Séminaires : Séminaire Théorie des Nombres

Equipe(s) : tn,
Responsables :Ziyang Gao, Marc Hindry, Bruno Kahn, João Pedro P. dos Santos
Email des responsables : ziyang.gao@imj-prg.fr
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http://www.imj-prg.fr/tn/STN/stnj.html

 


Orateur(s) Noriko Yui - ,
Titre The modularity of Calabi--Yau threefolds over Q
Date26/03/2007
Horaire15:30 à 17:30
Diffusion
RésumeWe will discuss the modularity of Galois representations associated toCalabi--Yau threefolds over $\mathbf Q$. We can show that any rigid Calabi--Yau threefold over $\mathbf Q$ is modulari.e., its two-dimensional Galois representation comes from a modular form of weight $4$ on some some congruence subgroup of $PSL_2(\mathbf Z)$. However, when a Calabi--Yauthreefold is non-rigid, the dimension of the Galois representation getsrather large, and the modularity question poses a serious challenge.We will construct explicit examples of non-rigidCalabi--Yau threefolds fibered over ${\bf P}^1$ by non-constantsemi-stable $K3$ surfaces and reaching the Arakelov--Yauupper bound. For these examples, we prove that the ``interesting'' partof their $L$-series do come from modular forms.
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