Finite Orbits in Surfaces with a Double Elliptic Fibration and torsion values of sections
[Orbites finies sur les surfaces à double fibration elliptique et valeurs de torsion de sections]
Corvaja, Pietro1 ; Tsimerman, Jacob2 ; Zannier, Umberto3
1Dipartimento di Scienze Matematiche, e Informatiche e Fisiche, Università di Udine, Via delle Scienze, 206, 33100 Udine (Italy)
2Department of Mathematics, University of Toronto (Canada)
3Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa (Italy)
Annales de l'Institut Fourier, Online first, 38 p.

We consider surfaces with two elliptic fibrations, each of them provided with a section. We study the orbits under the induced translation automorphisms proving that, under natural conditions, the finite orbits are confined to a curve. This goes in a similar direction of (and is motivated by) recent work by Cantat–Dujardin, although we use very different methods and obtain related but different results.

As a sample of application of similar arguments, we prove a new case of the Zilber–Pink conjecture, namely Theorem 1.5, for certain schemes over a 2-dimensional base, which was known to lead to substantial difficulties.

Most results rely, among other things, on recent theorems by Bakker and the second author of “Ax–Schanuel type”; we also relate a functional condition with a theorem of Shioda on unramified sections of the Legendre scheme. For one of our proofs, we also use recent height inequalities by Dimitrov–Gao–Habegger (or those by Yuan–Zhang).

Finally, in an appendix, we show that the Relative Manin–Mumford Conjecture over the complex number field is equivalent to its version over the field of algebraic numbers.

Nous considérons des surfaces munies de deux fibrations elliptiques, chacune avec une section. Les deux sections induisant des translations sur les fibres, nous nous intéressons aux orbites par cette action. Nous démontrons que, en dehors de cas exceptionnels classifiés, les orbites finies sont contenues dans une courbe algébrique. Alors que notre résultat s’insère dans la même veine d’un résultat semblable de Cantat–Dujardin, qui constitue la motivation principale de notre travail, les métodes que nous employons sont différentes et l’énoncé de finitude obtenu est aussi différent.

Notre approche mène aussi à la solution d’un cas de la conjecture de Zilber–Pink pour certains schémas abéliens sur une base de dimension deux, surmontant des difficultées bien connues.

Un outil essentiel pour la plus part de nos preuves consiste en un théorème récent de Bakker–Tsimerman de type «  Ax–Schanuel » ; nous utilisons aussi un théorème de Shioda sur les sections non-ramifiées du schéma de Legendre et une minoration pour la hauteur due à Dimitrov–Gao–Habegger et Yuan–Zhang.

Enfin, dans un appendice nous montrons que la conjecture de Manin–Mumford relative dans le cas complexe est équivalente à sa version dans le cas du corps des nombres algébriques.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3783
Classification : 11J, 14K15, 14G05
Keywords: elliptic schemes, Zilber–Pink conjecture
Mots-clés : schémas elliptiques, conjecture de Zilber Pink

Corvaja, Pietro  1   ; Tsimerman, Jacob  2   ; Zannier, Umberto  3

1 Dipartimento di Scienze Matematiche, e Informatiche e Fisiche, Università di Udine, Via delle Scienze, 206, 33100 Udine (Italy)
2 Department of Mathematics, University of Toronto (Canada)
3 Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa (Italy) 
@unpublished{AIF_0__0_0_A72_0,
     author = {Corvaja, Pietro and Tsimerman, Jacob and Zannier, Umberto},
     title = {Finite {Orbits} in {Surfaces} with a {Double} {Elliptic} {Fibration} and torsion values of sections},
     journal = {Annales de l'Institut Fourier},
     year = {2026},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     doi = {10.5802/aif.3783},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Corvaja, Pietro
AU  - Tsimerman, Jacob
AU  - Zannier, Umberto
TI  - Finite Orbits in Surfaces with a Double Elliptic Fibration and torsion values of sections
JO  - Annales de l'Institut Fourier
PY  - 2026
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3783
LA  - en
ID  - AIF_0__0_0_A72_0
ER  - 
%0 Unpublished Work
%A Corvaja, Pietro
%A Tsimerman, Jacob
%A Zannier, Umberto
%T Finite Orbits in Surfaces with a Double Elliptic Fibration and torsion values of sections
%J Annales de l'Institut Fourier
%D 2026
%V 0
%N 0
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3783
%G en
%F AIF_0__0_0_A72_0
Corvaja, Pietro; Tsimerman, Jacob; Zannier, Umberto. Finite Orbits in Surfaces with a Double Elliptic Fibration and torsion values of sections. Annales de l'Institut Fourier, Online first, 38 p.

[1] André, Yves Mumford–Tate groups of mixed Hodge structures and the theorem of the fixed part, Compos. Math., Volume 82 (1992) no. 1, pp. 1-24 | Numdam | MR | Zbl

[2] Bakker, Ben; Tsimerman, Jacob Functional Transcendence of Period Integrals (2022) | arXiv | Zbl

[3] Bertrand, Daniel Extensions de D-modules et groupes de Galois différentiels, p-adic analysis (Trento, 1989) Proceedings of the International Conference held in Trento, May 29-June 2, 1989 (Baldassarri, Francesco; Bosch, Siegfried; Dwork, Bernard, eds.) (Lecture Notes in Mathematics), Volume 1454, Springer (1989), pp. 125-141 | MR | DOI | Zbl

[4] Bierstone, Edward; Milman, Pierre D. Semianalytic and Subanalytic Sets, Publ. Math., Inst. Hautes Étud. Sci., Volume 67 (1998), pp. 5-42 | DOI | MR | Zbl

[5] Blázquez-Sanz, David; Casale, Guy; Freitag, James; Nagloo, Joel A differential approach to the Ax-Schanuel, I (2023) | arXiv

[6] Bombieri, Enrico; Gubler, Walter Heights in Diophantine Geometry, New Mathematical Monographs, 4, Cambridge University Press, 2006 | MR | Zbl

[7] Cantat, Serge Sur la dynamique du groupe d’automorphismes des surfaces K3, Transform. Groups, Volume 6 (2001) no. 3, pp. 201-214 | MR | DOI | Zbl

[8] Cantat, Serge; Dujardin, Romain Finite Orbits for Large Groups of Automorphisms of Projective Surfaces, Compos. Math., Volume 160 (2023) no. 1, pp. 120-175 | MR | DOI | Zbl

[9] Corvaja, Pietro; Masser, David W.; Demeio, Julian; Zannier, Umberto M. On the torsion values of sections of an abelian scheme, J. Reine Angew. Math., Volume 782 (2022), pp. 1-41 | MR | DOI | Zbl

[10] Corvaja, Pietro; Masser, David W.; Zannier, Umberto M. Torsion hypersurfaces on abelian schemes and Betti coordinates, Math. Ann., Volume 371 (2018) no. 3-4, pp. 1013-1045 | MR | DOI | Zbl

[11] Corvaja, Pietro; Zannier, Umberto M. On the Hilbert Property and fundamental group of algebraic varieties, Math. Z., Volume 286 (2017) no. 1-2, pp. 579-602 | MR | DOI | Zbl

[12] Corvaja, Pietro; Zannier, Umberto M. Unramified sections of the Legendre scheme and modular forms, J. Geom. Phys., Volume 166 (2021), 104266, 26 pages | MR | DOI | Zbl

[13] David, Sinnou Fonctions thêta et points de torsion des variétés abéliennes, Compos. Math., Volume 78 (1991) no. 2, pp. 121-160 | Numdam | MR | Zbl

[14] Dimitrov, Vesselin; Gao, Ziyang; Habegger, Philipp Uniformity in Mordell-Lang for curves, Ann. Math. (2), Volume 194 (2021) no. 1, pp. 237-298 | MR | DOI | Zbl

[15] Elkies, Noam D. On A 4 +B 4 +C 4 =D 4 , Math. Comput., Volume 51 (1988) no. 4, pp. 825-835 | MR | DOI | Zbl

[16] Fuchs, Elena; Litman, Matthew; Silverman, Joseph H.; Tran, Austin Orbits on K3 Surfaces of Markoff Type, Exp. Math., Volume 33 (2024) no. 4, pp. 663-700 | Zbl | MR | DOI

[17] Gao, Ziyang A special point problem of André–Pink–Zannier in the universal family of Abelian varieties, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume 17 (2017) no. 1, pp. 231-266 | DOI | MR | Zbl

[18] Gao, Ziyang Mixed Ax-Schanuel for the universal abelian varieties and some applications, Compos. Math., Volume 156 (2020) no. 11, pp. 2263-2297 | MR | DOI | Zbl

[19] Gao, Ziyang; Habegger, Philipp The relative Manin-Mumford Conjecture (2023) | arXiv | Zbl

[20] Habegger, Philipp Torsion points on elliptic curves in Weierstrass form, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume 12 (2013) no. 3, pp. 687-715 | Numdam | MR | DOI | Zbl

[21] Husemöller, Dale H. Elliptic Curves, Graduate Texts in Mathematics, 111, Springer, 2003 | MR | DOI | Zbl

[22] Manin, Yuriĭ I. Rational points on Curves over function fields, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 27 (1963) no. 6, pp. 1395-1440 | MR | Zbl

[23] Masser, David W.; Zannier, Umberto M. Torsion anomalous points and familes of elliptic curves, Am. J. Math., Volume 132 (2010), pp. 1677-1691 | DOI | MR | Zbl

[24] Masser, David W.; Zannier, Umberto M. Torsion points on familes of products of elliptic curves, Adv. Math., Volume 259 (2014), pp. 116-133 | MR | DOI | Zbl

[25] Pila, Jonathan O-minimality and the André–Oort conjecture on ℂ n , Ann. Math. (2), Volume 173 (2011) no. 3, pp. 1779-1840 | DOI | Zbl

[26] Pila, Jonathan Point counting and the Zilber–Pink conjecture, Cambridge Tracts in Mathematics, 228, Cambridge University Press, 2022 | MR | DOI | Zbl

[27] Pila, Jonathan; Wilkie, Alex J. The rational points of a definable set, Duke Math. J., Volume 133 (2005) no. 3, pp. 591-616 | MR | DOI | Zbl

[28] Swinnerton-Dyer, Peter A 4 +B 4 =C 4 +D 4 revisited, J. Lond. Math. Soc., Volume 43 (1968), pp. 149-161 | MR | DOI | Zbl

[29] Yuan, Xinyi; Zhang, Shou-Wu Adelic line bundles over quasi projective varieties (2022) | arXiv | Zbl

[30] Zannier, Umberto M. Some Problems of Unlikely Intersections in Arithmetic and Geometry, Annals of Mathematics Studies, 181, Princeton University Press, 2012 (with appendixes by David Masser) | MR | DOI | Zbl

Cité par Sources :