An infinitesimal p-adic multiplicative Manin–Mumford Conjecture
Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 2, pp. 393-408.

Nos résultats concernent certaines fonctions analytiques sur la boule ouverte unité dans p n centrée en 1. Nous montrons que celles-ci soit ne s’annulent en (ζ 1 -1,...,ζ n -1) que pour un nombre fini de racines de l’unité, soit s’annulent sur tout un translaté du groupe formel multiplicatif. Pour les fonctions polynômiales, cela suit de la conjecture de Manin–Mumford multiplicative. Or nous considérons un ensemble de fonctions bien plus vaste et en particulier déduisons un résultat de rigidité pour les tores formels. De plus, nous étendons ces résultats au-delà du groupe multiplicatif aux groupes formels de type Lubin–Tate.

Our results concern certain analytic functions on the open unit poly-disc in p n centered at the multiplicative unit and we show such functions only vanish at finitely many n-tuples of roots of unity (ζ 1 -1,...,ζ n -1) unless they vanish along a translate of the formal multiplicative group. For polynomial functions, this follows from the multiplicative Manin–Mumford conjecture. However we allow for a much wider class of analytic functions; in particular we establish a rigidity result for formal tori. Moreover, our methods apply to Lubin–Tate formal groups beyond just formal 𝔾 m and we extend the results to this setting.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1030
Classification : 11S31, 13H05, 13F25, 14L05
Mots clés : Manin–Mumford, $p$-adic rigidity
Vlad Serban 1

1 Fields Institute, 222 College Street, Toronto, Ontario M5T3J1, Canada
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{JTNB_2018__30_2_393_0,
     author = {Vlad Serban},
     title = {An infinitesimal $p$-adic multiplicative {Manin{\textendash}Mumford} {Conjecture}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {393--408},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {2},
     year = {2018},
     doi = {10.5802/jtnb.1030},
     zbl = {1443.11247},
     mrnumber = {3891318},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1030/}
}
TY  - JOUR
AU  - Vlad Serban
TI  - An infinitesimal $p$-adic multiplicative Manin–Mumford Conjecture
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2018
SP  - 393
EP  - 408
VL  - 30
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1030/
DO  - 10.5802/jtnb.1030
LA  - en
ID  - JTNB_2018__30_2_393_0
ER  - 
%0 Journal Article
%A Vlad Serban
%T An infinitesimal $p$-adic multiplicative Manin–Mumford Conjecture
%J Journal de théorie des nombres de Bordeaux
%D 2018
%P 393-408
%V 30
%N 2
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1030/
%R 10.5802/jtnb.1030
%G en
%F JTNB_2018__30_2_393_0
Vlad Serban. An infinitesimal $p$-adic multiplicative Manin–Mumford Conjecture. Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 2, pp. 393-408. doi : 10.5802/jtnb.1030. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1030/

[1] Ching-Li Chai Hecke orbits on Siegel modular varieties, Geometric methods in algebra and number theory (Progress in Mathematics), Volume 235, Birkhäuser, 2005, pp. 71-107 | DOI | MR | Zbl

[2] Ching-Li Chai A rigidity result for p-divisible formal groups, Asian J. Math., Volume 12 (2008) no. 2, pp. 193-202 | DOI | MR | Zbl

[3] Haruzo Hida The Iwasawa μ-invariant of p-adic Hecke L-functions, Ann. Math., Volume 172 (2010) no. 1, pp. 41-137 | DOI | MR | Zbl

[4] Haruzo Hida Constancy of adjoint -invariant, J. Number Theory, Volume 131 (2011) no. 7, pp. 1331-1346 | DOI | MR | Zbl

[5] Haruzo Hida Hecke fields of Hilbert modular analytic families, Automorphic forms and related geometry: assessing the legacy of I. I. Piatetski-Shapiro (Contemporary Mathematics), Volume 614, American Mathematical Society, 2014, pp. 97-137 | DOI | MR | Zbl

[6] Serge Lang Integral points on curves, Publ. Math., Inst. Hautes Étud. Sci., Volume 6 (1960), pp. 319-335 | Numdam

[7] Serge Lang Division points on curves, Ann. Mat. Pura Appl., Volume 70 (1965) no. 1, pp. 229-234 | DOI | MR | Zbl

[8] Jonathan Lubin; John Tate Formal complex multiplication in local fields, Ann. Math., Volume 81 (1965) no. 2, pp. 380-387 | DOI | MR | Zbl

[9] Paul Monsky On p-adic power series, Math. Ann., Volume 255 (1981) no. 2, pp. 217-227 | DOI | MR | Zbl

[10] Ana Raissa Berardo Neira Power series in p-adic roots of unity, University of Texas (USA) (2002) (Ph. D. Thesis) | MR

[11] Michel Raynaud Courbes sur une variété abélienne et points de torsion, Invent. Math., Volume 71 (1983) no. 1, pp. 207-233 | DOI | Zbl

[12] Thomas Scanlon p-adic distance from torsion points of semi-abelian varieties, J. Reine Angew. Math., Volume 499 (1998), pp. 225-236 | DOI | MR | Zbl

[13] John Tate; José Felipe Voloch Linear forms in p-adic roots of unity, Int. Math. Res. Not., Volume 1996 (1996) no. 12, pp. 589-601 | DOI | MR | Zbl

[14] Emmanuel Ullmo Positivité et discrétion des points algébriques des courbes, Ann. Math., Volume 147 (1998) no. 1, pp. 167-179 | DOI | MR | Zbl

[15] Emmanuel Ullmo Manin-Mumford, André-Oort, the equidistribution point of view, Equidistribution in number theory, an introduction (NATO Science Series II: Mathematics, Physics and Chemistry), Volume 237, Springer, 2007, pp. 103-138 | DOI | MR | Zbl

[16] Shouwu Zhang Positive line bundles on arithmetic varieties, J. Am. Math. Soc., Volume 8 (1995) no. 1, pp. 87-221 | MR | Zbl

[17] Shouwu Zhang Equidistribution of small points on abelian varieties, Ann. Math., Volume 147 (1998) no. 1, pp. 159-165 | DOI | MR | Zbl

Cité par Sources :