Algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, function fields; these properties, such as whether a ring admits unique factorization, the behavior of ideals, the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations. The beginnings of algebraic number theory can be traced to Diophantine equations, named after the 3rd-century Alexandrian mathematician, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers x and y such that their sum, the sum of their squares, equal two given numbers A and B, respectively: A = x + y B = x 2 + y 2. Diophantine equations have been studied for thousands of years.
For example, the solutions to the quadratic Diophantine equation x2 + y2 = z2 are given by the Pythagorean triples solved by the Babylonians. Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm. Diophantus' major work was the Arithmetica. Fermat's last theorem was first conjectured by Pierre de Fermat in 1637, famously in the margin of a copy of Arithmetica where he claimed he had a proof, too large to fit in the margin. No successful proof was published until 1995 despite the efforts of countless mathematicians during the 358 intervening years; the unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. One of the founding works of algebraic number theory, the Disquisitiones Arithmeticae is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat, Euler and Legendre and adds important new results of his own.
Before the Disquisitiones was published, number theory consisted of a collection of isolated theorems and conjectures. Gauss brought the work of his predecessors together with his own original work into a systematic framework, filled in gaps, corrected unsound proofs, extended the subject in numerous ways; the Disquisitiones was the starting point for the work of other nineteenth century European mathematicians including Ernst Kummer, Peter Gustav Lejeune Dirichlet and Richard Dedekind. Many of the annotations given by Gauss are in effect announcements of further research of his own, some of which remained unpublished, they must have appeared cryptic to his contemporaries. In a couple of papers in 1838 and 1839 Peter Gustav Lejeune Dirichlet proved the first class number formula, for quadratic forms; the formula, which Jacobi called a result "touching the utmost of human acumen", opened the way for similar results regarding more general number fields. Based on his research of the structure of the unit group of quadratic fields, he proved the Dirichlet unit theorem, a fundamental result in algebraic number theory.
He first used the pigeonhole principle, a basic counting argument, in the proof of a theorem in diophantine approximation named after him Dirichlet's approximation theorem. He published important contributions to Fermat's last theorem, for which he proved the cases n = 5 and n = 14, to the biquadratic reciprocity law; the Dirichlet divisor problem, for which he found the first results, is still an unsolved problem in number theory despite contributions by other researchers. Richard Dedekind's study of Lejeune Dirichlet's work was what led him to his study of algebraic number fields and ideals. In 1863, he published Lejeune Dirichlet's lectures on number theory as Vorlesungen über Zahlentheorie about which it has been written that: "Although the book is assuredly based on Dirichlet's lectures, although Dedekind himself referred to the book throughout his life as Dirichlet's, the book itself was written by Dedekind, for the most part after Dirichlet's death." 1879 and 1894 editions of the Vorlesungen included supplements introducing the notion of an ideal, fundamental to ring theory.
Dedekind defined an ideal as a subset of a set of numbers, composed of algebraic integers that satisfy polynomial equations with integer coefficients. The concept underwent further development in the hands of Hilbert and of Emmy Noether. Ideals generalize Ernst Eduard Kummer's ideal numbers, devised as part of Kummer's 1843 attempt to prove Fermat's Last Theorem. David Hilbert unified the field of algebraic number theory with his 1897 treatise Zahlbericht, he resolved a significant number-theory problem formulated by Waring in 1770. As with the finiteness theorem, he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers, he had little more to publish on the subject.
Alexander Grothendieck was a French mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory and category theory to its foundations, while his so-called "relative" perspective led to revolutionary advances in many areas of pure mathematics, he is considered by many to be the greatest mathematician of the 20th century. Born in Germany, Grothendieck was raised and lived in France. For much of his working life, however, he was, in effect, stateless; as he spelled his first name "Alexander" rather than "Alexandre" and his surname, taken from his mother, was the Dutch-like Low German "Grothendieck", he was sometimes mistakenly believed to be of Dutch origin. Grothendieck began his productive and public career as a mathematician in 1949. In 1958, he was appointed a research professor at the Institut des hautes études scientifiques and remained there until 1970, driven by personal and political convictions, he left following a dispute over military funding.
He became professor at the University of Montpellier and, while still producing relevant mathematical work, he withdrew from the mathematical community and devoted himself to political causes. Soon after his formal retirement in 1988, he moved to the Pyrenees, where he lived in isolation until his death in 2014. Grothendieck was born in Berlin to anarchist parents, his father, Alexander "Sascha" Schapiro, had Hasidic Jewish roots and had been imprisoned in Russia before moving to Germany in 1922, while his mother, Johanna "Hanka" Grothendieck, came from a Protestant family in Hamburg and worked as a journalist. Both had broken away from their early backgrounds in their teens. At the time of his birth, Grothendieck's mother was married to the journalist Johannes Raddatz and his birthname was recorded as "Alexander Raddatz." The marriage was dissolved in 1929 and Schapiro/Tanaroff acknowledged his paternity, but never married Hanka. Grothendieck lived with his parents in Berlin until the end of 1933, when his father moved to Paris to evade Nazism, followed soon thereafter by his mother.
They left Grothendieck in the care of a Lutheran pastor and teacher in Hamburg. During this time, his parents took part in the Spanish Civil War, according to Winfried Scharlau, as non-combatant auxiliaries, though others state that Sascha fought in the anarchist militia. In May 1939, Grothendieck was put on a train in Hamburg for France. Shortly afterwards his father was interned in Le Vernet, he and his mother were interned in various camps from 1940 to 1942 as "undesirable dangerous foreigners". The first was the Rieucros Camp, where his mother contracted the tuberculosis which caused her death and where Alexander managed to attend the local school, at Mende. Once Alexander managed to escape from the camp, intending to assassinate Hitler, his mother Hanka was transferred to the Gurs internment camp for the remainder of World War II. Alexander was permitted to live, separated from his mother, in the village of Le Chambon-sur-Lignon and hidden in local boarding houses or pensions, though he had to seek refuge in the woods during Nazis raids, surviving at times without food or water for several days.
His father was arrested under the Vichy anti-Jewish legislation, sent to the Drancy, handed over by the French Vichy government to the Germans to be sent to be murdered at the Auschwitz concentration camp in 1942. In Chambon, Grothendieck attended the Collège Cévenol, a unique secondary school founded in 1938 by local Protestant pacifists and anti-war activists. Many of the refugee children hidden in Chambon attended Cévenol, it was at this school that Grothendieck first became fascinated with mathematics. After the war, the young Grothendieck studied mathematics in France at the University of Montpellier where he did not perform well, failing such classes as astronomy. Working on his own, he rediscovered the Lebesgue measure. After three years of independent studies there, he went to continue his studies in Paris in 1948. Grothendieck attended Henri Cartan's Seminar at École Normale Supérieure, but he lacked the necessary background to follow the high-powered seminar. On the advice of Cartan and André Weil, he moved to the University of Nancy where he wrote his dissertation under Laurent Schwartz and Jean Dieudonné on functional analysis, from 1950 to 1953.
At this time he was a leading expert in the theory of topological vector spaces. From 1953 to 1955 he moved to the University of São Paulo in Brazil, where he immigrated by means of a Nansen passport, given that he refused to take French Nationality. By 1957, he set this subject aside in order to work in algebraic homological algebra; the same year he was invited to visit Harvard by Oscar Zariski, but the offer fell through when he refused to sign a pledge promising not to work to overthrow the United States government, a position that, he was warned, might have landed him in prison. The prospect did not worry him. Comparing Grothendieck during his Nancy years to the École Normale Supérieure trained students at that time: Pierre Samuel, Roger Godement, René Thom, Jacques Dixmier, Jean Cerf, Yvonne Bruhat, Jean-Pierre Serre, Bernard Malgrange, Leila Schneps says: He was so unknown to this group and to their professors, came from such a deprived and chaotic background, was, compared to them, so ignorant at the start of his research career
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to certain equations. For example, the equation 2 = − 9 has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem; the idea is to extend the real numbers with an indeterminate i, taken to satisfy the relation i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1: 2 = 2 = = 9 = − 9, 2 = 2 = 2 = 9 = − 9.
According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. In contrast, some polynomial equations with real coefficients have no solution in real numbers; the 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations. Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i; this means that complex numbers can be added and multiplied, as polynomials in the variable i, with the rule i2 = −1 imposed. Furthermore, complex numbers can be divided by nonzero complex numbers. Overall, the complex number system is a field. Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.
The complex number a + bi can be identified with the point in the complex plane. A complex number whose real part is zero is said to be purely imaginary. A complex number whose imaginary part is zero can be viewed as a real number. Complex numbers can be represented in polar form, which associates each complex number with its distance from the origin and with a particular angle known as the argument of this complex number; the geometric identification of the complex numbers with the complex plane, a Euclidean plane, makes their structure as a real 2-dimensional vector space evident. Real and imaginary parts of a complex number may be taken as components of a vector with respect to the canonical standard basis; the addition of complex numbers is thus depicted as the usual component-wise addition of vectors. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not available in a vector space. Based on the concept of real numbers, a complex number is a number of the form a + bi, where a and b are real numbers and i is an indeterminate satisfying i2 = −1.
For example, 2 + 3i is a complex number. This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i2 + 1 = 0 is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials; the relation i2 + 1 = 0 induces the equalities i4k = 1, i4k+1 = i, i4k+2 = −1, i4k+3 = −i, which hold for all integers k. The real number a is called the real part of the complex number a + bi. To emphasize, the imaginary part does not include a factor i and b, not bi, is the imaginary part. Formally, the complex numbers are defined as the quotient ring of the polynomia
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques from commutative algebra, for solving geometrical problems about these sets of zeros; the fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, parabolas, hyperbolas, cubic curves like elliptic curves, quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.
Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis and number theory. A study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, it becomes more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution. In the 20th century, algebraic geometry split into several subareas; the mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more to the points with coordinates in an algebraically closed field. Real algebraic geometry is the study of the real points of an algebraic variety. Diophantine geometry and, more arithmetic geometry is the study of the points of an algebraic variety with coordinates in fields that are not algebraically closed and occur in algebraic number theory, such as the field of rational numbers, number fields, finite fields, function fields, p-adic fields.
A large part of singularity theory is devoted to the singularities of algebraic varieties. Computational algebraic geometry is an area that has emerged at the intersection of algebraic geometry and computer algebra, with the rise of computers, it consists of algorithm design and software development for the study of properties of explicitly given algebraic varieties. Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space. One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way, similar to its use in the study of differential and analytic manifolds; this is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring.
This means that a point of such a scheme may be either a subvariety. This approach enables a unification of the language and the tools of classical algebraic geometry concerned with complex points, of algebraic number theory. Wiles' proof of the longstanding conjecture called Fermat's last theorem is an example of the power of this approach. In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that satisfy one or more polynomial equations. For instance, the two-dimensional sphere of radius 1 in three-dimensional Euclidean space R3 could be defined as the set of all points with x 2 + y 2 + z 2 − 1 = 0. A "slanted" circle in R3 can be defined as the set of all points which satisfy the two polynomial equations x 2 + y 2 + z 2 − 1 = 0, x + y + z = 0. First we start with a field k. In classical algebraic geometry, this field was always the complex numbers C, but many of the same results are true if we assume only that k is algebraically closed.
We consider the affine space of dimension n over denoted An. When one fixes a coordinate system, one may identify An with kn; the purpose of not working with kn is to emphasize that one "forgets" the vector space structure that kn carries. A function f: An → A1 is said to be polynomial if it can be written as a polynomial, that is, if there is a polynomial p in k such that f = p for every point M with coordinates in An; the property of a function to be polynomial does not depend on the choice of a coordinate system in An. When a coordinate system is chosen, the regular functions on the affine n-space may be identified with the ring of polynomial functions in n variables over k. Therefore, the set of the
The abc conjecture is a conjecture in number theory, first proposed by Joseph Oesterlé and David Masser. It is stated in terms of three positive integers, a, b and c that are prime and satisfy a + b = c. If d denotes the product of the distinct prime factors of abc, the conjecture states that d is not much smaller than c. In other words: if a and b are composed from large powers of primes c is not divisible by large powers of primes; the precise statement is given below. The abc conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the Szpiro conjecture about elliptic curves; the latter conjecture has more geometric structures involved in its statement in comparison with the abc conjecture. The abc conjecture and its versions express, in concentrated form, some fundamental feature of various problems in Diophantine geometry. A number of famous conjectures and theorems in number theory would follow from the abc conjecture or its versions. Goldfeld described the abc conjecture as "the most important unsolved problem in Diophantine analysis".
Various proofs of abc have been claimed but so far none is accepted by the mathematical community. Before we state the conjecture we introduce the notion of the radical of an integer: for a positive integer n, the radical of n, denoted rad, is the product of the distinct prime factors of n. For example rad = rad = 2, rad = 17, rad = rad = 2 · 3 = 6, rad = rad = 2 ⋅ 5 = 10. If a, b, c are coprime positive integers such that a + b = c, it turns out that "usually" c < rad. The abc conjecture deals with the exceptions, it states that: ABC Conjecture. For every positive real number ε, there exist only finitely many triples of coprime positive integers, with a + b = c, such that: c > rad 1 + ε. An equivalent formulation states that: ABC Conjecture II. For every positive real number ε, there exists a constant Kε such that for all triples of coprime positive integers, with a + b = c: c < K ε ⋅ rad 1 + ε. A third equivalent formulation of the conjecture involves the quality q of the triple, defined as q = log log .
For example, q = log / log = log / log = 0.46820... Q = log / log = log / log = 1.426565... A typical triple of coprime positive integers with a + b = c will have c < rad, i.e. q < 1. Triples with q > 1 such as in the second example are rather special, they consist of numbers divisible by high powers of small prime numbers. ABC Conjecture III. For every positive real number ε, there exist only finitely many triples of coprime positive integers with a + b = c such that q > 1 + ε. Whereas it is known that there are infinitely many triples of coprime positive integers with a + b = c such that q > 1, the conjecture predicts that only finitely many of those have q > 1.01 or q > 1.001 or q > 1.0001, etc. In particular, if the conjecture is true there must exist a triple which achieves the maximal possible quality q; the condition that ε > 0 is necessary as there exist infinitely many triples a, b, c with rad < c. For example let: a = 1, b = 2 6 n − 1, c = 2 6 n, n > 1. First we note that b is divisible by 9: b = 2 6 n − 1 = 64 n − 1 = = 9 ⋅ 7 ⋅ Using this fact we calculate: rad = rad rad rad = rad rad rad ( 2 6 n
Thoralf Albert Skolem was a Norwegian mathematician who worked in mathematical logic and set theory. Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skolem attended secondary school in Kristiania, passing the university entrance examinations in 1905, he entered Det Kongelige Frederiks Universitet to study mathematics taking courses in physics, chemistry and botany. In 1909, he began working as an assistant to the physicist Kristian Birkeland, known for bombarding magnetized spheres with electrons and obtaining aurora-like effects. In 1913, Skolem passed the state examinations with distinction, completed a dissertation titled Investigations on the Algebra of Logic, he traveled with Birkeland to the Sudan to observe the zodiacal light. He spent the winter semester of 1915 at the University of Göttingen, at the time the leading research center in mathematical logic and abstract algebra, fields in which Skolem excelled. In 1916 he was appointed a research fellow at Det Kongelige Frederiks Universitet.
In 1918, he became a Docent in Mathematics and was elected to the Norwegian Academy of Science and Letters. Skolem did not at first formally enroll as a Ph. D. candidate, believing that the Ph. D. was unnecessary in Norway. He changed his mind and submitted a thesis in 1926, titled Some theorems about integral solutions to certain algebraic equations and inequalities, his notional thesis advisor was Axel Thue though Thue had died in 1922. In 1927, he married Edith Wilhelmine Hasvold. Skolem continued to teach at Det kongelige Frederiks Universitet until 1930 when he became a Research Associate in Chr. Michelsen Institute in Bergen; this senior post allowed Skolem to conduct research free of administrative and teaching duties. However, the position required that he reside in Bergen, a city which lacked a university and hence had no research library, so that he was unable to keep abreast of the mathematical literature. In 1938, he returned to Oslo to assume the Professorship of Mathematics at the university.
There he taught the graduate courses in algebra and number theory, only on mathematical logic. Skolem's Ph. D. student Øystein Ore went on to a career in the USA. Skolem served as president of the Norwegian Mathematical Society, edited the Norsk Matematisk Tidsskrift for many years, he was the founding editor of Mathematica Scandinavica. After his 1957 retirement, he made several trips to the United States and teaching at universities there, he remained intellectually active until his unexpected death. For more on Skolem's academic life, see Fenstad. Skolem published around 180 papers on Diophantine equations, group theory, lattice theory, most of all, set theory and mathematical logic, he published in Norwegian journals with limited international circulation, so that his results were rediscovered by others. An example is the Skolem -- Noether theorem. Skolem published a proof in 1927. Skolem was among the first to write on lattices. In 1912, he was the first to describe a free distributive lattice generated by n elements.
In 1919, he showed that every implicative lattice is distributive and, as a partial converse, that every finite distributive lattice is implicative. After these results were rediscovered by others, Skolem published a 1936 paper in German, "Über gewisse'Verbände' oder'Lattices'", surveying his earlier work in lattice theory. Skolem was a pioneer model theorist. In 1920, he simplified the proof of a theorem Leopold Löwenheim first proved in 1915, resulting in the Löwenheim–Skolem theorem, which states that if a countable first-order theory has an infinite model it has a countable model, his 1920 proof employed the axiom of choice, but he gave proofs using Kőnig's lemma in place of that axiom. It is notable that Skolem, like Löwenheim, wrote on mathematical logic and set theory employing the notation of his fellow pioneering model theorists Charles Sanders Peirce and Ernst Schröder, including ∏, ∑ as variable-binding quantifiers, in contrast to the notations of Peano, Principia Mathematica, Principles of Mathematical Logic.
Skolem pioneered the construction of non-standard models of set theory. Skolem refined Zermelo's axioms for set theory by replacing Zermelo's vague notion of a "definite" property with any property that can be coded in first-order logic; the resulting axiom is now part of the standard axioms of set theory. Skolem pointed out that a consequence of the Löwenheim–Skolem theorem is what is now known as Skolem's paradox: If Zermelo's axioms are consistent they must be satisfiable within a countable domain though they prove the existence of uncountable sets; the completeness of first-order logic is an easy corollary of results Skolem proved in the early 1920s and discussed in Skolem, but he failed to note this fact because mathematicians and logicians did not become aware of completeness as a fundamental metamathematical problem until the 1928 first edition of Hilbert and Ackermann's Principles of Mathematical Logic articulated it. In any event, Kurt Gödel first proved this completeness in 1930. Skolem was one of the founders of finitism in mathematics.
Skolem sets out his primitive recursive arithm
In number theory, the Mordell conjecture is the conjecture made by Mordell that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. In 1983 it was proved by Gerd Faltings, is now known as Faltings's theorem; the conjecture was generalized by replacing Q by any number field. Let C be a non-singular algebraic curve of genus g over Q; the set of rational points on C may be determined as follows: Case g = 0: no points or infinitely many. Case g = 1: no points, or C is an elliptic curve and its rational points form a finitely generated abelian group. Moreover, Mazur's torsion theorem restricts the structure of the torsion subgroup. Case g > 1: according to the Mordell conjecture, now Faltings's theorem, C has only a finite number of rational points. Faltings's original proof used the known reduction to a case of the Tate conjecture, a number of tools from algebraic geometry, including the theory of Néron models. A different proof, based on diophantine approximation, was found by Vojta.
A more elementary variant of Vojta's proof was given by Bombieri. Faltings's 1983 paper had as consequences a number of statements, conjectured: The Mordell conjecture that a curve of genus greater than 1 over a number field has only finitely many rational points; the reduction of the Mordell conjecture to the Shafarevich conjecture was due to A. N. Paršin. A sample application of Faltings's theorem is to a weak form of Fermat's Last Theorem: for any fixed n > 4 there are at most finitely many primitive integer solutions to an + bn = cn, since for such n the curve xn + yn = 1 has genus greater than 1. Because of the Mordell–Weil theorem, Faltings's theorem can be reformulated as a statement about the intersection of a curve C with a finitely generated subgroup Γ of an abelian variety A. Generalizing by replacing C by an arbitrary subvariety of A and Γ by an arbitrary finite-rank subgroup of A leads to the Mordell–Lang conjecture, proved by Faltings. Another higher-dimensional generalization of Faltings's theorem is the Bombieri–Lang conjecture that if X is a pseudo-canonical variety over a number field k X is not Zariski dense in X.
More general conjectures have been put forth by Paul Vojta. The Mordell conjecture for function fields was proved by Manin and by Grauert. In 1990, Coleman found and fixed a gap in Manin's proof. Bombieri, Enrico. "The Mordell conjecture revisited". Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17: 615–640. MR 1093712. Coleman, Robert F.. "Manin's proof of the Mordell conjecture over function fields". L'Enseignement Mathématique. Revue Internationale. IIe Série. 36: 393–427. ISSN 0013-8584. MR 1096426. Archived from the original on 2011-10-02. Cornell, Gary. Arithmetic geometry. Papers from the conference held at the University of Connecticut, Connecticut, July 30 – August 10, 1984. New York: Springer-Verlag. Doi:10.1007/978-1-4613-8655-1. ISBN 0-387-96311-1. MR 0861969. → Contains an English translation of Faltings Faltings, Gerd. "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern". Inventiones Mathematicae. 73: 349–366. Doi:10.1007/BF01388432. MR 0718935. Faltings, Gerd. "Erratum: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern".
Inventiones Mathematicae. 75: 381. Doi:10.1007/BF01388572. MR 0732554. Faltings, Gerd. "Diophantine approximation on abelian varieties". Ann. of Math. 133: 549–576. Doi:10.2307/2944319. MR 1109353. Faltings, Gerd. "The general case of S. Lang's conjecture". In Cristante, Valentino. Barsotti Symposium in Algebraic Geometry. Papers from the symposium held in Abano Terme, June 24–27, 1991. Perspectives in Mathematics. San Diego, CA: Academic Press, Inc. ISBN 0-12-197270-4. MR 1307396. Grauert, Hans. "Mordells Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörper". Publications Mathématiques de l'IHÉS: 131–149. ISSN 1618-1913. MR 0222087. Hindry, Marc. Diophantine geometry. Graduate Texts in Mathematics. 201. New York: Springer-Verlag. Doi:10.1007/978-1-4612-1210-2. ISBN 0-387-98981-1. MR 1745599. → Gives Vojta's proof of Faltings's Theorem. Lang, Serge. Survey of Diophantine geometry. Springer-Verlag. Pp. 101–122. ISBN 3-540-61223-8. Manin, Ju. I.. "Rational points on algebraic curves over function fields".
Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya. 27: 1395–1440. ISSN 0373-2436. MR 0157971. Mordell, Louis J.. "On the rational solutions of the indeterminate equation of the third and fourth degrees". Proc. Cambridge Philos. Soc. 21: 179–192. Paršin, A. N.. "Quelques conjectures de finitude en géométrie diophantienne". Actes du Congrès International des Mathématiciens. Tome 1. Nice: Gauthier-Villars. Pp. 467–471. MR 0427323. Archived from the original on 2016-09-24. Retrieved 2016-06-11. Parshin, A. N. "Mordell conje