Modularity of rigid Calabi-Yau threefolds over ℚ

Abstract

We prove modularity for a huge class of rigid Calabi-Yau threefolds over Q. In particular we prove that every rigid Calabi-Yau threefold with good reduction at 3 and 7 is modular.

On the modularity of supersingular elliptic curves over certain totally real number fields

“…This theorem has been used by Dieulefait and the second author [7] to give a new criterion for the modularity of rigid Calabi-Yau threefolds. Of course, it is largely subsumed within recent work of Khare and Wintenberger; however, we need no hypothesis at 2.…”

“…The elliptic curve y 2 = x 3 + x has j -invariant 1728 and so has supersingular reduction. Taking a cyclic degree 3 isogeny of E if necessary, we can assume that X(ω 2 ⊕ ω 7 2 )(Q nr 7 ) contains an elliptic curve E having good supersingular reduction and with j -invariant 1728. Let us denote this point by P .…”

“…• L is defined over Q nr 7 , • L passes through four distinct points of X(ω 2 ⊕ ω 7 2 ) whose j -invariants are 1728.…”

Modularity of Calabi-Yau varieties and conformal field theory

“…Resolving singularities gives a smooth Calabi-Yau threefold X which is rigid. Schoen proved [5] showed the following Theorem 2.9. Let X be a rigid Calabi-Yau threefold over Q.…”

“…Then by the FontaineMazur conjecture [3] , also by Yui (see e.g., [4]), the L-series of the middle cohomology of a rigid Calabi-Yau threefold defined over Q is supposed to coincide with that of some cuspidal modular form of weight 4. Dieulefat and Manoharmayum [5] proved the modularity conjecture for rigid Calabi-Yau threefolds under certain restrictions on the primes of bad reduction. There exist examples of modular non-rigid Calabi-Yau threefolds as well, which can be found e.g., in [6,7].…”