View source for Mordell conjecture

A conjecture on the finiteness of the set of rational points on an [[Algebraic curve|algebraic curve]] of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m0649101.png" />. Advanced by L.J. Mordell [[#References|[1]]] for the case when the ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m0649102.png" /> is the field of rational numbers. At present (1982) Mordell's conjecture is taken to be the assertion of the finiteness of the set of rational points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m0649103.png" /> of an irreducible algebraic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m0649104.png" /> of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m0649105.png" /> defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m0649106.png" /> of finite type over the field of rational numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m0649107.png" /> in any finite extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m0649108.png" />. A reduction of Mordell's conjecture to the most difficult case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m0649109.png" /> is an algebraic number field has been obtained (see [[#References|[3]]]). A number of special results related to Mordell's conjecture are known. Thus, it has been proved [[#References|[2]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m06491010.png" /> is finite if the rank of the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m06491011.png" />-automorphisms from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m06491012.png" /> into an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m06491013.png" /> is larger than the rank of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m06491014.png" />. The finiteness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m06491015.png" /> has been established, [[#References|[7]]], for the broad class of modular curves (cf. [[Modular curve|Modular curve]]) and their fields of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m06491016.png" />. An estimate for the growth of the height,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m06491017.png" /></td> </tr></table>

of the rational points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m06491018.png" /> has been found, [[#References|[8]]], showing that they are situated more "sparsely" than on the curves of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m06491019.png" />. It has also been proved that Mordell's conjecture is a consequence of Shafarevich's conjecture on the finiteness of the number of algebraic curves having a given genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m06491020.png" />, a given field of definition (a finite extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m06491021.png" />) and a given set of points of bad reduction (see [[#References|[4]]], and also [[Siegel theorem|Siegel theorem]] on integer points).

The geometric analogue of Mordell's conjecture is the assertion of the finiteness of the number of sections of a bundle

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m06491022.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m06491023.png" /> is a non-singular projective surface, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m06491024.png" /> is a curve and the general fibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m06491025.png" /> is an irreducible curve of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m06491026.png" />. This assertion is true if the bundle is non-constant, that is, if it is not a direct product after a certain covering of the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m06491027.png" />, and if the characteristic of the ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m06491028.png" /> is equal to zero (see [[#References|[3]]], [[#References|[6]]]). For constant bundles it is possible only to assert the finiteness of the number of classes consisting of sections that are algebraically equivalent as curves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m06491029.png" />. If the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m06491030.png" /> is positive, the geometric analogue of Mordell's conjecture is false [[#References|[4]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L.J. Mordell, "On the rational solutions of the indeterminate equation of the third and fourth degrees" ''Proc. Cambridge Philos. Soc.'' , '''21''' (1922) pp. 179–192</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> V.A. Dem'yanenko, "Rational points of a class of algebraic curves" ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''30''' : 6 (1966) pp. 1373–1396 (In Russian) {{MR|}} {{ZBL|0162.24502}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> Yu.I. Manin, "Rational points of algebraic curves over function fields" ''Transl. Amer. Math. Soc. (2)'' , '''50''' (1966) pp. 189–234 ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''27''' : 6 (1963) pp. 1395–1440 {{MR|}} {{ZBL|0178.55102}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.N. Parshin, "Quelques conjectures de finitude en géométrie diophantienne" , ''Proc. Internat. Congress Mathematicians (Nice, 1970)'' , '''1''' , Gauthier-Villars (1971) pp. 467–471</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> H. Grauert, "Modell's Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörpern" ''Publ. Math. IHES'' : 25 (1965) pp. 131–149</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> S. Lang, "Diophantine geometry" , Interscience (1962) {{MR|0142550}} {{ZBL|0115.38701}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> B. Mazur, "Rational points on modular curves" J.-P. Serre (ed.) P. Deligne (ed.) W. Kuyk (ed.) , ''Modular functions of one variable. 5'' , ''Lect. notes in math.'' , '''601''' , Springer (1977) pp. 107–148 {{MR|0450283}} {{ZBL|0357.14005}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> D. Mumford, "A remark on Mordell's conjecture" ''Amer. J. Math.'' , '''87''' : 4 (1965) pp. 1007–1016 {{MR|186624}} {{ZBL|}} </TD></TR></table>



====Comments====
The article above was originally written in 1982. Since then much has happened, and the field of arithmetic algebraic geometry has taken on additional prominence.

The Mordell conjecture over a number field has been proved in 1983 by G. Faltings. In his fundamental paper [[#References|[a1]]], opening a new chapter in arithmetic algebraic geometry, Faltings proves at the same time two other famous conjectures, namely the Tate conjecture concerning endomorphisms of Abelian varieties over number fields (cf. [[Tate conjectures|Tate conjectures]]) and the Shafarevich conjecture concerning the finiteness of isomorphism classes of curves of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m06491031.png" /> (respectively, Abelian varieties) defined over a number field and with good reduction outside a finite number of given places. The relation of the Mordell conjecture with the Shafarevich conjecture was pointed out by A.N. Parshin in 1970 [[#References|[4]]]. Important for the work of Faltings were ideas of S.Yu. Arakelov, Yu.G. Zarkhin and L. Szpiro; essential in Faltings' proof is the theory of heights on the moduli space of Abelian varieties. For a good introduction to the proof of the Mordell conjecture see [[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G. Faltings, "Endlichkeitssätze für abelsche Varietäten über Zahlkörper" ''Invent. Math.'' , '''73''' (1983) pp. 349–366 (Erratum: Ibid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064910/m06491032.png" /> (1984), 381)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G. Faltings, "Die Vermutungen von Tate und Mordell" ''Jhber. Deutsch. Math. Verein.'' , '''86''' (1984) pp. 1–13 {{MR|0736624}} {{ZBL|0586.14012}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> P. Deligne, "Preuve et conjectures de Tate et de Shafarevich" ''Sém. Bourbaki'' , '''616''' (1983) {{MR|}} {{ZBL|0591.14026}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> L. Szpiro, "Le conjecture de Mordell" ''Sém. Bourbaki'' , '''619''' (1983)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> G. Faltings (ed.) G. Wüstholtz (ed.) , ''Rational points'' , Vieweg (1984) {{MR|0766568}} {{ZBL|0588.14027}} </TD></TR></table>