Mordell's guess

The conjecture of Mordell comes from the theory of numbers , was founded in 1922 by Louis Mordell erected and 1983 by Gerd Faltings in his article finite sets for abelian varieties over number fields (Faltings' set) proved.

Motivation and statement of the sentence

If a number field and a nonsingular curve are defined by , then the question arises, how many points on the curve itself have coordinates in . Of particular interest is the case of the field of rational numbers, for which the conjecture was originally formulated by Louis Mordell . ${\ displaystyle K}$ ${\ displaystyle C}$ ${\ displaystyle K}$ ${\ displaystyle C (K)}$ ${\ displaystyle K}$

If the gender of is equal , is isomorphic to the one-dimensional projective space over the algebraic closure of , that is . Hence it can be empty or an infinite amount . ${\ displaystyle g}$ ${\ displaystyle C}$ ${\ displaystyle 0}$ ${\ displaystyle C}$ ${\ displaystyle K}$ ${\ displaystyle C \ cong \ mathbb {P} ^ {1} ({\ overline {K}})}$ ${\ displaystyle C (K)}$ ${\ displaystyle \ left (C (K) \ cong \ mathbb {P} ^ {1} (K) \ right)}$

If the gender of the same , and if at least one point with coordinates has, then an elliptic curve and a finitely generated abelian group . The latter is common as Mordell-Weil's theorem and implies that it can be finite or infinite. ${\ displaystyle g}$ ${\ displaystyle C}$ ${\ displaystyle 1}$ ${\ displaystyle C (K)}$ ${\ displaystyle K}$ ${\ displaystyle C}$ ${\ displaystyle C (K)}$ ${\ displaystyle C (K)}$

If the gender of is greater than , then is finite.

{\ displaystyle g}

{\ displaystyle C}

{\ displaystyle 1}

{\ displaystyle C (K)}

According to the theorem, there are only finitely many rational points on the curve for curves of gender . Curves over the rational numbers show essentially different behavior for , and - a topological quantity determines the number-theoretic behavior. Carl Ludwig Siegel had already proven this for whole numbers in the 1920s. ${\ displaystyle g> 1}$ ${\ displaystyle g = 0}$ ${\ displaystyle g = 1}$ ${\ displaystyle g> 1}$

The third statement in the sentence is known as the “Mordell's Assumption” and was proven in 1983 by Gerd Faltings .

proof

Faltings first presented his proof at the mathematical workshop in Bonn on July 17 and 19, 1983. In 1986 he received the highest honor for mathematicians, the Fields Medal . Sometimes Mordell's conjecture , which is now a proven theorem, is named after Faltings's theorem .

In his work, Faltings also proved the Tate conjecture from John T. Tate and the conjecture from Igor Shafarevich by expanding the translation mechanism from function fields to number fields from Suren Arakelov . Alexei Nikolajewitsch Parschin proved in 1968 (lecture at the ICM 1970) that the Mordell conjecture follows from the Schafarewitsch conjecture .

According to Faltings, Paul Vojta proved the theorem in a different way . Vojta's proof was simplified by Faltings himself and Enrico Bombieri .

For functional bodies, the Mordell conjecture was already proven in 1963 by Yuri Manin , in 1965 by Hans Grauert and in 1968 by Alexei Nikolajewitsch Parschin .

Akshay Venkatesh gave a new proof with Brian Lawrence in 2018. The proof follows the basic lines of Falting's proof, but uses the analysis of the variation of p-adic Galois representations.

Applications

The theorem delivered an important partial result for the Fermat's conjecture , because according to him , the Fermat's equation has at most finitely many coprime solutions. However, with the proof of Fermat's Conjecture by Andrew Wiles in 1993, this statement is obsolete. However, Mordell's theorem remains important for other equations to which Wiles's method cannot be applied. ${\ displaystyle x ^ {n} + y ^ {n} = z ^ {n}}$ ${\ displaystyle n \ geq 4}$

The previously known evidence of the suspicion of murder is not effective, i.e. it does not provide any information about the number and size of the solutions. The murder presumption, however, follows from the unproven abc presumption ( Noam Elkies ) in an effective variant.

literature

Spencer Bloch : The proof of the Mordell conjecture . In: Mathematical Intelligencer , Volume 6, 1984, p. 41.
Gerd Faltings : The assumptions of Tate and Mordell . In: Annual report of the German Mathematicians Association , 1984, pp. 1–13.
Gerd Faltings: Finiteness theorems for Abelian varieties over number fields . In: Inventiones Mathematicae , Volume 73, 1983, pp. 349-366; Erratum , Volume 75, 1984, p. 381
Lucien Szpiro: La conjecture de Mordell . In: Séminaire Bourbaki , No. 619, 1983/84.
AN Parshin, Yu. G. Zarhin: Finiteness Problems in Diophantine Geometry. In: Eight papers translated from the Russian . In: American Mathematical Society Translations Ser. 2, Volume 143, 1989, pp. 35-102, revised version of the article originally published as an appendix in the Russian edition of Serge Lang Fundamentals of Diophantine Geometry . arxiv : 0912.4325
Barry Mazur : Arithmetic on curves . In: AMS (Ed.): Bulletin of the American Mathematical Society Volume 14 . April 2, 1986, p. 207–259 (English, ams.org [PDF; 4.8 MB ; accessed on November 3, 2014]).
Barry Mazur: The unity and breadth of mathematics - from Diophantus to today . Paul Bernays Lecture, ETH Zurich 2018, math.harvard.edu (PDF)

Individual evidence

↑ Finiteness theorems for Abelian varieties over number fields . In: Inventiones Mathematicae , 73 (3), pp. 349-366 and Erratum .
^ Siegel: On some applications of Diophantine approximations . In: Abh. Preuss. Akad. Wiss. , Phys-Math. Class 1929, No. 1, also in Siegel: Gesammelte Abhandlungen . Springer, 1966, Volume 1, pp. 209-266
^ Faltings: Diophantine approximation on abelian varieties . In: Annals of Mathematics , Volume 133, 1991, pp. 549-567
^ Bombieri: The Mordell conjecture revisited . In: Annali Scuola Normale Superiore di Pisa , Volume 17, 1990, pp. 615-640. Also shown in Bombieri, Gubler: Heights in Diophantine Geometry . Cambridge UP, 2006
↑ Manin: Rational points of algebraic curves over function fields . In: Izvestija Akad. Nauka SSSR , Volume 27, 1963, pp. 1397-1442
↑ Robert F. Coleman found a gap in Manin's proof and filled it in Coleman: Manin's proof of the Mordell conjecture over function fields . In: L'Enseignement Mathématique , Volume 36, 1990, pp. 393-427. retro.seals.ch
↑ Grauert: Mordell's conjecture about rational points on algebraic curves and function fields . In: Publ. Math. IHES , Volume 25, 1965, pp. 131-149
↑ Parshin: Algebraic curves over function fields I . In: Math. USSR Izvestija , 2, 1968, pp. 1145-1170
↑ Lawrence, Venkatesh: Diophantine problems and p-adic period mappings . arxiv : 1807.02721
↑ The greatest puzzles in mathematics. Spectrum of Science Dossier, 6/2009, ISBN 978-3-941205-34-5 , p. 8 (interview with Gerd Faltings ).

[1] Finiteness theorems for Abelian varieties over number fields . In: Inventiones Mathematicae , 73 (3), pp. 349-366 and Erratum .

[2] Siegel: On some applications of Diophantine approximations . In: Abh. Preuss. Akad. Wiss. , Phys-Math. Class 1929, No. 1, also in Siegel: Gesammelte Abhandlungen . Springer, 1966, Volume 1, pp. 209-266

[3] Faltings: Diophantine approximation on abelian varieties . In: Annals of Mathematics , Volume 133, 1991, pp. 549-567

[4] Bombieri: The Mordell conjecture revisited . In: Annali Scuola Normale Superiore di Pisa , Volume 17, 1990, pp. 615-640. Also shown in Bombieri, Gubler: Heights in Diophantine Geometry . Cambridge UP, 2006

[5] Manin: Rational points of algebraic curves over function fields . In: Izvestija Akad. Nauka SSSR , Volume 27, 1963, pp. 1397-1442

[6] Robert F. Coleman found a gap in Manin's proof and filled it in Coleman: Manin's proof of the Mordell conjecture over function fields . In: L'Enseignement Mathématique , Volume 36, 1990, pp. 393-427. retro.seals.ch

[7] Grauert: Mordell's conjecture about rational points on algebraic curves and function fields . In: Publ. Math. IHES , Volume 25, 1965, pp. 131-149

[8] Parshin: Algebraic curves over function fields I . In: Math. USSR Izvestija , 2, 1968, pp. 1145-1170

[9] Lawrence, Venkatesh: Diophantine problems and p-adic period mappings . arxiv : 1807.02721

[10] The greatest puzzles in mathematics. Spectrum of Science Dossier, 6/2009, ISBN 978-3-941205-34-5 , p. 8 (interview with Gerd Faltings ).