Taniyama-Shimura 2: Galois Representations where the standard modern approach to defining modularity for other types of varieties.

Fix some proper variety $latex {X/\mathbb{Q}}&fg=000000$. Our goal today will seem very strange, but it is to explain how to get a continuous representation of the absolute Galois group of $latex {\mathbb{Q}}&fg=000000$ from this data. I’m going to assume familiarity with etale cohomology, since describing Taniyama-Shimura is already going to take a bit of work. To avoid excessive notation, all cohomology in this post (including the higher direct image functors) are done on the etale site.

For those that are intimately familiar with etale cohomology, we’ll do the quick way first. I’ll describe a more hands on approach afterwards. Let $latex {\pi: X\rightarrow \mathrm{Spec} \mathbb{Q}}&fg=000000$ be the structure morphism. Fix an algebraic closure $latex {v: \mathrm{Spec} \overline{\mathbb{Q}}\rightarrow \mathrm{Spec}\mathbb{Q}}&fg=000000$ (i.e. a geometric point of the base). We’ll denote the base change of $latex {X}&fg=000000$ with respect to this morphism $latex {\overline{X}}&fg=000000$. Suppose the dimension of $latex {X}&fg=000000$ is $latex {n}&fg=000000$.

Let…

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