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      Questions tagged [shimura-varieties]

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      Which Shimura varieties admit or don't admit $p$-adic uniformization by Drinfeld spaces?

      $p$-adic uniformization is a powerful tool for studying Shimura curves and Shimura varieties. For instance, we know cohomology groups of Drinfeld spaces, so we know some information about the Shimura ...
      16
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      0answers
      417 views

      Shimura varieties and connected components

      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}_{\...
      4
      votes
      0answers
      180 views

      Higher dimensional generalization of an identity between traces of Hecke operators and number of elliptic curves over finite fields?

      In http://www.math.ubc.ca/~behrend/ladic.pdf, the author uses his generalization of Lefschetz trace formula to smooth algebraic stacks to prove an interesting identity (Proposition 6.4.11.): $\sum_{k}...
      2
      votes
      0answers
      89 views

      Euler characteristic of unitary shimura varieties

      Is there a general formula for the Euler characteristic of a unitary Shimura variety (with level structure)? Here we define the Euler characteristic using etale cohomology (of the constant sheaf). We ...
      14
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      3answers
      987 views

      Tower of moduli spaces in Scholze's theory

      My question is related to another one I read here in Overflow. I am reading Scholze's papers about moduli spaces of $p$-divisible groups and elliptic curves, and I am very interested in the formal ...
      1
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      0answers
      90 views

      Tate module is canonically isomorphic to a $\mathbb Z_p$-lattice on Shimura Variety of Hodge Type

      Let $(G, X)$ be a Shimura datum of Hodge type. Suppose that $K \le G(\mathbb A_f)$ is such a compact open subgroup that its $p$th component $K_p = \mathcal G(\mathbb Z_p)$ is a hyperspecial subgroup ...
      4
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      0answers
      233 views

      What is the analogy between the moduli of shtukas and Shimura varieties?

      I have heard that moduli spaces of shtukas are supposed to be the analogue of Shimura varieties in the setting of function fields. Could someone more knowledgeable about these objects explain how this ...
      3
      votes
      0answers
      199 views

      Shimura varieties of Hodge type

      I am trying to understand the theory of integral model of Shimura variety of Hodge type, like for example in Kisin's paper "Integral models for Shimura varieties of abelian type". I understand that ...
      1
      vote
      1answer
      307 views

      Points of infinite level modular curve

      Let us consider the anticanonical adic tower of modular curves which is described in Peter Scholze's paper "On torsion in the cohomology of locally symmetric varieties", and let us call $\mathcal{X}_{\...
      2
      votes
      1answer
      109 views

      Involution on false elliptic curve

      Let B be a indefinite quaternion algebra with discriminant $d>1$, maximal order $\mathcal{O}$ and standard involution $'$, then there exists $t\in{B}$ such that $t^2=-d$ and a new involution on B ...
      3
      votes
      1answer
      219 views

      Possible groups appearing in a Shimura datum

      Let $\mathbb{S}:=\text{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{G}_{m}$ be the Deligne torus. My question is the following: is there a sort of classification of real reductive algebraic groups $G$ for ...
      3
      votes
      1answer
      114 views

      What role do $(\mathfrak{g},K)$- modules play in the construction of automorphic vector bundles

      Looking at the connection between modular forms as sections and automorphic representations it is to me somewhat clear why automorphic representations are (demanded to be) admissible $G(\mathbb{A}^\...
      5
      votes
      0answers
      209 views

      What is the difference between Kisin's and Vasiu's work on models of Shimura varieties?

      My research is related to integral model of Shimura varieties. I realized there are two approaches building models for varieties of pre-abelian type and abelian type. I want to know what their ...
      2
      votes
      0answers
      84 views

      Compactification of symmetric spaces

      Let $G = GL_n$. In the literature, I see that the symmetric space associated to $G$ is of the form $G(\mathbb{R})/K_\infty$ with $K_\infty = O(n) Z(\mathbb{R})^\circ = O(n) \mathbb{R}^\times_{+}$. ...
      3
      votes
      2answers
      333 views

      What is wrong with this modification of the definition of Shimura datum?

      The definition of a "connected Shimura datum" (as in Milne's notes) is a pair $(G, X)$, where $G$ is a reductive algebraic group and $X$ is a $G(\mathbb{R})$-conjugacy class of morphisms $$ x: \mathbb{...

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