# Faltings sats - Faltings's theorem - qaz.wiki

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[1] Faltings G. Endlichkeitssätze für abelsche Varietäten über Zhalkörpern. Invent Math 1983, 73: Från Mordell-antagandet, bevisat av Faltings 1983, följer det att The Last Theorem, som han författade tillsammans med Frederick Paul. Don Zagier, Fieldsmedaljören Gerd Faltings, samt Günther Harder och med titeln An analytic approach to Briançon-Skoda type theorems. Gerd Faltings, direktör för Max Planck-institutet för ma- indices theorem. Ganska ”high indices theorems”, området där han alltså startade. 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 (pairwise coprime solutions) to a n + b n = c n, since for such n the Fermat curve x n + y n = 1 has genus greater than 1.

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From Wikipedia, the free encyclopedia In number theory, 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. Notes on the ˙niteness theorem of Faltings for abelian varieties Wen-Wei Li Peking University November 14, 2018 Abstract These are informal notes prepared for the seminar on Faltings’ proof of the Mordell conjecture organized by Xinyi Yuan and Ruochuan Liu at Beijing International Center for Mathematical Research, Fall 2018. by Faltings [1983] (which asserts that a curve of genus greater than 1 de ned over a number eld has only a nite number of points rational over that number eld). As an example of an application of this theorem, choose your favorite polynomial g(x) with rational coe cients, no multiple roots, and of degree 5, for example g(x) = x(x 1)(x 2)(x 3)(x 4); Faltings’ theorem Let K be a number field and let C / K be a non-singular curve defined over K and genus g . When the genus is 0 , the curve is isomorphic to ℙ 1 (over an algebraic closure K ¯ ) and therefore C ( K ) is either empty or equal to ℙ 1 ( K ) (in particular C ( K ) is infinite ). Faltings has been closely linked with the work leading to the final proof of Fermat's Last Theorem by Andrew Wiles. In 1983 Faltings proved that for every n > 2 n > 2 n > 2 there are at most a finite number of coprime integers x, y, z x, y, z x, y, z with x n + y n = z n x^{n} + y^{n} = z^{n} x n + y n = z n.

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In 1983 Faltings proved that for every n > 2 n > 2 n > 2 there are at most a finite number of coprime integers x, y, z x, y, z x, y, z with x n + y n = z n x^{n} + y^{n} = z^{n} x n + y n = z n. In this chapter we shall state the finiteness theorems of Faltings and give very detailed proofs of these results. In the second section we shall beginn with the finiteness theorem for isogeny classes of abelian varieties with good reduction outside a given set of primes. Faltings’ Theorem CollegeSeminar Summer2015 Wednesdays13.15-15.00in1.023 Benjamin Bakker 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 It was to do with Falting's Theorem and the geometrical representations of equations like x n + y n = 1.

### Faltings' Theorem: Surhone, Lambert M.: Amazon.se: Books

E. Faltings’s isogeny theorem. If Aand Bare two abelian varieties, then the natural map Hom K(A;B) Z Z ‘!

Let B!Abe a K-isogeny. Then exp(2[K: Q](h(B) h(A)) (read: \change in height under isogeny") is a rational number. Faltings' theorem In number theory, the Mordell conjecture stated a basic result regarding the rational number solutions to Diophantine equations. It was eventually proved by Gerd Faltings in 1983, about six decades after the conjecture was made; it is now known as Faltings' theorem. Faltings's theorem In number theory, 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. 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.

Ar talaren

The topos X o K 36 7. Computing compactly supported cohomology using Galois cohomology 44 8.

The Isogeny theorem that abelian varieties with isomorphic Tate modules (as Q ℓ-modules with Galois action) are isogenous.

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When the genus is 0 , the curve is isomorphic to ℙ 1 (over an algebraic closure K ¯ ) and therefore C ( K ) is either empty or equal to ℙ 1 ( K ) (in particular C ( K ) is infinite ). Faltings has been closely linked with the work leading to the final proof of Fermat's Last Theorem by Andrew Wiles. In 1983 Faltings proved that for every n > 2 n > 2 n > 2 there are at most a finite number of coprime integers x, y, z x, y, z x, y, z with x n + y n = z n x^{n} + y^{n} = z^{n} x n + y n = z n. In this chapter we shall state the finiteness theorems of Faltings and give very detailed proofs of these results. In the second section we shall beginn with the finiteness theorem for isogeny classes of abelian varieties with good reduction outside a given set of primes. Faltings’ Theorem CollegeSeminar Summer2015 Wednesdays13.15-15.00in1.023 Benjamin Bakker 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 It was to do with Falting's Theorem and the geometrical representations of equations like x n + y n = 1. I quote: "Faltings was able to prove that, because these shapes always have more than one hole, the associated Fermat equation could only have a finite number of whole number solutions." difﬁculty of the other theorems of yours, and in particular of the present theorem.— Chortasmenos, ˘1400.

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Faltings' theorem In number theory, the Mordell conjecture stated a basic result regarding the rational number solutions to Diophantine equations. It was eventually proved by Gerd Faltings in 1983, about six decades after the conjecture was made; it is now known as Faltings' theorem. Faltings's theorem In number theory, 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. 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.