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The origin of this question is related to the construction of Galois representations of Deligne attached to $f$ a new cuspidal form (of weight $k\geq 2$). To do this, we consider the fiber product $k$-fold $E^{k-2}$ where $E\rightarrow \overline{X}$ is the universal elliptic curve of the compactified modular curve.

Then Deligne's Kuga-Sato manifold is the desingularization of $E^{k-2}$.

Questions Is "desingularization" also what is called "resolution of singularities" in algebraic geometry? Why did Deligne need regular variety in his construction?

I think the goal was to then apply the smooth-proper base change theorem in order to get an L function that is the same at almost all primes. But I also feel like it's something else because "desingularization" should give you a regular scheme, not a smooth one. If I'm right about the proper-smooth base change theorem, that leads me to ask the following questions

Questions Is this the only reason we ask for smooth variety? Why is the K-S variety proper?

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  • $\begingroup$ $E^{k-2}$, as you have defined it, is smooth but not proper. You first need to compactify $E$ and then desingularize the fibre product. $\endgroup$
    – naf
    Commented Oct 20, 2022 at 3:57
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    $\begingroup$ I think here $E$ denotes the universal generalized elliptic curve, whose fibers over the cusps are Néron $N$-gons, hence $E$ is proper over $\overline{X}$. In fact $E$ is the minimal regular model of the universal elliptic curve over $X$. So $E^{k-2}$ is proper but not regular. One needs a smooth proper variety to get a pure motive / pure Galois representation. A detailed description of the construction of the KS has been given by Scholl, see here, Chapter 7. $\endgroup$
    – François Brunault
    Commented Oct 20, 2022 at 6:24
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    $\begingroup$ One of Deligne's goals was to prove the Ramanujan conjecture, $|a_p(f)| \le 2p^{(k-1)/2}$, by reducing it to the Weil conjectures. For this he needed to not just construct the Galois representation, but also show that it occurred as a subquotient of cohomology of a smooth variety. $\endgroup$
    – David Loeffler
    Commented Oct 20, 2022 at 6:32

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