Automorphy of Calabi-Yau threefolds of Borcea-Voisin type over ℚ

Yasuhiro Goto, Ron Livné, Noriko Yui

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Abstract

We consider certain Calabi-Yau threefolds of Borcea-Voisin type defined over ℚ.We will discuss the automorphy of the Galois representations associated to these Calabi-Yau threefolds. We construct such Calabi-Yau threefolds as the quotients of products of K3 surfaces S and elliptic curves by a specific involution. We choose K3 surfaces S over ℚ with non-symplectic involution σ acting by -1 on H2,0(S). We fish out K3 surfaces with the involution σ from the famous 95 families of K3 surfaces in the list of Reid [32], and of Yonemura [43], where Yonemura described hypersurfaces defining these K3 surfaces in weighted projective 3-spaces. Our first result is that for all but few (in fact, nine) of the 95 families of K3 surfaces S over ℚ in Reid-Yonemura's list, there are subsets of equations defining quasi-smooth hypersurfaces which are of Delsarte or Fermat type and endowed with non-symplectic involution σ. One implication of this result is that with this choice of defining equation, (S, σ) becomes of CM type. Let E be an elliptic curve over ℚ with the standard involution ι, and let X be a standard (crepant) resolution, defined over ℚ, of the quotient threefold E × S/ι × σ, where (S, σ) is one of the above K3 surfaces over ℚ of CM type. One of our main results is the automorphy of the L-series of X. The moduli spaces of these Calabi-Yau threefolds are Shimura varieties. Our result shows the existence of a CM point in the moduli space. We also consider the L-series of mirror pairs of Calabi-Yau threefolds of Borcea-Voisin type, and study how L-series behave under mirror symmetry.

Original languageAmerican English
Pages (from-to)581-670
Number of pages90
JournalCommunications in Number Theory and Physics
Volume7
Issue number4
DOIs
StatePublished - 2013

Bibliographical note

Funding Information:
ACKNOWLEDGEMENTS This Project was funded by King Abdulaziz City for Science and Technology (KACST) through National Science, Technology and Innovation Plan (NSTIP) under grant number 8-ENE198-3. The authors, therefore, acknowledge with thanks KACST for support for Scientific Research. Also, the authors are thankful to the Deanship of Scientific Research (DSR), King Abdulaziz University for their technical support.

Funding Information:
This Project was funded by King Abdulaziz City for Science and Technology (KACST) through National Science, Technology and Innovation Plan (NSTIP) under grant number 8-ENE198-3. The authors, therefore, acknowledge with thanks KACST for support for Scientific Research. Also, the authors are thankful to the Deanship of Scientific Research (DSR), King Abdulaziz University for their technical support.

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