Faltings theorem: lt;p|>In |number theory|, the |Mordell conjecture| is the conjecture made by |Mordell (1922|) th World Heritage Encyclopedia, the aggregation of the largest online encyclopedias available, and the most definitive collection ever assembled.

A CONTRIBUTION TO THE THEORY OF FORMAL. MEROMORPHIC FUNCTIONS. GERD FALTINGS. In my paper [F3] I more or less explicitly conjectured that if

In arithmetic geometry, the Mordell conjecture is the conjecture made by Mordell (1922) 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 (1983, 1984), and is now known as Faltings's theorem. The conjecture was later generalized by replacing Q by any number field. Last Theorem by R.Taylor and A.Wiles Gerd Faltings T he proof of the conjecture mentioned in the title was finally completed in Septem-ber of 1994. A. Wiles announced this result in the summer of 1993; however, there was a gap in his work. The paper of Taylor and Wiles does not close this gap but circumvents it.

Faltings theorem

Discover the world's research 20 View Faltings G. Lectures on the Arithmetic Riemann-Roch Theorem (PUP 1992)(ISBN 0691025444)(T)(107s).pdf from MATH 20 at Harvard University. LECTURES ON THE ARITHMETIC RIEMANN-ROCH THEOREM BY GERD In 1980, Faltings proved, by deep local algebra methods, a local result regarding formal functions which has the following global geometric fact as a consequence. Theorem: Let k be an use Faltings' Theorem to show the following. THEOREM 2. For any given integers p,q,r satisfying 1/p + l/q + 1/r < 1, the generalized Fermat equation Axp + Byq = Czr (2) has only finitely many proper integer solutions. (Our proofs of Theorems 1 and 2 are extended easily to proper solutions in any fixed number field, and even those that are S-units.) Salarian, Faltings’ theorem for the annihilation of local cohomology modules over a Gorenstein ring, Proc. Amer.

This is really just an elaboration of Emerton's comment: You should read Mark Kisins' review of Faltings's paper "Almost etale extensions". But I wanted to elaborate: Faltings regards the almost purity theorem as an analogue of Zariski-Nagata purity. In Faltings's original setup, it was formulated as follows. Consider the rings

Faltings, {\it \"Uber die Annulatoren lokaler Kohomologiegruppen}, Arch. Math. {\bf30} (1978) 473-476].

In arithmetic geometry, Faltings' product theorem gives sufficient conditions for a subvariety of a product of projective spaces to be a product of varieties in the projective spaces. It was introduced by Faltings ( 1991 ) in his proof of Lang's conjecture that subvarieties of an abelian variety containing no translates of non-trivial abelian subvarieties have only finitely many rational points.

Den bevisades senare av Gerd Faltings 1983 och är numera känt som Faltings sats. Källor [ redigera | redigera wikitext ] Den här artikeln är helt eller delvis baserad på material från engelskspråkiga Wikipedia , Faltings' theorem , 19 januari 2014 . E. Faltings’s isogeny theorem. If Aand Bare two abelian varieties, then the natural map Hom K(A;B) Z Z ‘! Hom G K (T ‘(A);T ‘(B)) is an isomorphism. This is what we mean when we say that the Tate module is almost a complete invariant: if two Tate modules are isomorphic, then there is an isogeny between the abelian varieties they are de ned from. This Last Theorem by R.Taylor and A.Wiles Gerd Faltings T he proof of the conjecture mentioned in the title was finally completed in Septem-ber of 1994.

Awards and honours. Fields Medal (1986) Guggenheim Fellowship (1988/89) Gottfried Wilhelm Leibniz Prize (1996) King Faisal International Prize (2014) Shaw Prize (2015) Foreign Member of the Royal Society (2016) Cantor Medal (2017) For a finitely generated field over a number field the theorem holds by work of Faltings, if I am not mistaken. See the book by Faltings-Wüstholz on the Mordell conjecture (I don't have it at hand). $\endgroup$ – Damian Rössler Dec 20 '12 at 21:15
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A. Wiles announced this result in the summer of 1993; however, there was a gap in his work. The paper of Taylor and Wiles does not close this gap but circumvents it.

With this convention, we have the following.
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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.

A more elementary variant of Vojta's proof was given by Enrico Bombieri. Faltings theorem: lt;p|>In |number theory|, the |Mordell conjecture| is the conjecture made by |Mordell (1922|) th World Heritage Encyclopedia, the aggregation of the largest online encyclopedias available, and the most definitive collection ever assembled. Faltings’ Finiteness Theorems Michael Lipnowski Introduction: How Everything Fits Together This note outlines Faltings’ proof of the niteness theorems for abelian varieties and curves. Let Kbe a number eld and Sa nite set of places of K:We will demonstrate the following, in order: The main goal of the semester is to understand some aspects of Faltings' proofs of some far--reaching finiteness theorems about abelian varieties over number fields, the highlight being the Tate conjecture, the Shafarevich conjecture, and the Mordell conjecture. There are a … especially Theorem 4. (The quotation marks above are because all of this function field work came first: the above link is to Samuel's 1966 paper, whereas Faltings' theorem was proved circa 1982.) The statement is the same as the Mordell Conjecture, except that there is an extra hypothesis on "nonisotriviality", i.e., one does not want the curve have constant moduli. Faltings has been closely linked with the work leading to the final proof of Fermat's Last Theorem by Andrew Wiles.