIP Awards Ceremony Sunday 30/3/2014 (29/5/1435H.)

Your Royal Highness Prince Salman Bin Abd Al-Aziz, Crown Prince, Deputy Premier, Minister of Defence, Your Highnesses, Your Eminences, Your Excellencies, Distinguished Guests.

I am very much honoured to receive this years King Feisal prize for science. I am from a country which was culturally backward during the times when the Roman and Arabic civilisations already flourished. But we managed to catch up and are part of the European culture which dominated the world for some time, mainly because of advances in science and engineering. Now this era comes to an end as other nations are catching up, and it does not take them a thousand years. I am very pleased that Saudi Arabia strikes hard to develop from a country known for its mineral wealth to become a centre of learning and research, and I see the King Faisal prize as a sign for the great esteem to which science is held in this country.

7.1. The Shaw Prize.

Mr Run Run Shaw, a visionary philanthropist, held a steadfast conviction in the power of knowledge. He recognised the critical role that scientists play in illuminating the intricate mysteries of nature, and understood that their tireless efforts are fundamental to the advancement of civilisation.

In 2002, under the auspice of Mr Shaw, the Shaw Prize Foundation was established. The inaugural Shaw Prize was presented two years later in 2004. The Shaw Prize consists of three annual awards, namely the Prize in Astronomy, the Prize in Life Science and Medicine, and the Prize in Mathematical Sciences. Each prize carries a monetary award, which has been set at of one million two hundred thousand US dollars since 2016.

The Shaw Prize honours individuals, regardless of race, nationality, gender, and religious belief, who are currently active in their respective fields and who have recently achieved distinguished and significant advances, who have made outstanding contributions in academic and scientific research or applications, or who in other domains have achieved excellence. The Shaw Prize is dedicated to furthering societal progress, enhancing quality of life, and enriching humanity's spiritual civilisation.

Since 2004, the Shaw Prize has recognised over a hundred exceptional individuals who have made ground-breaking contributions to their respective fields, many of whom have gone on to receive other prestigious international awards. The Shaw Prize Foundation has also taken a proactive role in advancing scientific literacy through a range of engaging activities, including lectures, public forums, exhibitions, and other outreach programmes, in partnership with esteemed local and international universities and institutions.

The Shaw Prize Foundation, committed to upholding Mr Shaw's vision, remains dedicated to the promotion of excellence and innovation, and aspires to serve as a beacon of inspiration for those who seek to make a positive impact on society. Through its unwavering commitment to this mission, the Shaw Prize Foundation is poised to encourage and elevate the next generation of scientists and innovators, for the benefit of humankind.

7.2. Gerd Faltings awarded the Shaw Prize.

The Shaw Prize in Mathematical Sciences 2015 was awarded to Gerd Faltings, Managing Director at Max Planck Institute for Mathematics in Bonn, Germany for his introduction and development of fundamental tools in number theory, allowing him as well as others to resolve some longstanding classical problems.

Number theory concerns whole numbers, prime numbers, and polynomial equations involving them. The central problems are often easy to state but extraordinarily difficult to resolve. Success, when it is achieved, relies on tools from many fields of mathematics. This is no coincidence since some of these fields were introduced in attempts to resolve classical problems in number theory. Faltings has developed many of the most powerful modern tools in algebra, analysis, algebraic and arithmetic geometry, automorphic forms, and the theory of zeta functions. He and others have used these tools to resolve longstanding problems in number theory.

A polynomial equation of degree n in one variable with coefficients which are rational numbers has just n complex numbers as solutions. Such an equation has a symmetry group, its Galois group, that describes how these complex solutions are related to each other.

A polynomial equation in two variables with rational coefficients has infinitely many complex solutions, forming an algebraic curve. In most cases (that is, when the curve has genus 2 or more) only finitely many of these solutions are pairs of rational numbers. This well-known conjecture of Mordell had defied resolution for sixty years before Faltings proved it. His unexpected proof provided fundamental new tools in Arakelov and arithmetic geometry, as well as a proof of another fundamental finiteness theorem - the Shaferavich and Tate Conjecture - concerning polynomial equations in many variables. Later, developing a quite different method of Vojta, Faltings established a far-reaching higher dimensional finiteness theorem for rational solutions to systems of equations on Abelian Varieties (the Lang Conjectures). In order to study rational solutions of polynomial equations by geometry, one needs arithmetic versions of the tools of complex geometry. One such tool is Hodge theory. Faltings' foundational contributions to Hodge theory over the $p$-adic numbers, as well as his introduction of other related novel and powerful techniques, are at the core of some of the recent advances connecting Galois groups (from polynomial equations in one or more variables) and the modern theory of automorphic forms (a vast generalisation of the theory of periodic functions). The recent striking work of Peter Scholze concerning Galois representations is a good example of the power of these techniques.

7.3. Gerd Faltings' autobiography.

I was born 28 July 1954 in Gelsenkirchen, an industrial town in the then coal mining region of Germany called "Ruhrgebiet". My parents originate from the Hamburg region and had PhD's in Physics and in Chemistry. After primary school I attended the Max-Planck-Gymnasium, a high school, in Gelsenkirchen which I finished in 1972 with my Abitur. During my last years in high school I participated twice successfully in the "Bundeswettbewerb Mathematik", a competition for high school students interested in mathematics. As a result I became a member of the "Studienstiftung des deutschen Volkes", a foundation dedicated to the support of talented students.

In the fall of 1972 I started to study mathematics at the University of Münster near Gelsenkirchen. Interrupted by 15 months of obligatory military service I finished my studies in 1978 with a diploma (in local cohomology) and a PhD (in Macaulayfication). My advisor was Professor H J Nastold who was specialising in commutative algebra, and so the topics are from that field. He co-organised a regular Oberwolfach meeting (together with Berger, Kunz, Szpiro) on commutative algebra.

The PhD enabled me to obtain a stipend from the Deutsche Forschungsgemeinschaft (a German NSF) to spend one year at Harvard University. My host was Professor Hironaka, to whom I was recommended by Professor Matsumura, an old friend of Professor Nastold. At Harvard I first learned about toroidal embeddings, a subject which became important to me later. Returning to Münster I became assistant to Professor Nastold and got my Habilitation in 1981. This allowed me to apply for professorships and to my surprise, the first application was successful. From 1982-84 I was a full professor at the University of Wuppertal. There I managed to prove the Mordell conjecture over number fields, and this success changed my personal circumstances considerably. For example, I received my first prize, the Dannie-Heinemann award from the Academy in Göttingen. It involved a considerable amount of cash. Usually, such prizes are only given to established researchers who do not need the money anymore, so this was a pleasant exception. Also, in Wuppertal I met my future wife Angelika, and we got married in December 1984.

The Mordell conjecture was an old open problem and had been solved for function fields by (among others) Parshin and Arakelov. Szpiro had extended their theory to positive characteristics and tried to use Arakelov theory (another invention by Arakelov) to extend this to number fields. Unfortunately, one ingredient (the Kodaira-Spencer class) was missing. I was very fortunate to find that it could be replaced by a tool from the theory of Galois representations. Also, I profited very much from an Oberwolfach meeting (Arbeitsgemeinschaft) on the paper by Harris-Mumford proving that the moduli space of curves is usually of general type. So far, my knowledge of the theory of moduli spaces covered only the construction, after which everybody seemed to be exhausted. That they could be used for something was new to me.

From 1985-1994 I was full professor at Princeton University. In 1986 I was awarded a Fields Medal at the ICM in Berkeley, and especially recently this has been followed by quite a number of awards. Also, during my stay at Princeton my two daughters Christina and Ulrike were born (1985 and 1988). My mathematics at Princeton centred around two topics where an ad hoc solution was sufficient for the Mordell, but the full picture required more work. These were toroidal compactifications of the Siegel moduli space, and $p$-adic Hodge theory. On the first topic I wrote a book jointly with C L Chai, and on the second I extended ideas of J Tate to define "almost étale coverings". In addition, I learned about a new idea of Vojta leading to a new proof of Mordell via diophantine approximation. Everybody talked highly about it but nobody seemed to be ready to declare it correct. Out of a sense of duty I studied it, found it to be correct, and as a reward also realised that the method allowed a vast generalisation (via the "product theorem"). At Princeton I also heard lectures from E Witten about string theory. As a result, I concluded firstly that physics is not a branch of mathematics, and secondly, was inspired to study moduli spaces of vector bundles on curves where I could show some new results. At Princeton I also was awarded a fellowship from the Guggenheim Foundation (1988).

In 1994 I accepted an offer from the Max Planck Society to become one of the directors of the Max Planck Institute for Mathematics in Bonn, and I am still there. Shortly after resettling I was awarded a Leibniz-Preis (1996). I continued my work and watched my daughters grow up. Unfortunately, my wife died in 2011 so I am a widower. In recent years I have been honoured with a number of awards: von Staudt Preis 2008, Heinz Gumin Preis 2010, honorary degree from Münster 2012, King Faisal prize 2014, and finally the Shaw Prize 2015.

24 September 2015
Hong Kong

8.1. The Georg Cantor Medal.

The Deutsche Mathematiker-Vereinigung (DMV) is the German Mathematical Society. Founded in 1890, the DMV promotes and supports mathematics in its full breadth; mathematics as a science in the full range from theory to applications, mathematics in schools, universities, and industry, as well as its presentation in the media and society. In memory of its first chairperson, the DMV awards the Georg Cantor Medal, along with a certificate, for outstanding scientific achievements in mathematics. The DMV awards its Georg Cantor Medal every two years. Recipients should have distinguished themselves through outstanding scientific achievements in mathematics and have ties to the German-speaking world. All DMV members may submit nominations. Recipients are invited to deliver a keynote address at the next DMV annual meeting. The first award was made in 1990.

8.2. Gerd Faltings awarded the Georg Cantor Medal.

Gerd Faltings, Scientific Member and Director at the Max Planck Institute for Mathematics in Bonn, Honorary Professor of Mathematics at the University of Bonn, and member of the Hausdorff Centre for Mathematics, the Bonn Cluster of Excellence for Mathematics and Mathematical Economics, has been awarded the Cantor Medal of the German Mathematical Society (DMV) by decision of the DMV's Executive Committee. Faltings' work revolutionised algebraic geometry and also had a significant impact on other areas of mathematics, such as number theory. Faltings gained international recognition in 1983 when, as a 28-year-old mathematics professor, he proved Mordell's Conjecture - a landmark proof of a problem formulated in 1922 by the American-British mathematician Louis Joel Mordell. With the Cantor Medal, the DMV honours Gerd Faltings' lifetime achievement. The medal was presented to Faltings in September 2017 at the DMV Annual Meeting in Salzburg.

9.1. The Order Pour le Mérite.

The Order Pour le Mérite stretches back to the 19th century. King Friedrich Wilhelm IV of Prussia founded the Order in 1842 as the civil class of the military order of the same name which had been established in 1740. The realisation of the civil class of conferrals for science and the arts was the responsibility of German natural scientist Alexander von Humboldt (1769-1859), the Order's first Chancellor. The Order was to bring together leading intellectuals, from Germany and abroad, from the most diverse disciplines and arts. The Order Pour le Mérite for Sciences and the Arts was re-established in 1952 by Federal President Theodor Heuss. Ever since then, the Federal President in office has been the Protector, or patron, of the Order. Its 2015 statutes declare it to be a cross-disciplinary association of scientists and artists "who have made an outstanding name for themselves through the widespread recognition of their achievements in science or the arts".

9.2. Gerd Faltings awarded the Order Pour le Mérite.

Gerd Faltings was admitted to the Order Pour le Mérite in 2024. On 1 June 2025 at the Spring Conference at Bellevue Palace, Berlin, he received the award of the Grand Order of Merit of the Federal Republic of Germany with Star.

Faltings was born on 28 July 1954 in Gelsenkirchen-Buer, the son of a physicist and a chemist. During his school years, he participated twice in the Federal Mathematics Competition of the Stifterverband (Donors' Association for the Promotion of Sciences and Humanities in Germany) and was accepted as a national winner into the German National Academic Foundation. After graduating from high school, he studied mathematics and physics at the University of Münster. In 1978/79, he was a visiting scholar at Harvard University in Cambridge, Massachusetts. Returning to Münster, he became an assistant to Professor Nastold in 1979 and received his habilitation in 1981. As a professor in Wuppertal, he achieved great success and moved to Princeton University in New Jersey, USA, as a full professor in early 1985.

Among his early awards were the Danny Heinemann Prize of the Göttingen Academy in 1984 and the Fields Medal in Berkeley in 1986, an award presented by the International Mathematical Union only every four years at its congress to young mathematicians under the age of 40. As his daughters grew older, he returned to Germany and was a scientific member of the Max Planck Society at the Max Planck Institute for Mathematics in Bonn from 1994 until his retirement in 2023.

Mathematically, he began his research in the field of commutative algebra, the specialty of his teacher, Nastold. Nastold also facilitated contact with Professor L Szpiro in Paris, who had ideas related to the Mordell Conjecture. Faltings found this very interesting and worked on it, hoping to achieve some useful partial result. To his surprise, in 1983 he was able to prove the conjecture in his paper "Finality Theorems for Abelian Varieties over Number Fields" (Faltings' Theorem) and became a star overnight. Subsequently, he worked on compactifications of moduli spaces and $p$-adic Hodge theory. Both areas played a crucial role in the Mordell conjecture and were initially addressed with ad hoc constructions, which he then replaced with a more systematic theory. Next, fate brought him a paper by P Vojta on Diophantine approximation, which he was able to generalise significantly. Finally, he attended a lecture by E Witten at the Institute for Advanced Study at Princeton. The lecture contained interesting statements on modulo spaces of bundles, and he was able to achieve a whole series of mathematical results in this area.

Gerd Faltings is a member of the academies in Düsseldorf, Göttingen, Berlin, and Halle, the European Academy, the Royal Society (London), and the National Academy of Sciences (Washington). In Germany, he received the Leibniz Prize in 1996, the von Staudt Prize in 2008, the Heinz Gumin Prize in 2010, and the Georg Cantor Medal in 2017. International awards included the King Faisal International Prize in 2014 and the Shaw Prize in 2015.

10.1. The Abel Prize.

The Abel Prize was awarded for the first time in 2003 but it was first suggested over 100 years earlier. Sophus Lie, when he saw that Nobel's plans for annual prizes did not include one for mathematics, proposed the setting up of an Abel Prize which would be awarded every five years. He contacted mathematicians world-wide and gathered wide support. However he had not set up any machinery to carry the idea forward and when he died soon after this, in 1899, nothing further happened.

The year 1902 was one in which the centenary of Abel's birth was celebrated. A decision was again taken to establish an international Abel Prize but again the plan did not come to fruition. With the bicentenary of Abel's birth approaching, Arild Stubhaug, who had written a major new biography of Abel, made another attempt to set up an Abel Prize.

A committee was set up which gathered support both within Norway and also international support. They put their proposals before the Norwegian government in May 2001 and in a speech on the campus of the University of Oslo in August 2001, the Norwegian Prime Minister announced that the Government would establish an Abel Fund.

The Norwegian Academy of Science and Letters announces the winners of the Abel prize, the first being awarded in 2003.

10.2. Gerd Faltings awarded the Abel Prize.

The Norwegian Academy of Science and Letters awards the Abel Prize 2026 to Gerd Faltings of the Max Planck Institute for Mathematics, Bonn, Germany

... 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).

In 1989, Paul Vojta found another proof of the Mordell conjecture, following the more traditional lines initiated by André Weil (1906-1998) and Carl Ludwig Siegel (1896-1981). In 1991, Faltings adapted this approach to prove a vast generalisation of the Mordell conjecture, namely, the Mordell-Lang conjecture on subvarieties of abelian varieties. An abelian variety, generalising an elliptic curve, is a complete projective variety having a group structure. Mordell's finite generation result was extended to rational points of abelian varieties by Weil. Here, rational may be taken to be over any fixed number field.

The Mordell-Lang conjecture describes the distribution of rational points in any subvariety of an abelian variety. More precisely, it says that all such rational points are contained in the union of finitely many subsets of the given subvariety, each of which is a translate of an abelian subvariety by a rational point.

To prove the Mordell-Lang conjecture, Faltings established a diophantine approximation result known as Faltings's product theorem. This generalizes a key result of Klaus F Roth (1925-2015) used in the proof of his celebrated theorem on the approximation of algebraic numbers by rational numbers. In 1994, using the product theorem, Faltings and Gisbert Wüstholz gave a new proof of Roth's theorem and its multi-dimensional generalisation known as the Schmidt subspace theorem. In his 1991 paper, Faltings also proved the finiteness of integral points on affine subvarieties of abelian varieties as conjectured by Serge Lang (1927-2005). Faltings's work still stands as the central pillar in modern diophantine geometry.

Classical Hodge theory relates the topology of complex manifolds to their differential geometry. In a similar spirit, $p$-adic Hodge theory studies the natural structures carried by the cohomology of algebraic varieties over $p$-adic fields, where Galois and Frobenius actions encode arithmetic and geometric information, respectively.

Faltings made major contributions to $𝑝$-adic Hodge theory, giving proofs of the main conjectures formulated by Tate and Jean-Marc Fontaine (1944-2019), and extending its scope to the non-abelian setting under the name of $p$-adic Simpson correspondence. The conjectures of Tate and Fontaine relate $p$-adic étale cohomology (which plays the role played by Betti cohomology in the classical theory) and de Rham cohomology; the non-abelian version relates $p$-adic representations of the fundamental group and Higgs bundles.

Tools introduced by Faltings have proved crucial to subsequent developments in $p$-adic Hodge theory and commutative algebra. These include the purity theorem and his notion of almost étale extensions (clarified through the work of Ofer Gabber and Lorenzo Ramero, and subsequently strengthened by Peter Scholze).

Elliptic curves over the complex numbers are parameterised up to isomorphism by points of the modular curve. The modular curve arises as the quotient of the upper-half plane by the group of two-by-two integral matrices of determinant one, acting by linear fractional transformations. Abelian varieties are similarly parameterised by the points of Siegel modular varieties. Faltings, in a 1990 monograph with Ching-Li Chai, constructed an arithmetic compactification of these varieties. Their work became a cornerstone for subsequent developments in the theory of integral models and compactifications of Shimura varieties.

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.

10.3. Faltings' biography written for The Abel Prize by Timandra Harkness.

Gerd Faltings was born on 28 July 1954 in the Buer district of Gelsenkirchen, an industrial town in Germany's Ruhrgebiet region. His parents both held Ph.D.s in science - his father in physics and his mother in chemistry. He later said that Physics was his first interest, but then he preferred mathematics "because things are either true or false, it's not a matter of opinion."

In Gelsenkirchen he attended the Max-Planck-Gymnasium secondary school, where he won two national prizes in the Bundeswettbewerb Mathematik.

From 1972 he studied mathematics at the University of Münster, interrupted by 18 months of compulsory military service. With Hans Joachim Nastold as his supervisor he gained his Ph.D. in 1978 in Commutative Algebra, with a thesis titled "Über Macaulayfizierung" (On Macaulayfication).

Then followed a year as a research fellow at Harvard, thanks to a stipend from the German science foundation (DFG), specialising in algebraic geometry, and toroidal embeddings in particular.

Moving back to Münster in 1979, Faltings assisted H J Nastold until his habilitation in 1981, with a focus on formal geometry and local cohomology.

He then moved to Wuppertal as a full Professor in 1982, aged 28. The following year, he proved the Mordell Conjecture - thus turning it into the Faltings Theorem - and "somehow became famous overnight." This theorem asserts the finiteness of rational points on algebraic curves of genus greater than one. The Göttingen Academy of Sciences and Humanities awarded him the Dannie-Heineman Prize.

While working at Wuppertal, he met fellow mathematician Angelika Tschimmel, and they married in 1984. In 1985 Gerd took up a full Professorship at Princeton University in New Jersey, where their two daughters Christina and Ulrike were born. There he continued his research into toroidal compactifications and $p$-adic Hodge theory. With Gisbert Wüstholz he re-proved Roth's Theorem on Diophantine Approximations of algebraic numbers.

In 1986, the International Congress of Mathematicians awarded him the Fields Medal for his work in algebraic geometry. His work on the Mordell Conjecture - now the Faltings Theorem - continued, with the help of a Guggenheim Fellowship in 1988. Building on a new proof by Paul Vojta in 1989, Faltings proved the Mordell-Lang Conjecture, further generalising the Mordell results.

In 1994 Faltings and his family moved back to Germany where he took up a post as Scientific Member at the Max Planck Institute for mathematics in Bonn.

As a director at the Max Planck Institute since 1995, Faltings has enjoyed exceptional freedom to pursue his research. He continued his work on moduli spaces, introducing the concept of 'almost étale coverings' and collaborating with Chiang-Li Chai on the book "The Geometry of Moduli Spaces of Abelian Varieties".

Many more honours followed: The Gottfried Wilhelm Liebniz Prize in 1996; the Karl Georg Christian von Staudt Prize in 2008; the Heinz Gumin Prize in 2010; and the Georg Cantor Medal in 2017. International Prizes too: the King Faisal International Prize for Science in 2014 for "seminal contributions to mathematics, particularly to algebraic geometry, number theory and arithmetic," and the Shaw Prize for Mathematical Sciences in 2015, shared with Henryk Iwaniec "for their introduction and development of fundamental tools in number theory, allowing them as well as others to resolve some longstanding classical problems."

Faltings is a member of the Academies of Düsseldorf, Gõttingen, Berlin and Halle, and in 2024 was elected a member of the Order Pour le Mérite, founded in 1842 and now awarded by the German federal government. In 2016 he became a Fellow of the UK Royal Society, and in 2018 a member of the National Academy of Sciences in the US.

In 2011 his wife Angelika sadly lost her life to cancer.

An emeritus director at the Max Planck Institute since 2023, Faltings continues his research in arithmetic geometry. Alongside his work, he enjoys opera, gardening, and collecting fine wines (and, presumably, drinking them).

10.4. From Mordell's conjecture to Falting's theorem.

From a historical perspective, mathematical science is founded on two pillars: number theory and geometry. The core of number theory is the natural numbers and their extension to the rational numbers. In primary school, children learn about addition and multiplication, and the connection between them; multiplication can be considered as repeated addition. But this somewhat naive approach does not reveal the more mysterious side of numbers. An example of how combining addition and multiplication can lead to difficult problems is the classic question of whether it is possible to add two squares and get a square as the answer. This question is closely related to the Pythagorean theorem, which provides a simple connection between the (squares of) the sides of a right-angled triangle. The number-theoretic question behind the theorem concerns integer solutions $(a, b, c)$ of the equation $a^{2} + b^{2} = c^{2}$, often referred to as Pythagorean triples.

It has been known since ancient times that there are infinitely many Pythagorean triples. The simplest example is reflected in the so-called carpenter's triangle. The carpenter's triangle is a right-angled triangle with sides of lengths 3, 4, and 5. Since we have that $3^{2} + 4^{2} = 5^{2}$, one of the angles in this triangle will be right, and we can use this to make sure that the corners of the house are right-angled. A probability-based explanation for the existence of infinitely many Pythagorean triples is that the number of perfect squares is a sufficiently dense subset of the natural numbers. Thus, the set of natural numbers that are a sum of two squares and the set of squares have an infinite overlap.

Moving on to the cubic numbers, the frequency of such numbers within the set of natural numbers will be far more rare than the set of squares, and it is not obvious that the analogous problem of finding Pythagorean triples now has any positive solution. In fact, according to Fermat's Last Theorem, it is not possible at all. This was proved by the Briton Andrew Wiles, a result for which he received the Abel Prize in 2016. Fermat's Last Theorem concerns integer solutions of the equation $x^{d} + y^{d} = z^{d}$ for $d ≥ 3$, and Wiles proved that there are no non-trivial integer solutions of this equation. Mordell's conjecture from 1922 suggests that for all quadratic equations of a certain form, the equation has only finitely many rational solutions.

The solution of an equation of degree 3 has a geometric interpretation as the points on an elliptic curve. An elliptic curve has genus 1, which refers to a certain property of the shape of the curve. Curves of genus ≥ 2 are geometrically more complex and defined by equations of even higher degree. Among the points on a curve there may be some whose coordinates are rational numbers, referred to as rational points. The Mordell conjecture states that there are only a finite number of rational points on a curve of genus ≥ 2. Mordell himself was only able to prove a slightly less strict result; that there can be infinitely many rational points on an elliptic curve, but only a finite number of them are needed to construct the rest. The construction refers to the fact that elliptic curves are endowed with a natural addition operation, which turns the set of points on the curve into what mathematicians call an abelian group. Abelian groups are named after Niels Henrik Abel, from whom the Abel Prize takes its name.

The Mordell conjecture for curves of genus ≥ 2 remained unproven for more than 60 years. During this period, some results with connections to the conjecture were established, among them a conjecture by Shafarevich concerning the finiteness of a family of curves. Through a result referred to as Parshin's trick, there is a close connection between the aforementioned family of curves and a curve underlying that family. In the early 1980s, it was known that a proof of the Shafarevich conjecture would automatically give a proof of the Mordell conjecture.

In 1983, Gerd Faltings succeeded in proving the Shafarevich conjecture. An important move in his proof is the introduction of what is now called the Faltings height of a curve. Faltings shows a finiteness result for the number of certain curves with finite Faltings height and also a finiteness result for the Faltings height itself. Combining the two yields a finiteness result for the number of curves involved, and thus the Shafarevich conjecture follows. Using Parshin's trick, it follows that the Mordell conjecture is no longer a conjecture, but now a result called Faltings' theorem.

10.5. Rational solutions to Diophantine equations.

The expression Diophantine equation refers to Diophantus of Alexandria, a Hellenistic mathematician of the 3rd century. Diophantus was a pioneer in finding integer solutions to polynomial equations with integer coefficients, and his name has ever since been connected to finding integer and rational solutions to such equations.

The problem of finding rational solutions of polynomial equations dates several hundreds of years back in time. Already in ancient time, it was known that the Pythagorean equation $x^{2} + y^{2} = z^{2}$ has infinitely many integer solutions. The solutions are completely described by the formula

          $x = p^{2} − q^{2}, y = 2pq, z = p^{2} + q^{2}$

for two arbitrary positive integers $p$ and $q$.

The Pythagorean equation is a quadratic equation, and the fact that there are infinitely many integer solutions can be viewed as a sign of the relatively frequent occurrence of squares among the integers. Increasing the degree of the equation will in general lead to a decreasing number of integer solutions. The number might drop from infinite to finite, and maybe also to zero, meaning that for a general equation with integer coefficients there are no integer solutions at all.

At the International Congress of Mathematicians in Paris in 1900, the German mathematician David Hilbert put forth a list of 10 unsolved problems in mathematics. Later he published an extended list of 23 problems, all considered to be very influential for 20th-century mathematics. Some of the problems are still open, some has been solved. Hilbert introduces his 23 problems with the following words: "Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?"

One of Hilbert's problems, known as Hilbert's 10th problem, concerns the solutions of Diophantine equations. "Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers."

Hilbert is not concerned with finding solutions to the equation, his question is more in the direction of deciding if there are any solutions at all. The French mathematician Poincaré also shows interest in this type of problems, and in 1901 he formulates a conjecture concerning rational solutions of elliptic curves, i.e. the solution set of a cubic equation. His claim is that the set of rational points on an elliptic curve forms a finitely generated abelian group, a claim the American-born British mathematician Louis Mordell should prove some twenty years later.

In the meantime, the number theorists published several results concerning solutions of Diophantine equations. In 1909, the Norwegian mathematician Axel Thue showed (among other things) that the equation $y^{3} − 2x^{2} = −1$ has only finitely many integer solutions. It was later proved that the equation has exactly one solution, given by $x = 78$ and $y = 23$.

Thue's more general result is known as Thue's theorem. It states that if $f(x, y)$ is a homogeneous polynomial with integer coefficients, irreducible over the rational numbers and of degree ≥ 3, then the equation $f(x, y) = k$ will have only finitely many integer solutions for an arbitrary choice of the integer $k$. Mordell himself showed already in 1913 that the equation

          $y^{2} = x^{3} + k$

where $k$ is an integer, has only a finite number of integer solutions.

The main result of Mordell's 1922 paper is what has since been called Mordell's theorem: The group $E(Q)$ of $Q$-points on an elliptic curve $E$ is finitely generated. This result came as an answer to the problem formulated by Poincaré in 1901.

Mordell's theorem refers to the fact that an elliptic curve, typically defined as the solution set of an equation of the form

          $y^{2} = x^{3} + px + q$

has an additional structure as an abelian group. The group law is often called the chord and tangent rule, since the construction is given by a purely geometric recipe, involving chords and tangents. Mordell proves his result by the classical method of infinite descent. If there are infinitely many solutions, they can all be traced back by the chord and tangent rule to only finitely many generating solutions. Mordell also shows the result which is crucial for his theorem, namely that the sum of two rational points on the elliptic curve again is a rational point.

Based on the shape of the solution set of a general equation of degree 3, elliptic curves are also referred to as curves of genus 1. Curves of higher genuses are defined by equations of higher degree, and the solution sets of these equations are also increasingly more complicated.

Mordell's conjecture, also presented in his 1922 paper "On the Rational Solutions of the Indeterminate Equations of the Third and Fourth Degree" claims that the set of rational points on curves of genus ≥ 2 is finite. This is not true for elliptic curves. Finitely generated is not the same as finite, due to the fact that rational points on the curve might have infinite order.

The Norwegian mathematician Trygve Nagell was a student of Axel Thue. He found a criterion for a rational point on an elliptic curve to be of finite order. The result is known as the Nagell-Lutz Theorem, honouring Nagell and Elisabeth Lutz. Elisabeth Lutz was a French mathematician who discovered the theorem independently of Nagell. The Nagell-Lutz Theorem describes rational points of finite order on elliptic curves over the integers: Assume that the equation

          $y^{2} = x^{3} + px + q, p, q \in \mathbb{Z}$

defines a non-singular elliptic curve $E$ with discriminant

          $\Delta = −4p^{3} − 27q^{2} ≠ 0$

If $P = (x, y)$ is a rational point of finite order on $E$, then $x$ and $y$ are integers and either $y = 0$, in which case $P$ has order 2, or else $y$ divides $\Delta$, which immediately implies that $y^{2}$ divides $\Delta$.

Let $E$ be the elliptic curve given by

          $y^{2} = x^{3} − 4x + 9, \Delta = −256 + 2187 = 1931$

It is easily seen that $P = (2, 3)$ is a rational point on $E$. Obviously 3 is not a factor of 1931 and $P$ has therefore infinite order by the Nagell-Lutz theorem.

As another example, the equation $y^{2} = x^{3} − x$ has only 4 rational points, including the additive identity 0 at infinity; $P = (1, 0), Q = (−1, 0), P + Q = (0, 0)$, and $2P = 2Q = 0$, i.e. $E(Q) <simeq> \mathbb{Z}_{2} \times \mathbb{Z}_{2}$. Together the two examples illustrate the fact that elliptic curves may or may not have infinitely many rational points.

Axel Thue and Jean Mordell were both number theorists. Their methods are either purely arithmetic or more approximative. After Faltings and Wiles, we have realised that the solutions to number theory problems are likely to be found in other fields of mathematics than number theory itself.