Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field $${\displaystyle \mathbb {Q} }$$ of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordell, and known as the Mordell conjecture until its 1983 proof … See more Igor Shafarevich conjectured that there are only finitely many isomorphism classes of abelian varieties of fixed dimension and fixed polarization degree over a fixed number field with good reduction outside a fixed finite set of See more Faltings's 1983 paper had as consequences a number of statements which had previously been conjectured: • The Mordell conjecture that a curve of genus greater than … See more Webtheorem is known ([8] for details). Theorem 3. Let Rbe a regular local ring of mixed characteristic p>0 and let S be a torsion free module- nite R-algebra such that the localization R[1 p] !S[p] is nite etale. Then Shas a balanced big Cohen-Macaulay algebra. The proof of this theorem is based on the almost purity theorem. We have the following ...
Faltings
WebJan 13, 2024 · Summary. Chapter 5 is devoted to giving a detailed proof of Faltings’s theorem (Theorem 5.1), asserting that "any algebraic curve of genus at least two … WebJan 13, 2024 · Summary. Chapter 4 is devoted to several fundamental results of Diophantine geometry such as Siegel's lemma (Lemma 4.1 and Proposition 4.3) and Roth's lemma (Theorem 4.20). Besides them, we also introduce Guass’s lemma, the Mahler measure, the height of a polynomial, Gelfond’s inequality, the index with respect to a … tenda w300a setup
Faltings
WebFaltings was a monumental achievement in twentieth-century mathematics. In this book, we will call the Mordell conjecture Faltings s theorem. Perhaps Faltings s success lifted a … WebFeb 9, 2024 · Faltings’ theorem. Let K K be a number field and let C/K C / K be a non-singular curve defined over K K and genus g g. When the genus is 0 0, the curve is … WebBased on the OP's comment clarifying his question, I fear that the answer is no, there are no concrete special cases in which one can follow the approach of Faltings' proof that yield any significant simplifications. Faltings' proof is very indirect. First one uses rational points in C ( K) to construct coverings of C that have good reduction ... tenda w300a