no. 8
Number theory
Signature (n − 2,2) CM types and the unitary Colmez conjecture
[Type CM signature (n − 2,2) et la conjecture unitaire de Colmez]

Présenté par : the Editorial Board

Parenti, Solly1
1 Department of Mathematics, University of Wisconsin–Madison, Madison, WI 53706, USA
Comptes Rendus. Mathématique, Tome 356 (2018) no. 8, pp. 833-838.

Colmez a conjecturé une formule établissant une relation entre la hauteur de Faltings d'une variété abélienne à multiplication complexe et une combinaison linéaire particulière de dérivées logarithmiques de fonctions L. Dans cet article, nous restreindrons notre étude aux corps CM unitaires et, par l'étude des fonctions centrales qui se présentent, nous réduirons la conjecture à un cas particulier. En utilisant des actions de Galois, nous démontrerons la conjecture de Colmez pour différents cas.

Colmez conjectured a formula relating the Faltings height of CM abelian varieties to a certain linear combination of logarithmic derivatives of L-functions. In this paper, we study the case of unitary CM fields and by studying the class functions that arise, we reduce the conjecture to a special case. Using the Galois action, we prove more cases of the Colmez Conjecture.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.06.008
Parenti, Solly 1

1 Department of Mathematics, University of Wisconsin–Madison, Madison, WI 53706, USA 
@article{CRMATH_2018__356_8_833_0,
     author = {Parenti, Solly},
     title = {Signature (\protect\emph{n} \ensuremath{-} 2,2) {CM} types and the unitary {Colmez} conjecture},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {833--838},
     publisher = {Elsevier},
     volume = {356},
     number = {8},
     year = {2018},
     doi = {10.1016/j.crma.2018.06.008},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2018.06.008/}
}
TY  - JOUR
AU  - Parenti, Solly
TI  - Signature (n − 2,2) CM types and the unitary Colmez conjecture
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 833
EP  - 838
VL  - 356
IS  - 8
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2018.06.008/
DO  - 10.1016/j.crma.2018.06.008
LA  - en
ID  - CRMATH_2018__356_8_833_0
ER  - 
%0 Journal Article
%A Parenti, Solly
%T Signature (n − 2,2) CM types and the unitary Colmez conjecture
%J Comptes Rendus. Mathématique
%D 2018
%P 833-838
%V 356
%N 8
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2018.06.008/
%R 10.1016/j.crma.2018.06.008
%G en
%F CRMATH_2018__356_8_833_0
Parenti, Solly. Signature (n − 2,2) CM types and the unitary Colmez conjecture. Comptes Rendus. Mathématique, Tome 356 (2018) no. 8, pp. 833-838. doi : 10.1016/j.crma.2018.06.008. http://www.numdam.org/articles/10.1016/j.crma.2018.06.008/

[1] Andreatta, F.; Goren, E.; Howard, B.; Madapusi Pera, K. Faltings heights of abelian varieties with complex multiplication, Ann. of Math. (2), Volume 187 (2018), pp. 391-531

[2] Bray, J.; Holt, D.; Roney-Dougal, C. The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, London Mathematical Society, 2013

[3] Bruinier, J.; Howard, B.; Kudla, S.; Rapoport, M.; Yang, T. Modularity of generating series of divisors on unitary Shimura varieties II: arithmetic applications | arXiv

[4] Colmez, P. Périodes des variétés abéliennes à multiplication complexe, Ann. of Math. (2), Volume 138 (1993), pp. 625-683

[5] Colmez, P. Sur la hauteur de Faltings des variétés abéliennes à multiplication complexe, Compos. Math., Volume 111 (1998), pp. 359-368

[6] Dixon, J.; Mortimer, B. Permutation Groups, Graduate Texts in Mathematics, 1996

[7] Kalyanswamy, S. Inverse Galois Problem for Totally Real Number Fields, Cornell University, Mathematics Department, 2012 http://pi.math.cornell.edu/files/Research/SeniorTheses/kalyanswamyThesis.pdf (Senior Thesis, Accessed 2018-6)

[8] The LMFDB Collaboration The L-functions and modular forms database, 2017 http://www.lmfdb.org (Accessed 2017-12)

[9] Tsimerman, J. The André–Oort conjecture for Ag, Ann. of Math. (2), Volume 187 (2018) no. 2, pp. 379-390

[10] Yang, T.; Yin, H. CM fields of dihedral type and the Colmez conjecture, Manuscr. Math., Volume 156 (2018) no. 1–2, pp. 1-22

[11] Yuan, X.; Zhang, S. On the averaged Colmez conjecture, Ann. of Math. (2), Volume 187 (2018), pp. 533-638

Cité par Sources :