Feuilleter
Strong minimality of triangle functions
Guy Casale1  ; Matthew DeVilbiss2  ; James Freitag3  ; Joel Nagloo3
1Univ Rennes, CNRS, IRMAR-UMR 6625, F-35000 Rennes, France
2Saint Xavier University, Department of Mathematics, 3700 W 103rd St, Chicago, IL 60655, USA
3University of Illinois Chicago, Department of Mathematics, Statistics, and Computer Science, 851 S. Morgan Street, Chicago, IL 60607-7045, USA
Publications mathématiques de Besançon. Algèbre et théorie des nombres (2026), pp. 53-59

In this manuscript, we give a new proof of strong minimality of certain automorphic functions, originally results of Freitag and Scanlon (2017), Casale, Freitag and Nagloo (2020), Blázquez-Sanz, Casale, Freitag and Nagloo (2020). Our proof is shorter and conceptually different than those presently in the literature.

Dans ce manuscrit, nous présentons une nouvelle preuve de la forte minimalité de certaines fonctions automorphes, résultats établis à l’origine par Freitag et Scanlon (2017), Casale, Freitag et Nagloo (2020), ainsi que Blázquez-Sanz, Casale, Freitag et Nagloo (2020). Notre preuve est plus courte et conceptuellement différente de celles qui figurent actuellement dans la littérature.

Publié le :
DOI : 10.5802/pmb.69
Classification : 11F03, 12H05, 03C60

Guy Casale  1   ; Matthew DeVilbiss  2   ; James Freitag  3   ; Joel Nagloo  3

1 Univ Rennes, CNRS, IRMAR-UMR 6625, F-35000 Rennes, France
2 Saint Xavier University, Department of Mathematics, 3700 W 103rd St, Chicago, IL 60655, USA
3 University of Illinois Chicago, Department of Mathematics, Statistics, and Computer Science, 851 S. Morgan Street, Chicago, IL 60607-7045, USA
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Guy Casale; Matthew DeVilbiss; James Freitag; Joel Nagloo. Strong minimality of triangle functions. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2026), pp. 53-59. doi: 10.5802/pmb.69
@article{PMB_2026____53_0,
     author = {Guy Casale and Matthew DeVilbiss and James Freitag and Joel Nagloo},
     title = {Strong minimality of triangle functions},
     journal = {Publications math\'ematiques de Besan\c{c}on. Alg\`ebre et th\'eorie des nombres},
     pages = {53--59},
     year = {2026},
     publisher = {Presses universitaires de Franche-Comt\'e},
     doi = {10.5802/pmb.69},
     language = {en},
     url = {https://pmb.centre-mersenne.org/articles/10.5802/pmb.69/}
}
TY  - JOUR
AU  - Guy Casale
AU  - Matthew DeVilbiss
AU  - James Freitag
AU  - Joel Nagloo
TI  - Strong minimality of triangle functions
JO  - Publications mathématiques de Besançon. Algèbre et théorie des nombres
PY  - 2026
SP  - 53
EP  - 59
PB  - Presses universitaires de Franche-Comté
UR  - https://pmb.centre-mersenne.org/articles/10.5802/pmb.69/
DO  - 10.5802/pmb.69
LA  - en
ID  - PMB_2026____53_0
ER  - 
%0 Journal Article
%A Guy Casale
%A Matthew DeVilbiss
%A James Freitag
%A Joel Nagloo
%T Strong minimality of triangle functions
%J Publications mathématiques de Besançon. Algèbre et théorie des nombres
%D 2026
%P 53-59
%I Presses universitaires de Franche-Comté
%U https://pmb.centre-mersenne.org/articles/10.5802/pmb.69/
%R 10.5802/pmb.69
%G en
%F PMB_2026____53_0

[1] Vahagn Aslanyan Ax–Schanuel and strong minimality for the j-function, Ann. Pure Appl. Logic, Volume 172 (2021) no. 1, 102871, 24 pages | Zbl | MR | DOI

[2] David Blázquez-Sanz; Guy Casale; James Freitag; Joel Nagloo Some functional transcendence results around the Schwarzian differential equation, Ann. Fac. Sci. Toulouse, Math. (6), Volume 29 (2020) no. 5, pp. 1265-1300 | Zbl | Numdam | DOI | MR

[3] Guy Casale; James Freitag; Joel Nagloo Ax–Lindemann–Weierstrass with derivatives and the genus 0 Fuchsian groups, Ann. Math. (2), Volume 192 (2020) no. 3, pp. 721-765 | Zbl | MR | DOI

[4] James Freitag Not Pfaffian (2021) | arXiv | Zbl

[5] James Freitag; Rémi Jaoui; David Marker; Joel Nagloo On the equations of Poizat and Liénard, Int. Math. Res. Not., Volume 2023 (2023) no. 19, pp. 16478-16539 | Zbl | DOI | MR

[6] James Freitag; Rahim Moosa Finiteness theorems on hypersurfaces in partial differential-algebraic geometry, Adv. Math., Volume 314 (2017), pp. 726-755 | Zbl | DOI | MR

[7] James Freitag; Rahim Moosa Bounding nonminimality and a conjecture of Borovik–Cherlin, J. Eur. Math. Soc., Volume 27 (2025) no. 2, pp. 589-613 | Zbl | DOI | MR

[8] James Freitag; Thomas Scanlon Strong minimality and the $j$-function, J. Eur. Math. Soc., Volume 20 (2017) no. 1, pp. 119-136 | Zbl | DOI | MR

[9] Ehud Hrushovski The Mordell–Lang conjecture for function fields, J. Am. Math. Soc., Volume 9 (1996) no. 3, pp. 667-690 | Zbl | DOI | MR

[10] Ehud Hrushovski; Anand Pillay Effective bounds for the number of transcendental points on subvarieties of semi-abelian varieties, Am. J. Math., Volume 122 (2000) no. 3, pp. 439-450 | Zbl | DOI | MR

[11] Tosihusa Kimura On Riemann’s equations which are solvable by quadratures, Funkc. Ekvacioj, Ser. Int., Volume 12 (1969), pp. 269-281 | Zbl | MR

[12] Jerald J. Kovacic An algorithm for solving second order linear homogeneous differential equations, J. Symb. Comput., Volume 2 (1986) no. 1, pp. 3-43 | DOI | Zbl | MR

[13] Kurt Mahler On algebraic differential equations satisfied by automorphic functions, J. Aust. Math. Soc., Volume 10 (1969) no. 3-4, pp. 445-450 | Zbl | DOI | MR

[14] Rahim Moosa; Anand Pillay Some model theory of fibrations and algebraic reductions, Sel. Math., New Ser., Volume 20 (2014) no. 4, pp. 1067-1082 | Zbl | DOI | MR

[15] Rahim Moosa; Thomas Scanlon Model theory of fields with free operators in characteristic zero, J. Math. Log., Volume 14 (2014) no. 2, 1450009, 43 pages | Zbl | MR | DOI

[16] Keiji Nishioka A conjecture of Mahler on automorphic functions, Arch. Math., Volume 53 (1989) no. 1, pp. 46-51 | DOI | Zbl

[17] Keiji Nishioka Painlevé’s theorem on automorphic functions, Manuscr. Math., Volume 66 (1990) no. 4, pp. 341-349 | Zbl | DOI | MR

[18] Keiji Nishioka Painlevé’s Theorem on Automorphic Functions II, Funkc. Ekvacioj, Ser. Int., Volume 35 (1992), pp. 597-602 | Zbl | MR

[19] Paul Painlevé Leçons sur la théorie analytique des équations différentielles (Leçons de Stockholm, 1895) (Hermann, Paris, 1897), Œuvres de Paul Painlevé. Tome I, Éditions du Centre National de la Recherche Scientifique, 1973

Cité par Sources :