Alex
  • Member for 11 years
Which Shimura varieties are known to be automorphic?
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14 votes

As far as I understand, what you need to do to prove that the zeta function of a Shimura variety is automorphic is (I'm ignoring the bad primes here, but I think that we now know enough about them too ...

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snake lemma in category of groups
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13 votes

The snake lemma doesn't make sense in the category of groups in general, because maps don't always have cokernels. If you assume that the three vertical maps in your snake lemma diagram do have ...

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are irreducible representations with large fixed subspaces trivial?
13 votes

I don't know the answer, but here are a few remarks : (0) As has been pointed out before, "irreducible" and "indecomposable" are not the same for representations in positive characteristic. "...

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What's an example of an intersection cohomology sheaf whose stalks are pure but not pointwise pure?
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10 votes

Here is an example. Sorry it's so complicated. (There's probably a simpler one, but my mind works in complicated ways, it seems.) Consider a Siegel modular threefold $U$, i.e., a Shimura variety for ...

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Finiteness of normalization of Noetherian normal domain
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8 votes

No, it's not true. You can say some things about B even if L/K is not separable : B is a dvr if A is, B is a Dedekind ring if A is (that is called the Krull-Akizuki theorem). But it's not true that B ...

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Is there an easy proof of the fact that the intermediate image functor respects weights?
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8 votes

The proof in BBD is not that complicated, and it doesn't matter much whether $j$ is affine or not. It uses the three following facts : If $f$ is a morphism of schemes, then $f_*$ sends a complex of ...

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Are two conjectural descriptions of the motivic t-structure (via cohomology and via affine varieties) known to be equivalent?
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7 votes

Okay, I'm not familiar with Beilinson's paper, but here's my take on this. First let's recall the two definitions. I will denote the triangulated category of motives over a field $k$ by $DM(k)$ (for ...

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If one wants to work with $Q_l$-adic sheaves, should the scheme be of finite type over a 1-dimensional one?
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7 votes

Gabber has announced a proof of the finiteness theorem for (direct images of constructible sheaves by) morphisms of finite type between general noetherian schemes years ago, but, being Gabber, he has ...

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Bad behaviour of perverse sheaves over 'general' bases?
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6 votes

The answer is very likely "yes", but you will need to put together some technical articles (and unpublished results) that may not have yet been put together. Here are the key ingredients, as I see it :...

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When do adjunctions preserve equivalence?
5 votes

I think that in this generality the answer is "no". For example, take $\mathcal{C}'=\mathcal{D}'$ equal to some additive category, take the identity as the equivalence, take $\mathcal{C}=\mathcal{C}'$ ...

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Functoriality properties of the perverse $t$-structure for torsion (constructible complexes of) sheaves
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5 votes

If you want only $\mathbb{Z}/\ell\mathbb{Z}$ coefficients (not general $\mathbb{Z}/\ell^m\mathbb{Z}$), then there is only one middle perverse t-structure, which is good. The way the exactness ...

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Which statements in section 5 of BBD will fail if we consider $\mathbb{Q}_l$-adic sheaves there?
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5 votes

I think that all the statements are true, except for 5.3.9 (ii). Remark 5.3.10 says that all the statements in 5 up to and including 5.3.8 are true for $\mathbb{Q}_\ell$-coefficients with the same ...

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semisimplicity of automorphic Galois representations
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4 votes

Do you mean the local Galois representations or the local Galois representations ? The global Galois representations they are constructing correspond to cuspidal automorphic representations of GL(n). ...

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The category of l-adic sheaves
4 votes

Zheng and Liu are using $\infty$-categories to study constructible sheaves on stacks, and they have a $\ell$-adic version too. (Though most of the details for the $\ell$-Adic version should appear in ...

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Is it easy to define weights for $Q_l$-sheaves over finite type $Z[1/l]$-schemes?
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4 votes

The answer to the question in your title is, I think : "in general, no". The answer to your last question is : "well, it depends how you have defined the objects, and you will have to be very careful ...

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Do etale neighhbourhoods of a subvariety descend along base field extensions; does normalization commute with etale base change?
3 votes

I don't know. Do you have a reference for the affine case ? The variety obtained is called "integral closure" (or "normalization") of $V$ in $L$. ;-) A reference is EGA II 6.3, which seems pretty ...

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Reference for Numerical vs Homological equivalence
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3 votes

There's a quick proof in Yves André's book "Une introduction aux motifs" (proposition 3.4.6.1). Note that a stronger result is true : actually, algebraic equivalence coincides with numerical ...

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Finiteness of étale Cohomology Groups
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3 votes

(This was going to be a comment, but it's too long.) I don't see how the big étale site appears, even in the proof of cor IV.2.8 of Milne. Seems like he's just base changing to the integral closure ...

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Is there a $k$-structure for Hodge modules over a $k$-variety?
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2 votes

I think that the answer is "yes". If you denote by $MFW(X)$ (resp. $MFW(X_\mathbb{C})$) the category of regular holonomic $D$-modules on $X$ (resp. $X_\mathbb{C}$) with a good filtration $F$ and a ...

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Could the Kunneth decomposition of a motif depend on the choice of $l$?
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2 votes

Let me develop YBL's answer a bit. (I wanted to make this a comment but it was too long...) Consider a smooth variety $U$ over $\mathbb{F}_p$ with function field $K$ such that your motive and its two ...

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Equivalent forms of the proper base change isomorphism
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1 votes

(1) is not always an isomorphism when $f$ is an open immersion. (Take $X=Y$ equal to an open subscheme of $Z$, with the obvious maps.) Here is why : when you try to show that the restriction of $g_*q_!...

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Conjugacy class of a full Jordan block over integers
1 votes

The answer to your first (less general) question is this : Let $A$ be a $n\times n$ matrix with coefficients in $\mathbb{Z}$. Then $A$ is similar over $\mathbb{Z}$ to a full Jordan block if and only ...

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When does the equivariant homology of the fixed part of a $G$-space surject onto the equivariant homology of the whole space?
0 votes

Is equivariant homology dual to equivariant cohomology ? Because there is a paper of Goresky-Kottwitz-MacPherson that gives condition for the equivariant cohomology of $X$ to inject into that of $X^G$;...

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