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Description of a Shimura variety

Unless I misunderstood, this seems straightforward. The Shimura variety you are asking for is $$(U\times V_N)\backslash (X\times G(\mathbb{A}_f)\times \mathbb{G}_m(\mathbb{A}_f))/(G(\mathbb Q)\times \...
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4 votes
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Purity for proper varieties

The claim that alteration induces an injection on cohomology is wrong, as the example of the resolution of a nodal cubic curve by $\mathbb P^1$ shows (resolutions being a special case of alterations). ...
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1 vote

Bounded torsion of quotients of affine formal models

As it turns out, the quotient $C$ does not necessarily have bounded $\pi$-torsion. For example, let $X=Sp(K\langle t\rangle)$ and $Y=Sp(K\langle \frac{t}{\pi}\rangle)$ be the disc of radius $\pi$. ...
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8 votes
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What's a right parameter space of abelian varieties over a non algebraically closed fields?

This is true in a Zariski-local sense. Any elliptic curve together with a nowhere vanishing differential can be put in this form. For an elliptic curve over $S$, the differentials form a line bundle ...
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2 votes
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Pairwise intersecting circles in the plane

First, it is sufficient to show that there is always an empty 3-cell inside each circle. This is because we can always do a Möbius transformation to move the outside of a circle inside of it, which ...
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3 votes

Special value of Hecke $L$ function

Not clear what you mean by $\varphi$ or $\psi$. At any rate, $E$ is a CM curve, hence $L(s,E)$ equals $L(s,\psi)$ for a suitable Hecke character $\psi$ of the number field $\mathbb{Q}(i)$. You can ...
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