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Let $G$ be a connected reductive algebraic group over $\mathbf{Q}$. I've seen two slightly different definitions in the literature of the Shimura variety of level $U$, for $U \subseteq G(\mathbf{A}_{\ …

Let $G$ be the quasi-split unitary similitude group $GU(2, 1)$, for some choice of imaginary quadratic field $E$; and let $T = GU(1)$ be the torus $Res_{E/Q}\mathbf{G}_m$. Then there's a morphism $\th …

Let $F$ be a totally real field, and $E \subseteq F$ a subfield. Choose a quaternion algebra $B$ over $F$ satisfying the following condition: there is a distinguished infinite place $\tau$ of $E$ su …

This is an algebraic question disguised as a rigid-analytic one: what you're asking for is a description of the $\mathbf{Z}_p$-scheme $X^{\star} - X$, the boundary of the arithmetic minimal compactifi …

Yes, this is true. It works with arbitrary algebraic varieties, no need to be specific to Shimura varieties. Let $X \to^{\pi} Spec(K)$ be an algebraic variety, $\mathcal{F}$ an etale sheaf on $X$, and …

If $K$ is 'everything 1 mod N' for some N, then the canonical model of $\mathbf{Q}^\times_+ \backslash \mathbf{A}^\times_{\mathrm{f}} / K$ is exactly $\mu_N / \mathbf{Q}$, the scheme of $N$-th roots o …

You might like to read the introduction of Harris' 2013 Crelle paper "The Taylor-Wiles method for coherent cohomology" (see link). Here is an excerpt: In practice, all the higher-dimensional results, …

This question already has multiple nice answers, but I am going to add one more thing which isn't quite covered by the existing posts. One distinctive advantage of the $\Gamma_0(N)$ and $\Gamma_1(N)$ …

There is a very natural, intrinsic definition of a "symmetric space", as a manifold (Riemannian or Hermitian) with an extra symmetry of a certain prescribed type. It is then a theorem, not a definitio …