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The Modularity Conjecture for Calabi-Yau Threefolds

A Calabi-Yau variety, X, of dimension n, is defined to be a smooth projective variety over a field k that satisfies ωX := ∧n Ω ' OX and H(X,OX) = 0 for 0 < j < n. This can be seen as a higher dimensional “cohomological” analogue of an elliptic curve. A Calabi-Yau threefold is a Calabi-Yau variety of dimension three. The main goal of this paper is to introduce two equivalent notions of modularity for rigid Calabi-Yau threefolds defined over Q. They are both defined via the L-series of the variety. One way is to produce a certain `-adic Galois representation and the other is to define the L-series directly from Frobenius actions. There are several more equivalent notions of modularity, but we’ll focus on these two. We should probably start by noting that modularity of one-dimensional Calabi-Yau varieties was the famous Taniyama-Shimura conjecture. This took alone was very difficult to prove. Wiles and Taylor published the case of semi-stable elliptic curves in 1995, but it wasn’t until 2001 that the full conjecture was published by Breuil, Conrad, Diamond, and Taylor. It may seem curious that we immediately jump to threefolds instead of focusing on two-dimensional varieties next. This has to do with the fact that threefolds can be rigid and hence produce certain small cohomology whereas in the two-dimensional case the cohomology groups have dimensions 1, 0, 22, 0, and 1 respectively. To talk about modularity most easily we will want a twodimensional representation. We won’t dwell on this point. Instead, it should become clearer after a thorough analysis of the three-dimensional case.