Number theory
Signature (n − 2,2) CM types and the unitary Colmez conjecture
Comptes Rendus. Mathématique, Volume 356 (2018) no. 8, pp. 833-838.

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.

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.

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DOI: 10.1016/j.crma.2018.06.008
Parenti, Solly 1

1 Department of Mathematics, University of Wisconsin–Madison, Madison, WI 53706, USA
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Parenti, Solly. Signature (n − 2,2) CM types and the unitary Colmez conjecture. Comptes Rendus. Mathématique, Volume 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/

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