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The modularity conjecture for rigid Calabi-Yau threefolds over $\mathbf{Q}$
We formulate the modularity conjecture for rigid Calabi-Yau threefolds defined over the field $\mathbf{Q}$ of rational numbers. We establish the modularity for the rigid Calabi-Yau threefold arising from the root lattice $A_{3}$. Our proof is based on geometric analysis.