Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field Q {\displaystyle \mathbb {Q} }...

Gerd Faltings (German pronunciation: [ɡɛʁt ˈfaltɪŋs] ; born 28 July 1954) is a German mathematician known for his work in arithmetic geometry. From 1972...

{\displaystyle C} with A ( K ) {\displaystyle A(K)} be infinite? Because of Faltings's theorem, this is false unless C = A {\displaystyle C=A} . In the same context...

theorems in Diophantine geometry that are of fundamental importance include: Mordell–Weil theorem Roth's theorem Siegel's theorem Faltings's theorem Serge...

than two. This conjecture was proved in 1983 by Gerd Faltings, and is now known as Faltings's theorem. In the latter half of the 20th century, computational...

form of the equations. For g > 1 it was superseded by Faltings's theorem in 1983. Siegel's theorem on integral points: For a smooth algebraic curve C of...

heights in Arakelov theory. In 1983, Faltings developed his theory of Faltings heights in his proof of Faltings's theorem. Classical or naive height is defined...

geometry) Faltings's theorem (Diophantine geometry) Fulton–Hansen connectedness theorem (algebraic geometry) Grauert–Riemenschneider vanishing theorem (algebraic...

important mathematical results and conjectures, including Roth's theorem, Faltings's theorem, Fermat–Catalan conjecture, and a negative solution to the Erdős–Ulam...

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 abstract algebra (specifically commutative ring theory), Faltings' annihilator theorem states: given a finitely generated module M over a Noetherian...

Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were believed to be...

Z} with Z → ∞ {\displaystyle Z\rightarrow \infty } ). Roth's theorem Faltings's theorem A. Thue (1909). "Über Annäherungswerte algebraischer Zahlen"....

is finite, or from the related Birch–Swinnerton-Dyer conjecture. Faltings's theorem (formerly the Mordell conjecture) says that for any curve X of genus...

1007/s002220050059. MR 1369424. Faltings, Gerd (1983). "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" [Finiteness theorems for abelian varieties over...

is consistent with the known results on rational points, notably Faltings's theorem on subvarieties of abelian varieties. More precisely, let X be a projective...

conjecture. The Bombieri–Lang conjecture is an analogue for surfaces of Faltings's theorem, which states that algebraic curves of genus greater than one only...

m ≥ 2 and n ≥ 3 from work by Kraus. The Darmon–Granville theorem uses Faltings's theorem to show that for every specific choice of exponents (x, y,...

values (am, bn, ck). It is known by the Darmon–Granville theorem, which uses Faltings's theorem, that for any fixed choice of positive integers m, n and...

include: Roth's theorem on Diophantine approximation of algebraic numbers. The Mordell conjecture (already proven in general by Gerd Faltings). As equivalent...

conjecture for function fields in all characteristics, which generalizes Faltings's theorem about counting rational points on curves and the Manin-Mumford conjecture...

as of September 2022[update]. The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic...

g)} K {\displaystyle K} -rational points. This is a refinement of Faltings's theorem, which asserts that the set of K {\displaystyle K} -rational points...

theorems asserting that there are no solutions (for example Fermat's Last Theorem) or that the number of solutions is finite (for example Falting's theorem)...

emphasize the connections to modern algebraic geometry (for example, in Faltings's theorem) rather than to techniques in Diophantine approximations. Probabilistic...

its complexity introduced by Faltings in his proof of the Mordell conjecture. Fermat's Last Theorem Fermat's Last Theorem, the most celebrated conjecture...

elliptic curves. Such curves defined over the rational numbers, by Faltings's theorem, can have only a finite number of rational points, and they may be...

The Hirzebruch–Riemann–Roch theorem, Lipschitz continuity, the Petri net, the Schönhage–Strassen algorithm, Faltings's theorem and the Toeplitz matrix are...

University of Münster, where he was awarded his doctorate in 1954. Faltings's theorem Levi problem Grauert, Hans (1994), Selected papers. Vol. I, II, Berlin...

of the associated curve base changed to an algebraic closure. See Faltings's theorem for details on the arithmetic implications. 3.  Classification of...