Skip to main content
    • Aa
    • Aa

Mordell's finite basis theorem revisited

  • J. W. S. Cassels (a1)

0. Mordell proved his ‘Finite Basis Theorem’ in the paper [31] ‘On the rational solutions of the indeterminate equations of the third and fourth degrees’ which appeared in 1922 in Volume 21 of these Proceedings. It had been assumed, rather than conjectured, by Poincaré some 20 years previously, but it was not what he had set out to prove. The theorem and its generalizations are at the heart of many of the most interesting achievements and problems of the theory of numbers and also of algebraic geometry. Mordell himself had virtually no part in these developments: his great work was to lie elsewhere ([5]).

Hide All
[1]Baker A. and Coates J.. Integer points on curves of genus 1. Proc. Cambridge Philos. Soc. 67 (1970), 595602.
[2]Birch B. J.. Heegner points of elliptic curves. Symposia Mathematica 15 (Academic Press 1975), 441445.
[3]Birch B. J. and Swinnerton-Dyer H. P. F.. Notes on elliptic curves II. J. Reins Angew. Math. 218 (1965), 79108.
[4]Cassels J. W. S.. Diophantine equations with special reference to elliptic curves. J. London Math. Soc. 41 (1966), 193291.
Cassels J. W. S.. Diophantine equations with special reference to elliptic curves. J. London Math. Soc. Crrigenda 42 (1967), 183.
[5]Cassels J. W. S.. Mordell L. J.. Biog. Mem. Roy. Soc. 19 (1973), 493520.
Cassels J. W. S.. Bull. London Math. Soc. 6 (1974), 6996.
[6]Cassels J. W. S.. The Mordell–Weil group and curves of genus 2. Arithmetic and Geometry. Papers dedicated to I. R. Shafarevich on the occasion of his sixtieth birthday, I (Birkhäuser, 1983), 2960.
[7]Coates J. and Wiles A.. On the conjectures of Birch and Swinnerton-Dyer. Invent. Math. 39 (1977), 223251.
[8]Dickson L. E.. History of the Theory of Numbers. (3 vols.) Carnegie Institute of Washington Pub. 256, 1919 (Reprint: G. E. Steehert, New York, 1934).
[9]Faltings G.. Endllichkeitssätze für abelsehe Varietäten über Zahlkörpern. Invent. Math. 73 (1983), 349366.
[10]Grauert H.. Mordells Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörper. Inst. Hautes Études Sci. Publ. Math. 25 (1965), 131149.
[11]Gross B. and Zagier D.. Points do Heegner et dérivées do fonctions L. C. R. Acad. Sci. Paris, Sér I. Math. 297 (1983), 8587.
[12]Heegner K.. Diophantische Analysis und Modulfunktionen. Math. Z. 56 (1952), 227253.
[13]Hilbert D.. Gesammelte Abhandlungen (3 vols.) (Springer, Göttingen, 1932) (Reprint: Chelsea, New York, 1965).
[14]Hilbert D. and Hurwitz A.. Über die diophantischen Gleichungen vom Geschlecht Null. Acta Math. 14 (18901891), 217224(= Hubert, Ges. Abh. II, 258–263; = Hurwitz, Math. Werke II, 116–121).
[15]Hurwitz A.. Über ternäre diophantische Gleichungen dritten Grades. Vierteljahrschrift d. Naturf. Ges. in Zürich 62 (1917), 207229 (= Math. Werke II, 446–468).
[16]Hurwitz A.. Mathematische Werke. (2 vols) (Birkhäuser, Basel, Stuttgart, 1963).
[17]Landau E. and Ostrowski A.. On the diophantine equation ay 2+by+c = dx n. Proc. London Math. Soc. (2) 19 (1920), 276280.
[18]Lang S.. Diophantine Geometry (Interscience, 1962).
[19]Lang S.. Fundamentals of Diophantine Geometry (Springer-Verlag, 1983).
[20]Lang S. and Néron A.. Rational points of abelian varieties over function fields. Amer. J. Math. 81 (1959), 95118.
[21]Lang S. and Tate J.. Principal homogeneous spaces over abelian varieties. Amer. J. Math. 80 (1958), 659684.
[22]Levi B., Saggio per una teoria aritmetica delle forme cubiche ternari. Atti B. Accad. Sc. Torino 41 (1906), 739764 and 43 (1908), 99–120; 413–434; 672–681.
[23]Levi B.. Sull'equazione indeterminata del 3° ordine. Atti IV Congresso Mat., Roma 1909, II, 173–7. (Reprint: Kraus, Liechtenstein, 1967).
[24]Manin Ju. I.. Rational points on algebraic curves over function fields (in Russian). Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963), 13951440.
[25]Manin Ju. I.. Cyclotomic fields and modular curves (in Russian). Uspekhi Mat. Nauk 26:6 (1971), 771.
[26]Mordell L. J.. The diophantine equation y 2k = x 3. Proc. London Math. Soc. (2) 13 (1913), 6080.
[27]Mordell L. J.. Indeterminate equations of the third and fourth degrees. Quart. J. Pure Appl. Math. 45 (1914), 170186.
[28]Mordell L. J.. A statement by Fermat (Records of proceedings). Proc. London Math. Soc. (2) 18 (1919), v–vi.
[29]Mordell L. J.. Three Lectures on Fermat's Last Theorem (Cambridge University Press, 1921) (Reprinted in: Two papers on number theory (ed. Neumann O.), VEB Deutscher Verlag d. Wiss., 1973).
[30]Mordell L. J.. Note on the integer solutions of the equation Ey 2 = Ax 3 + Bx 2 + Cx + D. Mess. Math. 51 (1922), 169171.
[31]Mordell L. J.. On the rational solutions of the indeterminate equations of the third and fourth degrees. Proc. Cambridge Philos. Soc. 21 (1922), 179182.
[32]Mordell L. J.. On the integer solutions of the equation ey 2 = ax 3 + bx 2 + cx + d. Proc. London Math. Soc. (2) 21 (1923), 415419.
[33]Mordell L. J.. Indeterminate equations of the third degree. Science Progress 18 (1923), 3955.
[34]Néron A.. Problémes arithmétiques et géometriques rattachés a Ia notion de rang d'une courbe algébrique dans un corps. Bull. Soc. Math. France 80 (1952), 101166.
[35]Newton I.. Mathematical Papers (8 vols) (Cambridge University Press, 19671981).
[36]Noether M.. Über Flächen, weiche Schaaren rationaler Curven besitzen. Math. Ann. 3 (1871), 161227.
[37]Poincaré H.. Sur les propriétés arithmétiques des courbes algébriques. J. math. pures appl. (5) 7 (1901), 161233. (= Oeuvres 5, 483550).
[38]Poincaré H.. Oeuvres (10 vols.) (Gauthier-Villars, Paris, 1916).
[39]Šfarevič I. R.. The group of principal homogeneous algebraic manifolds (in Russian). Dokl. Akad. Nauk SSSR 124 (1959), 4243.
[40]Samuel P.. Compléments à un article de Hans Grauert sur la conjecture de Mordell. Inst. Hautes Études Sci. Publ. Math. 29 (1966), 5562.
[41]Serre J. -P.. Autour du théorème de Mordell-Weil (2 vols). (Publ. Math. de l'Université Pierre et Marie Curie, 1983).
[42] X[ = Siegel C. L.]. The integer solutions of the equation y 2 = ax n + bx n–1 + … + k. J. London Math. Soc. 1(1926), 6668 (= Ges. Abh. I, 207–208).
[43]Siegel C. L.. Über einige Anwendungen diophantischer Approximationen. Abh. Preuss. Akad. Wiss. 1929, Nr 1 (= Ges. Abh. I, 209–266).
[44]Siegel C. L.. Gesammelte Abhandlungen (4 vols) (Springer, 1966).
[45]Skolem Th.. Diophantische Gleichungen. Erg. Math. 54, 1938. (Reprint: Chelsea, New York, 1950).
[46]Tate J.. WC groups over p-adic fields. Sém. Bourbaki 10(1957) no. 156 (13 pp.).
[47]Tate J.. The arithmetic of elliptic curves. Invent. Math. 23 (1974), 179206.
[48]Thue A.. Om en generel i store hele tal uløsbar ligning. Kristiania Videns. Selsk. Skrifter. I Mat. Nat. Kl., 1908, no. 7 (= Selected Math. Pap., 219231).
[49]Thue A.. Über Annäherungswerte algebraischer Zahien. J. Reine Angew. Math. 135 (1909), 284305 (= Selected Math. Pap., 232–253).
[50]Thue A.. Über die Unlösbarkeit der Gleichung ax 2 + bx + c = dy n in grossen ganzen Zahlen. Arch. math. og naturv., Kristiania 34 (1917) no. 16 (= Selected Math. Pap., 561564).
[51]Thue A.. Selected Mathematical Papers (Universitetsforlaget. Oslo, Bergen, Tromsø, 1977).
[52]Weil A.. L'arithmétique sur les courbes algébriques. Acta Math. 52 (1928), 281315 (= Coll. Papers I, 11–45).
[53]Weil A.. Sur un théorème de Mordell. Bull. Sci. Math. (2) 54 (1929), 182191 (= Coll. Papers I, 47–56).
[54]Weil A.. Oeuvres scientifiques. Collected papers. (3 vols) (Springer, 1980).
[55]Weil A.. Number Theory: An Approach Through History (Birkhäuser, 1983).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 12 *
Loading metrics...

Abstract views

Total abstract views: 99 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd October 2017. This data will be updated every 24 hours.