HEXAGONbeta

Four exponentials at CM and Beta periods

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

Contributed by Shawn Krisman ↗

Version 1 / Nov 07, 2026 / CC BY 4.0

Abstract

For every nonzero eigen-period PP of a CM abelian variety over Q‾\overline{\mathbb{Q}}, we show that the four exponentials conjecture and its strong form hold at t∈Q‾×P±1t\in\overline{\mathbb{Q}}^\times P^{\pm1} and t∈Q‾×(P/π)±1t\in\overline{\mathbb{Q}}^\times(P/\pi)^{\pm1}; these include Beta values such as t=Γ(1/5)2/Γ(2/5)t=\Gamma(1/5)^2/\Gamma(2/5). For an elliptic curve EE over Q‾\overline{\mathbb{Q}} with complex multiplication, we also show that eπ2e^{\pi^2} and ℘E(1/π)\wp_E(1/\pi) are not both algebraic. Further results concern two CM elliptic curves: for example, π,ϖ,Ωρ,ϖΩρ\pi,\varpi,\Omega_\rho,\varpi\Omega_\rho are linearly independent over Q‾\overline{\mathbb{Q}}, where ϖ\varpi and Ωρ\Omega_\rho are periods of y2=4x3−4xy^2=4x^3-4x and y2=4x3−4y^2=4x^3-4. These are explicit consequences of published theorems of Waldschmidt (1988) and Roy–Waldschmidt (1993), together with classical results of Baker, Schneider, Masser and Wüstholz; the contribution is the statements and the proofs written out at these periods. Cases at specific periods, such as that 2ϖ2^{\varpi} and 3ϖ3^{\varpi} are not both algebraic, are checked in Lean 4 and reported separately.

References

  1. A. Baker; G. Wüstholz. Logarithmic Forms and Diophantine Geometry. Cambridge Univ.\ Press, vol. 9. 2007
  2. D. Bertrand. Valeurs de fonctions thêta et hauteurs pp-adiques. Progr.\ Math., vol. 22, pp. 1–11. 1982
  3. D. Bertrand. Endomorphismes de groupes algébriques; applications arithmétiques. Approximations diophantiennes et nombres transcendants (Luminy 1982), vol. 31, pp. 1–45. 1983
  4. D. Bertrand; M. Laurent. Propriétés de transcendance de nombres liés aux fonctions thêta. C. R. Acad.\ Sci.\ Paris Sér.\ I, vol. 292, pp. 747–749. 1981
  5. G. V. Chudnovsky. Contributions to the Theory of Transcendental Numbers. Amer.\ Math.\ Soc., vol. 19. 1984
  6. 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
  7. G. Diaz. Produits et quotients de combinaisons linéaires de logarithmes de nombres algébriques: conjectures et résultats partiels. J. Théor.\ Nombres Bordeaux, vol. 19, pp. 373–391. 2007
  8. 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
  9. S. Krisman. \textttfour-exponentials-cm-periods: a Lean 4 certificate for the statements of this note. depending on \citeLSG, commit a02679d. https://github.com/telaviv/four-exponentials-cm-periodslink
  10. S. Krisman. The Roy–Waldschmidt theorem with period term and inner weight: a written proof from Waldschmidt's subgroup theorem. in preparation
Show all 34 referencesShow fewer references
  1. 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
  2. S. Krisman. A Lean formalization of Waldschmidt's subgroup theorem with period term for groups with elliptic factors. 2026. preprint
  3. M. Laurent. Indépendance linéaire de valeurs de fonctions doublement quasi-périodiques. C. R. Acad.\ Sci.\ Paris Sér. A, vol. 290, pp. 397–399. 1980
  4. M. Laurent. Transcendance de périodes d'intégrales elliptiques II. J. reine angew.\ Math., vol. 333, pp. 144–161. 1982
  5. D. W. Masser. Elliptic Functions and Transcendence. Springer, vol. 437. 1975
  6. B. Mazur; W. Messing. Universal Extensions and One Dimensional Crystalline Cohomology. Springer, vol. 370. 1974
  7. P. Philippon. Lemmes de zéros dans les groupes algébriques commutatifs. Bull.\ Soc.\ Math.\ France, vol. 114, pp. 355–383. 1986. errata, ibid.\ 115 (1987), 397–398
  8. D. Roy. Matrices whose coefficients are linear forms in logarithms. J. Number Theory, vol. 41, pp. 22–47. 1992
  9. D. Roy. Transcendance et questions de répartition dans les groupes algébriques. Approximations diophantiennes et nombres transcendants (Luminy 1990), pp. 249–274. 1992
  10. D. Roy; M. Waldschmidt. Autour du théorème du sous-groupe algébrique. Canad.\ Math.\ Bull., vol. 36, pp. 358–367. 1993
  11. Th. Schneider. Arithmetische Untersuchungen elliptischer Integrale. Math.\ Ann., vol. 113, pp. 1–13. 1937
  12. G. Shimura. Abelian Varieties with Complex Multiplication and Modular Functions. Princeton Univ.\ Press, vol. 46. 1998
  13. M. Waldschmidt. Nombres transcendants et groupes algébriques. vol. 69–70. 1979. with appendices by D. Bertrand and J.-P. Serre
  14. M. Waldschmidt. Sous-groupes analytiques de groupes algébriques. Ann.\ of Math., vol. 117, pp. 627–657. 1983
  15. M. Waldschmidt. On the transcendence methods of Gel'fond and Schneider in several variables. New Advances in Transcendence Theory (Durham 1986), pp. 375–398. 1988. Ch. 24
  16. M. Waldschmidt. Dependence of logarithms of algebraic points. Number Theory, Vol. II (Budapest 1987), vol. 51, pp. 1013–1035. 1990
  17. M. Waldschmidt. Sur la nature arithmétique des valeurs de fonctions modulaires. Astérisque, vol. 245, pp. 105–140. 1997. Séminaire Bourbaki, exposé 824
  18. M. Waldschmidt. Dependence of logarithms on commutative algebraic groups. 2003. slides, International Conference on Algebra and Number Theory, Hyderabad
  19. M. Waldschmidt. Variations on the six exponentials theorem. Algebra and Number Theory (Hyderabad 2003), pp. 338–355. 2005
  20. M. Waldschmidt. Elliptic functions and transcendence. Surveys in Number Theory, vol. 17, pp. 143–188. 2008
  21. M. Waldschmidt. The role of complex conjugation in transcendental number theory. Diophantine Equations (DION 2005, Mumbai), vol. 20, pp. 297–308. 2008
  22. M. Waldschmidt. The four exponentials problem and Schanuel's conjecture. Mathematics Going Forward, vol. 2313, pp. 579–592. 2023
  23. A. Weil. Sur les périodes des intégrales abéliennes. Comm.\ Pure Appl.\ Math., vol. 29, pp. 813–819. 1976
  24. G. Wüstholz. Algebraische Punkte auf analytischen Untergruppen algebraischer Gruppen. Ann.\ of Math., vol. 129, pp. 501–517. 1989

Version history

  1. v1 · 26 pagesSubmitted by Shawn KrismanInitial depositCurrentNov 07, 2026