HEXAGONbeta

Period relations as period terms: linear independence at CM and Beta periods

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by Shawn Krisman ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

We prove linear independence results for periods of CM abelian varieties and Beta values by using a known relation between periods as the period term in the algebraic subgroup theorem. For example, 11, B(27,27)B(\frac27,\frac27), B(37,37)B(\frac37,\frac37) and B(37,37)/B(17,17)B(\frac37,\frac37)/B(\frac17,\frac17) are linearly independent over Q‾\overline{\mathbb{Q}}, and families of such relations attached to the CM types of Q(ζN)\mathbb{Q}(\zeta_N) give the linear independence of 11, a common ratio and κ≥2\kappa\ge2 periods. The same method proves Waldschmidt's Conjecture 42 at six classes of CM ratios (for instance, ℘(log⁡2/ϖ)\wp(\log 2/\varpi) and ℘(log⁡3/ϖ)\wp(\log 3/\varpi) are not both algebraic on the lemniscatic curve) and several four exponentials statements at CM periods. All results are explicit consequences of theorems of Waldschmidt and of Roy and Waldschmidt.

Tools used

Anthropic
Claude OpusVersion 5.5

References

  1. G. W. Anderson. Logarithmic derivatives of Dirichlet LL-functions and the periods of abelian varieties. Compositio Math., vol. 45, pp. 315–332. 1982
  2. A. Baker; G. Wüstholz. Logarithmic Forms and Diophantine Geometry. Cambridge Univ.\ Press, vol. 9. 2007
  3. D. Bertrand. La théorie de Baker revisitée. The reference list of \citePW88 gives the title as La méthode de Baker revisitée. Publ.\ Math.\ Univ.\ P. et M. Curie 73, Groupe d'Étude sur les Problèmes Diophantiens 1984–85, no. 2, 25 pp.
  4. D. Bertrand. Endomorphismes de groupes algébriques; applications arithmétiques. Approximations diophantiennes et nombres transcendants (Luminy 1982), vol. 31, pp. 1–45. 1983
  5. D. Blasius. On the critical values of Hecke LL-series. Ann.\ of Math., vol. 124, pp. 23–63. 1986
  6. P. Deligne. Hodge cycles on abelian varieties. P. Deligne, J. S. Milne, A. Ogus, K.-Y. Shih, Hodge Cycles, Motives, and Shimura Varieties, vol. 900, pp. 9–100. 1982. notes by J. S. Milne
  7. G. Diaz. Utilisation de la conjugaison complexe dans l'étude de la transcendance de valeurs de la fonction exponentielle usuelle. J. Théor.\ Nombres Bordeaux, vol. 16, pp. 535–553. 2004
  8. Z. Gao; E. Ullmo. Hodge cycles and quadratic relations between holomorphic periods on CM abelian varieties. J. Inst.\ Math.\ Jussieu, vol. 25, no. 1, pp. 215–249. 2026. arXiv:2411.12249arXivDOI
  9. Z. Gao; E. Ullmo; A. Yafaev. Bi-Q‾\overline{\mathbb Q}-structures on Hermitian symmetric spaces and quadratic relations between CM periods. Doc.\ Math., vol. 31, no. 4, pp. 935–981. 2026. arXiv:2401.10488arXivDOI
  10. B. H. Gross. On the periods of abelian integrals and a formula of Chowla and Selberg. Invent.\ Math., vol. 45, pp. 193–211. 1978. with an appendix by D. E. Rohrlich
Show all 37 referencesShow fewer references
  1. A. Huber; G. Wüstholz. Transcendence and Linear Relations of 1-Periods. Cambridge Univ.\ Press, vol. 227. 2022. we cite the numbering of the arXiv version, arXiv:1805.10104arXiv
  2. N. Koblitz; A. Ogus. Algebraicity of some products of values of the Γ\Gamma function. Automorphic Forms, Representations and LL-Functions, vol. 33, part 2, pp. 343–346. 1979. appendix to P. Deligne, Valeurs de fonctions LL et périodes d'intégrales
  3. S. Krisman. \textttfour-exponentials-cm-periods: a Lean 4 certificate. depending on \citeLSG, commit a02679d. https://github.com/telaviv/four-exponentials-cm-periodslink
  4. S. Krisman. \textttwaldschmidt-subgroup-theorem: a Lean formalization of Waldschmidt's subgroup theorem with period term for groups with elliptic factors. commit adcfce9. https://github.com/telaviv/waldschmidt-subgroup-theoremlink
  5. S. Krisman. A Lean formalization of Waldschmidt's subgroup theorem with period term for groups with elliptic factors. 2026. Hexagon preprint 2611.00170v1, https://hexagonmath.org/2611.00170v1hexagon
  6. S. Krisman. Four exponentials at CM and Beta periods. 2026. Hexagon preprint 2611.00171v1, https://hexagonmath.org/2611.00171v1hexagon
  7. M. Laurent. Transcendance de périodes d'intégrales elliptiques II. J. reine angew.\ Math., vol. 333, pp. 144–161. 1982
  8. D. W. Masser. Elliptic Functions and Transcendence. Springer, vol. 437. 1975
  9. B. Mazur; W. Messing. Universal Extensions and One Dimensional Crystalline Cohomology. Springer, vol. 370. 1974
  10. P. Philippon. Lemmes de zéros dans les groupes algébriques commutatifs. Bull.\ Soc.\ Math.\ France, vol. 114, pp. 355–383. 1986. errata et addenda, ibid.\ 115 (1987), 397–398
  11. P. Philippon; M. Waldschmidt. Formes linéaires de logarithmes sur les groupes algébriques commutatifs. Illinois J. Math., vol. 32, pp. 281–314. 1988
  12. D. Roy. Transcendance et questions de répartition dans les groupes algébriques. Approximations diophantiennes et nombres transcendants (Luminy 1990), pp. 249–274. 1992
  13. D. Roy; M. Waldschmidt. Autour du théorème du sous-groupe algébrique. Canad.\ Math.\ Bull., vol. 36, pp. 358–367. 1993
  14. G. Shimura. Abelian Varieties with Complex Multiplication and Modular Functions. Princeton Univ.\ Press, vol. 46. 1998
  15. M. Waldschmidt. Propriétés arithmétiques des valeurs de fonctions méromorphes algébriquement indépendantes. Acta Arith., vol. 23, pp. 19–88. 1973
  16. M. Waldschmidt. Nombres transcendants et groupes algébriques. Soc. Math. France, vol. 69–70. 1979. with appendices by D. Bertrand and J.-P. Serre
  17. M. Waldschmidt. Sous-groupes analytiques de groupes algébriques. Ann.\ of Math., vol. 117, pp. 627–657. 1983
  18. M. Waldschmidt. On the transcendence methods of Gelfond and Schneider in several variables. New Advances in Transcendence Theory (Durham 1986), pp. 375–398. 1988. Ch. 24
  19. M. Waldschmidt. Dependence of logarithms of algebraic points. Number Theory, Vol. II (Budapest 1987), vol. 51, pp. 1013–1035. 1990
  20. M. Waldschmidt. Variations on the six exponentials theorem. Algebra and Number Theory (Hyderabad 2003), pp. 338–355. 2005
  21. M. Waldschmidt. Elliptic functions and transcendence. Surveys in Number Theory, vol. 17, pp. 143–188. 2008
  22. M. Waldschmidt. The role of complex conjugation in transcendental number theory. Diophantine Equations (DION 2005, Mumbai), vol. 20, pp. 297–308. 2008
  23. M. Waldschmidt. On Ramachandra's contributions to transcendental number theory. Hardy-Ramanujan J., vol. 34–35, pp. 18–46. 2013
  24. M. Waldschmidt. The four exponentials problem and Schanuel's conjecture. Mathematics Going Forward, vol. 2313, pp. 579–592. 2023
  25. A. Weil. Sur les périodes des intégrales abéliennes. Comm.\ Pure Appl.\ Math., vol. 29, pp. 813–819. 1976
  26. J. Wolfart; G. Wüstholz. Der Überlagerungsradius gewisser algebraischer Kurven und die Werte der Betafunktion an rationalen Stellen. Math.\ Ann., vol. 273, pp. 1–15. 1985
  27. G. Wüstholz. Algebraische Punkte auf analytischen Untergruppen algebraischer Gruppen. Ann.\ of Math., vol. 129, pp. 501–517. 1989

Version history

  1. v1 · 30 pagesSubmitted by Shawn KrismanInitial depositCurrentNov 10, 2026