An example of a theorem from geometric model theory is Hrushovski's proof of the Mordell – Lang conjecture for function fields.

In number theory, the Mordell conjecture is the conjecture made by that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points.

* Case g > 1: according to the Mordell conjecture, now Faltings ' Theorem, C has only a finite number of rational points.

The reduction of the Mordell conjecture to the Shafarevich conjecture was due to.

Because of the Mordell – Weil theorem, Faltings ' 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, which has been proved.

The Mordell conjecture for function fields was proved by and by.

A special case of the conjectures, which are open in the general case, was involved in the proof of the Mordell conjecture by Gerd Faltings.

The fields of diophantine approximation and transcendence theory have expanded, to the point that the techniques have been applied to the Mordell conjecture.

The development of Arakelov theory and its exploitation by Gerd Faltings in his proof of the Mordell conjecture is a dramatic example.

There are more general statements ; this one is most clearly motivated by the Mordell conjecture, where such a curve C should intersect J ( K ) only in finitely many points.

These turned out to involve some close parallels, and to lead to fresh points of view on the Mordell conjecture and related questions.

The general approach of diophantine geometry is illustrated by Faltings ' theorem ( a conjecture of L. J. Mordell ) starting that an algebraic curve C of genus g > 1 over the rational numbers has only finitely many rational points.

He was awarded the Fields Medal in 1986 for proving the Mordell conjecture, which states that any non-singular projective curve of genus g > 1 defined over a number field K contains only finitely many K-rational points.

He made a number of conjectures in diophantine geometry: Mordell – Lang conjecture, Bombieri – Lang conjecture, Lang's integral point conjecture, Lang – Trotter conjecture, Lang conjecture on Gamma values, Lang conjecture on analytically hyperbolic varieties.

His basic work on Mordell's theorem is from 1921 / 2, as is the formulation of the Mordell conjecture.

This conjecture was proved by Gerd Faltings as a step in his proof of the Mordell conjecture.

* Case g = 1: no points, or C is an elliptic curve and its rational points form a finitely generated abelian group ( Mordell's Theorem, later generalized to the Mordell – Weil theorem ).

The structural result of Mordell, that the rational points on an elliptic curve E form a finitely-generated abelian group, used an infinite descent argument based on E / 2E in Fermat's style.

# proved that the average rank of the Mordell – Weil group of an elliptic curve over Q is bounded above by 7 / 6.

Mordell's theorem on the finite generation of the group of rational points on an elliptic curve is in Chapter 16, and integer points on the Mordell curve in Chapter 26.

Having taken third place in the Mathematical Tripos, he began independent research into particular diophantine equations: the question of integer points on the cubic curve, and special case of what is now called a Thue equation, the Mordell equation

Numerous direct calculations were done, and the proof of the Mordell – Weil theorem had to proceed by some surrogate of a finiteness proof for a particular H < sup > 1 </ sup > group.

The basic result ( Mordell – Weil theorem ) says that A ( K ), the group of points on A over K, is a finitely-generated abelian group.

In 1930-1960 research on the geometry of numbers was conducted by many number theorists ( including Louis Mordell, Harold Davenport and Carl Ludwig Siegel ).

In mathematics, the Chowla – Mordell theorem is a result in number theory determining cases where a Gauss sum is the square root of a prime number, multiplied by a root of unity.

Louis Joel Mordell ( 28 January 1888 – 12 March 1972 ) was a British mathematician, known for pioneering research in number theory.

Manin's early work included papers on the arithmetic and formal groups of abelian varieties, the Mordell conjecture in the function field case, and algebraic differential equations.

The ' if ' part was known to Gauss: the contribution of Chowla and Mordell was the ' only if ' direction.

Chapman quotes Louis Mordell as saying " His result is very pretty, and there are many applications of it.

This began in his doctoral work leading to the Mordell – Weil theorem ( 1928, and shortly applied in Siegel's theorem on integral points ).

In 1958 Bing proved a weak version of the Poincaré conjecture: if every simple closed curve of a compact 3-manifold is contained in a 3-ball, then the manifold is homeomorphic to the 3-sphere .< ref > Bing also described some of the pitfalls in trying to prove the Poincaré conjecture.

The green bar shows the failure of the Pòlya conjecture ; the blue curve shows the oscillatory contribution of the first Riemann zero.

Goro Shimura and Taniyama worked on improving its rigor until 1957. rediscovered the conjecture, and showed that it would follow from the ( conjectured ) functional equations for some twisted L-series of the elliptic curve ; this was the first serious evidence that the conjecture might be true.

A well-known example is the Taniyama – Shimura conjecture, now the modularity theorem, which proposed that each elliptic curve over the rational numbers can be translated into a modular form ( in such a way as to preserve the associated L-function ).

Combining this work with a result of William Thurston, Cameron Gordon assembled a proof of the Smith conjecture: for any cyclic group acting on a sphere, the set of fixed points is not a knotted curve.

The Manin – Mumford conjecture of Yuri Manin and David Mumford, proved by Michel Raynaud, states that a curve C in its Jacobian variety J can only contain a finite number of points that are of finite order in J, unless C = J.

The conjecture relates arithmetic data associated to an elliptic curve E over a number field K to the behaviour of the Hasse – Weil L-function L ( E, s ) of E at s = 1.

The X-axis is log ( log ( X )) and Y-axis is in a logarithmic scale so the conjecture predicts that the data should form a line of slope equal to the rank of the curve, which is 1 in this case.

# showed that for elliptic curves defined over an imaginary quadratic field K with complex multiplication by K, if the L-series of the elliptic curve was not zero at s = 1, then the p-part of the Tate – Shafarevich group had the order predicted by the Birch and Swinnerton-Dyer conjecture, for all primes p > 7.

In mathematics, the Sato – Tate conjecture is a statistical statement about the family of elliptic curves E < sub > p </ sub > over the finite field with p elements, with p a prime number, obtained from an elliptic curve E over the rational number field, by the process of reduction modulo a prime for almost all p. If N < sub > p </ sub > denotes the number of points on E < sub > p </ sub > and defined over the field with p elements, the conjecture gives an answer to the distribution of the second-order term for N < sub > p </ sub >.

The chronology protection conjecture should be distinguished from chronological censorship under which every closed timelike curve passes through an event horizon, which might prevent an observer from detecting the causal violation.

The epsilon conjecture is a relative statement: assuming that a given elliptic curve E over Q is modular, it predicts the precise level of E.

By the Taniyama – Shimura conjecture, E is a modular elliptic curve.

The Chisini conjecture in algebraic geometry is a uniqueness question for morphisms of generic smooth projective surfaces, branched on a cuspidal curve.