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BRUDNO’S FAMILIES OF EQUAL SUMS OF SIXTH POWERS AND BREMNER’S K3 SURFACE

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theorymath.AG — Algebraic Geometry

Contributed by Alon Amit ↗

Version 1 / Oct 11, 2026 / CC BY 4.0

Abstract

Brudno (1970, 1976) and Brudno–Kaplansky (1974) found an elliptic family and a degree-four polynomial family of solutions of x^6 + y^6 + z^6 = u^6 + v^6 + w^6. We explain both in terms of the K3 surface V ⊂P5, the complete intersection of three quadrics studied by Bremner (1981) and Kuwata (2007). The Brudno–Kaplansky curve is Cremona’s curve 480f4 and is the fibre at λ=∞ of the elliptic pencil on V that Bremner calls the Brudno pencil; the curve z^2 = x^4 + 38x^2y^2 + y^4 of 1970 is 480f1, and the classical recursions of Lebesgue and Desboves that Brudno iterated are the multiplication maps [2] and [−3] on it. Brudno’s 1976 quartic is the unique rational curve of degree four on V (Bremner’s Theorem 4.4) and a section of the Brudno pencil; its parameter is the fibre coordinate λ, and the involution Brudno noticed is a linear symmetry of V. Using the Mordell–Weil lattice of the pencil we prove a closed formula for the degree of a section, deg C_S = 2⟨S−γ/4, S−γ/4⟩, enumerate all sections of degree ≤12 (four of them new, of degrees 7,10,11,11), and enumerate, by a lattice computation, all smooth rational curves of degree ≤12 on V over Q(i) and over Q, reproducing Bremner’s theorems and testing his conjecture. A census of the 74 primitive solutions with all terms ≤ 1343 identifies which of them lie on V, on known curves, on sections, and on fibres of jumping rank, and records some phenomena off V. In the second part we compute equations and parametrisations of the 48 new curves that are not sections and show that seven of the eight census solutions on no known curve lie on them, leaving a single unexplained point; we determine the geometry over C of the sextic fourfold M6, of the threefold W= M6 ∩M2 (which has exactly 710 nodes and defect 95) and of V (transcendental lattice U(2) ⊕⟨12⟩, zeta function through the newform 320f2), and discuss in what sense the geometry drives the arithmetic; and we show that Bremner’s conjecture follows, with non-explicit finite sets, from the theorems of Sterk and Bright–Logan–van Luijk, describe how to make it explicit (Borcherds–Shimada), and run the first step of Bremner’s own cone procedure, which identifies the degree-8 pencil involutions that the proof requires.

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References

  1. Artebani, Michela; Laface, Antonio; Testa, Damiano. On Büchi's K3 surface. Math. Z., vol. 278, no. 3–4, pp. 1113–1131. 2014. DOI 10.1007/s00209-014-1348-9; arXiv:1212.1426arXiv
  2. Baragar, Arthur. Rational points on K3 surfaces in P1×P1×P1\mathbb P^1\times\mathbb P^1\times\mathbb P^1. Math. Ann., vol. 305, pp. 541–558. 1996
  3. Baragar, Arthur. Canonical vector heights on algebraic K3 surfaces with Picard number two. Canad. Math. Bull., vol. 46, no. 4, pp. 495–508. 2003DOI
  4. Baragar, Arthur. Orbits of curves on certain K3 surfaces. Compos. Math., vol. 137, no. 2, pp. 115–134. 2003DOI
  5. Baragar, Arthur. Canonical vector heights on K3 surfaces with Picard number three–an argument for nonexistence. Math. Comp., vol. 73, no. 248, pp. 2019–2025. 2004DOI
  6. Baragar, Arthur. The ample cone for a K3 surface. Canad. J. Math., vol. 63, no. 3, pp. 481–499. 2011DOI
  7. Baragar, Arthur; McKinnon, David. K3 surfaces, rational curves, and rational points. J. Number Theory, vol. 130, no. 7, pp. 1470–1479. 2010DOI
  8. Baragar, Arthur; van Luijk, Ronald. K3 surfaces with Picard number three and canonical vector heights. Math. Comp., vol. 76, pp. 1493–1498. 2007
  9. Batyrev, Victor V.; Manin, Yuri I. Sur le nombre des points rationnels de hauteur bornée des variétés algébriques. Math. Ann., vol. 286, pp. 27–43. 1990
  10. Bogomolov, Fedor; Hassett, Brendan; Tschinkel, Yuri. Constructing rational curves on K3 surfaces. Duke Math. J., vol. 157, no. 3, pp. 535–550. 2011DOI
Show all 94 referencesShow fewer references
  1. Bogomolov, Fedor; Tschinkel, Yuri. Density of rational points on elliptic K3 surfaces. Asian J. Math., vol. 4, pp. 351–368. 2000
  2. Bogomolov, Fedor; Tschinkel, Yuri. Rational curves and points on K3 surfaces. Amer. J. Math., vol. 127, no. 4, pp. 825–835. 2005DOI
  3. Borcherds, Richard. Automorphism groups of Lorentzian lattices. J. Algebra, vol. 111, pp. 133–153. 1987DOI
  4. Borcherds, Richard E. Coxeter groups, Lorentzian lattices, and K3 surfaces. Int. Math. Res. Not. IMRN, vol. 1998, no. 19, pp. 1011–1031. 1998DOI
  5. Brandhorst, Simon; Mezzedimi, Giacomo. Borcherds lattices and K3 surfaces of zero entropy. 2022. arXiv:2211.09600
  6. Brandhorst, Simon; Shimada, Ichiro. Automorphism groups of certain Enriques surfaces. Found. Comput. Math., vol. 22, no. 5, pp. 1463–1512. 2022DOI
  7. Brandhorst, Simon; Shimada, Ichiro. Borcherds' method for Enriques surfaces. Michigan Math. J., vol. 71, no. 1, pp. 3–18. 2022DOI
  8. Braun, Andreas P.; Fortin, Hugo; Lopez Garcia, Daniel; Villaflor Loyola, Roberto. More on GG-flux and general Hodge cycles on the Fermat sextic. J. High Energy Phys. 2024. arXiv:2401.00470arXiv
  9. Bremner, Andrew. A geometric approach to equal sums of fifth powers. J. Number Theory, vol. 13, pp. 337–354. 1981
  10. Bremner, Andrew. A geometric approach to equal sums of sixth powers. Proc. London Math. Soc. (3), vol. 43, pp. 544–581. 1981
  11. Bright, Martin; Logan, Adam; van Luijk, Ronald. Finiteness results for K3 surfaces over arbitrary fields. European J. Math., vol. 6, no. 2, pp. 336–366. 2020. arXiv:1810.04905arXivDOI
  12. Brudno, Simcha. Some new results on equal sums of like powers. Math. Comp., vol. 23, pp. 877–880. 1969
  13. Brudno, Simcha. On generating infinitely many solutions of the Diophantine equation A6+B6+C6=D6+E6+F6A^6+B^6+C^6=D^6+E^6+F^6. Math. Comp., vol. 24, pp. 453–454. 1970
  14. Brudno, Simcha. Triples of sixth powers with equal sums. Math. Comp., vol. 30, pp. 646–648. 1976
  15. Brudno, Simcha; Kaplansky, Irving. Equal sums of sixth powers. J. Number Theory, vol. 6, pp. 401–403. 1974
  16. Campana, Frédéric. Orbifolds, special varieties and classification theory. Ann. Inst. Fourier (Grenoble), vol. 54, pp. 499–630. 2004
  17. Charles, François. The Tate conjecture for K3 surfaces over finite fields. Invent. Math., vol. 194, pp. 119–145. 2013
  18. Chen, Xi; Gounelas, Frank; Liedtke, Christian. Curves on K3 surfaces. Duke Math. J., vol. 171, no. 16, pp. 3283–3362. 2022DOI
  19. Choudhry, Ajai. On equal sums of sixth powers. Indian J. Pure Appl. Math., vol. 25, no. 8, pp. 837–841. 1994
  20. Choudhry, Ajai. On equal sums of sixth powers. Rocky Mountain J. Math., vol. 30, no. 3, pp. 843–848. 2000
  21. Clemens, C. Herbert. Double solids. Adv. in Math., vol. 47, pp. 107–230. 1983
  22. Conway, J. H. The automorphism group of the 2626-dimensional even unimodular Lorentzian lattice. J. Algebra, vol. 80, pp. 159–163. 1983DOI
  23. Cremona, J. E. Algorithms for modular elliptic curves. Cambridge University Press. 1997. Tables of elliptic curves over Q\mathbb{Q} (files \textttallcurves, \textttallgens, \textttallisog, \textttallbsd), https://github.com/JohnCremona/ecdatalink
  24. Cynk, Sławomir. Defect of a nodal hypersurface. Manuscripta Math., vol. 104, pp. 325–331. 2001
  25. Cynk, Sławomir. Defect formula for nodal complete intersection threefolds. 2016. arXiv:1607.08106arXiv
  26. Cynk, Sławomir; Rams, Sławomir. Defect via differential forms with logarithmic poles. Math. Nachr., vol. 284. 2011. arXiv:0909.4655. \todoissue and pages not checkedarXiv
  27. Deligne, Pierre. La conjecture de Weil. II. Inst. Hautes Études Sci. Publ. Math., vol. 52, pp. 137–252. 1980
  28. Delorme, Jean-Joël. On the Diophantine equation x16+x26+x36=y16+y26+y36x_1^6+x_2^6+x_3^6=y_1^6+y_2^6+y_3^6. Math. Comp., vol. 59, no. 200, pp. 703–715. 1992
  29. Desboves, A. Mémoire sur la résolution en nombres entiers de l'équation aXm+bYm=cZnaX^m+bY^m=cZ^n. Nouv. Ann. Math. (2), vol. 18, pp. 265–279, 398–410, 433–444, 481–499. 1879. \todotitle not checked against the original
  30. Dickson, Leonard Eugene. History of the theory of numbers. Vol. II: Diophantine analysis. Chelsea Publishing Co. 1952. Reprint of the 1920 edition; Ch. XXII, p. 637
  31. Dolgachev, Igor V. Reflection groups in algebraic geometry. Bull. Amer. Math. Soc. (N.S.), vol. 45, no. 1, pp. 1–60. 2008DOI
  32. Dolgachev, Igor V. Classical algebraic geometry. A modern view. Cambridge University Press. 2012. \todosection number for the Segre cubic not checked
  33. Elkies, Noam D. On A4+B4+C4=D4A^4+B^4+C^4=D^4. Math. Comp., vol. 51, no. 184, pp. 825–835. 1988DOI
  34. Goresky, Mark; MacPherson, Robert. Stratified Morse theory. Springer-Verlag, vol. 14. 1988
  35. Guy, Richard K. Unsolved problems in number theory. Springer-Verlag. 1994. Problem D1
  36. Hirzebruch, Friedrich. Topological methods in algebraic geometry. Springer-Verlag, vol. 131. 1966. Appendix I by R. L. E. Schwarzenberger
  37. Humphreys, James E. Reflection groups and Coxeter groups. Cambridge University Press, vol. 29. 1990
  38. Huybrechts, Daniel. Lectures on K3 surfaces. Cambridge University Press, vol. 158. 2016
  39. Kani, Ernst. Elliptic curves on abelian surfaces. Manuscripta Math., vol. 84, pp. 199–223. 1994
  40. Kawamata, Yujiro. On the cone of divisors of Calabi–Yau fiber spaces. Internat. J. Math., vol. 8, no. 5, pp. 665–687. 1997DOI
  41. Keum, Jong Hae. Automorphisms of Jacobian Kummer surfaces. Compos. Math., vol. 107, no. 3, pp. 269–289. 1997DOI
  42. Keum, Jonghae; Kondō, Shigeyuki. The automorphism groups of Kummer surfaces associated with the product of two elliptic curves. Trans. Amer. Math. Soc., vol. 353, no. 4, pp. 1469–1487. 2001DOI
  43. Kondō, Shigeyuki. Algebraic K3 surfaces with finite automorphism groups. Nagoya Math. J., vol. 116, pp. 1–15. 1989DOI
  44. Kondō, Shigeyuki. The automorphism group of a generic Jacobian Kummer surface. J. Algebraic Geom., vol. 7, no. 3, pp. 589–609. 1998
  45. Kondō, Shigeyuki; Mukai, Shigeru. The automorphism groups of Kummer surfaces in characteristic two and their complex analogues. 2024. arXiv:2412.18902
  46. Kovács, Sándor J. The cone of curves of a K3 surface. Math. Ann., vol. 300, no. 4, pp. 681–691. 1994DOI
  47. Kuwata, Masato. Equal sums of sixth powers and quadratic line complexes. Rocky Mountain J. Math., vol. 37, no. 2, pp. 497–517. 2007
  48. Lander, L. J.; Parkin, T. R.; Selfridge, J. L. A survey of equal sums of like powers. Math. Comp., vol. 21, pp. 446–459. 1967
  49. Lang, Serge. Hyperbolic and Diophantine analysis. Bull. Amer. Math. Soc. (N.S.), vol. 14, pp. 159–205. 1986
  50. Lebesgue, V. A. Résolution des équations biquadratiques z2=x4±2my4z^2=x^4\pm 2^m y^4, z2=2mx4−y4z^2=2^m x^4-y^4, 2mz2=x4±y42^m z^2=x^4\pm y^4. J. Math. Pures Appl. (1), vol. 18, pp. 73–86. 1853. \todotitle taken from the Numdam summary, not from the paper
  51. Lieblich, Max; Maulik, Davesh. A note on the cone conjecture for K3 surfaces in positive characteristic. Math. Res. Lett., vol. 25, no. 6, pp. 1879–1891. 2018DOI
  52. Looijenga, Eduard. Discrete automorphism groups of convex cones of finite type. Compos. Math., vol. 150, no. 11, pp. 1939–1962. 2014. arXiv:0908.0165arXivDOI
  53. Lutz, Élisabeth. Sur l'équation y2=x3−Ax−By^2=x^3-Ax-B dans les corps pp-adiques. J. Reine Angew. Math., vol. 177, pp. 237–247. 1937
  54. Madapusi Pera, Keerthi. The Tate conjecture for K3 surfaces in odd characteristic. Invent. Math., vol. 201, pp. 625–668. 2015
  55. Morrison, David R. On K3 surfaces with large Picard number. Invent. Math., vol. 75, pp. 105–121. 1984
  56. Morrison, David R. Compactifications of moduli spaces inspired by mirror symmetry. Journées de géométrie algébrique d'Orsay (Orsay, 1992), vol. 218, pp. 243–271. 1993
  57. Nagell, Trygve. Solution de quelques problèmes dans la théorie arithmétique des cubiques planes du premier genre. Skr. Norske Vid.-Akad. Oslo I, no. 1, pp. 1–25. 1935. \todotitle and bibliographic data of this entry not checked against the original
  58. Namikawa, Yukihiko. Periods of Enriques surfaces. Math. Ann., vol. 270, pp. 201–222. 1985DOI
  59. Nikulin, V. V. Kummer surfaces. Math. USSR Izv., vol. 9, pp. 261–275. 1975
  60. Nikulin, V. V. On factor groups of the automorphism groups of hyperbolic forms modulo subgroups generated by 22-reflections. Soviet Math. Dokl., vol. 20, pp. 1156–1158. 1979. Russian original: Dokl. Akad. Nauk SSSR 248 (1979) 1307–1309
  61. Nikulin, V. V. Integral symmetric bilinear forms and some of their applications. Math. USSR Izv., vol. 14, pp. 103–167. 1980
  62. Nikulin, V. V. Factor groups of groups of automorphisms of hyperbolic forms with respect to subgroups generated by 22-reflections. Algebro-geometric applications. J. Soviet Math., vol. 22, pp. 1401–1475. 1983. Russian original: Itogi Nauki i Tekhniki, Sovrem. Probl. Mat. 18 (1981) 3–114DOI
  63. Nikulin, V. V. Surfaces of type K3 with finite group of automorphisms and Picard group of rank three. Proc. Steklov Inst. Math., vol. 165, pp. 131–155. 1985. Russian original: Trudy Mat. Inst. Steklov. 165 (1984) 119–142
  64. OSCAR. K3 surfaces \textup(functions \textttK3_surface_automorphism_group, \textttborcherds_method\textup), in: OSCAR – Open Source Computer Algebra Research system. accessed October 2026. Documentation, https://docs.oscar-system.org/stable/AlgebraicGeometry/Surfaces/K3Surfaces/link
  65. Piezas, III, Tito. Sixth powers (4 or 6 terms). accessed October 2026. Online collection of algebraic identities, https://sites.google.com/view/tpiezas/023-sixth-powers-4-or-6-termslink
  66. Pjateckiĭ-Šapiro, I. I.; Šafarevič, I. R. A Torelli theorem for algebraic surfaces of type K3. Math. USSR Izv., vol. 5, pp. 547–588. 1971. Russian original: Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971)DOI
  67. Shimada, Ichiro. An algorithm to compute automorphism groups of K3 surfaces and an application to singular K3 surfaces. Int. Math. Res. Not. IMRN, vol. 2015, no. 22, pp. 11961–12014. 2015. arXiv:1304.7427arXivDOI
  68. Shioda, Tetsuji. The Hodge conjecture and the Tate conjecture for Fermat varieties. Proc. Japan Acad. Ser. A Math. Sci., vol. 55, pp. 111–114. 1979
  69. Shioda, Tetsuji. The Hodge conjecture for Fermat varieties. Math. Ann., vol. 245, pp. 175–184. 1979
  70. Shioda, Tetsuji. On the Mordell–Weil lattices. Comment. Math. Univ. St. Pauli, vol. 39, no. 2, pp. 211–240. 1990
  71. Shioda, Tetsuji; Katsura, Toshiyuki. On Fermat varieties. Tôhoku Math. J. (2), vol. 31, pp. 97–115. 1979
  72. Silverman, Joseph H. Rational points on K3 surfaces: a new canonical height. Invent. Math., vol. 105, pp. 347–373. 1991
  73. Silverman, Joseph H. The arithmetic of elliptic curves. Springer, vol. 106. 2009
  74. Sterk, Hans. Finiteness results for algebraic K3 surfaces. Math. Z., vol. 189, pp. 507–513. 1985DOI
  75. Subba Rao, K. On sums of sixth powers. J. London Math. Soc., vol. 9, pp. 172–173. 1934
  76. Swinnerton-Dyer, H. P. F. A4+B4=C4+D4A^4+B^4=C^4+D^4 revisited. J. London Math. Soc., vol. 43, pp. 149–151. 1968DOI
  77. Totaro, Burt. The cone conjecture for Calabi–Yau pairs in dimension 2. Duke Math. J., vol. 154, no. 2, pp. 241–263. 2010DOI
  78. Vinberg, È. B. Discrete linear groups generated by reflections. Math. USSR Izv., vol. 5, pp. 1083–1119. 1971DOI
  79. Vinberg, È. B. On groups of unit elements of certain quadratic forms. Math. USSR Sb., vol. 16, pp. 17–35. 1972DOI
  80. Vinberg, È. B. The two most algebraic K3 surfaces. Math. Ann., vol. 265, pp. 1–21. 1983DOI
  81. Vinberg, È. B. Classification of 22-reflective hyperbolic lattices of rank 4. Trans. Moscow Math. Soc., vol. 2007, pp. 39–66. 2007DOI
  82. Wehler, Joachim. K3-surfaces with Picard number 2. Arch. Math. (Basel), vol. 50, no. 1, pp. 73–82. 1988DOI
  83. Weil, André. Numbers of solutions of equations in finite fields. Bull. Amer. Math. Soc., vol. 55, pp. 497–508. 1949
  84. Werner, Jürgen. Kleine Auflösungen spezieller dreidimensionaler Varietäten. Universität Bonn, Mathematisches Institut, vol. 186. 1987

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