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Frobenius eigenvalues of abelian variety

2012-07-25T12:49:57Z

Let $A$ be an abelian variety over a finite field $\mathbb{F}_q$ and $x_i$ the Frobenius eigenvalues on $H^1$. Does $x_i \mapsto q/x_i$ permute the $x_i$, and why? It should follow from Poincare duality.

cocharacters of GL(V)

2011-10-02T13:28:09Z

Can someone give me a hint how to determine the cocharacters of $GL(V)$?

formally smooth => smooth

2011-09-05T10:25:02Z

A morphism of set-valued functors $\eta: F \to G$ on $\mathcal{C}$ is called smooth if for all epimorphisms $B \to A$, the natural morphism $F(B) \to F(A) \times_{G(A)} G(B)$ is surjective.

Obviusly "smooth => formally" smooth for $\mathcal{C} = \mathrm{Sch}$.

Now my question: Does the converse hold?

My thoughts: 1. Assume the morphism is of finite presentation.

Assume it is in the local standard form: an étale morphism followed by an affine projection
It is clear that an affine projection is smooth in the above sense, so we have reduced the problem to étale morphisms.

cohomological dimension for coarser/finer topologies

2011-08-02T15:37:59Z

Given a sheaf $\mathcal{F}$ with respect to some Grothendieck topology, is the cohomological dimension for this sheaf less than or equal to the cohomological dimension of a finer topology?

Example: $cd_{Zar} \leq cd_{Nis} \leq cd_{ét}$.

Book on linear algebraic groups in scheme language

2011-07-14T15:02:42Z

Is there a book on linear algebraic groups using the scheme language (i.e. not Springer or Borel, but like Waterhouse, but more in-depth)?

The book should discuss topics like Borel subgroups etc.

Why does one consider the dual of the Steenrod algebra?

2011-06-13T18:00:02Z

Why does one consider the dual of the Steenrod algebra?

$H_2$ of a simply connected Lie group vanishes

2011-05-26T19:54:19Z

How do I show that the $H_2$ of a simply connected Lie group vanishes? (I don't want to use that $\pi_2(Lie group) = 0$, since this is what I want to prove. And I don't want to use the classification of compact simply-connected Lie groups.)

applications of Postnikov approximation

2011-05-23T19:56:40Z

What are applications of Postnikov approximation? As I understand it, it is dual to a cell decomposition and can, e.g. be used for computing some homotopy groups of spheres.

map defined by element of function field of a variety

2011-05-21T16:46:41Z

Given a smooth proper variety $V/K$ of dimension $n$ and an element $f \in K(V)$, does this define a map $V \to \mathbf{P}^n$? Probably only a rational map outside a set of codimension $> 1$?

Albanese variety

2011-05-20T19:04:20Z

What are good references for the Albanese variety and its properties?

Quotient category $Coh(X)_d/Coh(X)_{d-1}$

2011-05-15T15:16:48Z

Let $X$ be a regular scheme, and $Coh(X)_d$ be the category of coherent sheaves of $\leq d$ dimensional support.

Why is $Coh(X)_d/Coh(X)_{d-1}$ equivalent to $\bigoplus_{x \in X_d} \mathcal{A}(\mathcal{O}_{X,x})$ ?

See: http://www.math.uni-bielefeld.de/~mseverit/algkalgc.pdf page 2, 6th equation.

(cross post from http://math.stackexchange.com/questions/39186/quotient-category-coh-d-coh-d-1 )

structure of $T_\ell A$ for $A/\mathbf{F}_q$ an abelian variety

2011-05-15T14:29:28Z

Can someone give me references for the structure of the $G_{\mathbf{F}_q}$ -module $T_\ell A$, $A/\mathbf{F}_q$ an abelian variety?

Does every finite flat group scheme become constant after finite base change?

2011-05-12T09:16:41Z

Does every finite flat group scheme $G/X$ become constant after finite base change? Which additional properties of the base change morphism can we impose?

Edit: Which conditions do we have to impose on $G/X$ so that the answer becomes "yes"?

prove statement in Galois cohomology by étale cohomology

2011-04-01T14:31:37Z

According to Milne's Arithmetic Duality Theorems, Proposition I.6.4: $0 \to H^1(G_S, A[m]) \to H^1(K, A[m]) \to \oplus_{v \not\in S}H^1(K_v, A)$ for an abelian variety $A$ and a nonempty set of primes $S$ containing all infinite primes and the primes of bad reduction.

I want to prove this using étale cohomology. My idea was to extend $A$ to an abelian scheme and use the (excision) long exact sequence, but this lead me nowhere. Can someone give me some hints?

tensor product of motivic complexes $\mathbf{Z}(n)$

2011-04-10T15:30:09Z

Is the morphism $\mathbf{Z}(n) \otimes^L \mathbf{Z}(m) \to \mathbf{Z}(n+m)$ from the Beilinson-Lichtenbaum conjectures a quasi-isomorphism?

Galois cohomology groups given by étale cohomology

2011-04-01T17:10:03Z

What are cases when Galois cohomology groups are given by étale cohomology?

Example: $S = Spec(K)$ the spectrum of a field, $F \in Sh(K)$, then $H^p(K, F) = H^p(G_K, F_{\bar{K}})$.

What if $G = \pi_1(X)$ and $F \in Sh(X)$? Under what conditions do we have $H^p(X, F) = H^p(G, [F])$, where $[F]$ denotes a suitable $\pi_1(X)$-module associated with $F$? (Example for this: $X = Spec(O_K)\setminus S$)

questions regarding modular forms

2011-02-08T19:15:35Z

Let $f$ be modular of level $p^nN$, $(p,n) = 1$, $p > 2$ with character $\chi\psi\eta$, where $\chi$ has conductor dividing $N$, $\psi$ conductor power of $p$ and order power of $p$, and $\eta$ conductor $p$ and order dividing $p-1$. Since $p$ is odd, one can write $\psi = \xi^{-2}$. (i) Why is the character of $f \otimes \xi$ equal to $\chi\eta$; (ii) why is the reduction mod $p$ of this equal to the reduction of $f$; (iii) why is for $r \gg n$ $f \otimes \xi$ modular with respect to $\Gamma_0(p^r) \cap \Gamma_1(pN)$? Can this be proven using the converse theorem?

2.a Why is the twisted Eisenstein series $G = a_0 + \sum_{n=1}^\infty\sum_{d \mid n}\eta^{-1}(d)d^{i-1}q^n$ of Nebentypus $\eta^{-1}$ with respect to $\Gamma_0(p)$?

2.b Why is $fG$ modular with respect to $\Gamma_0(p^r) \cap \Gamma_1(N)$, wenn $f \in S_k(\Gamma_0(p^r) \cap\ \Gamma_1(pN), \chi\eta)$ ist?

(The article is here: math.berkeley.edu/~ribet/Articles/motives.pdf )

Comment by

2011-09-05T11:08:33Z

I think my definition of "smooth" is a priori different from this.

Comment by

2011-08-03T13:28:33Z

"Suppose we have one category with two Grothendieck topologies, one finer than the other. Suppose some presheaf is a sheaf with respect to both of them. If Hn=0 for all n>d in the finer case, must the same be true in the other case?" Yes, that is what i meant.

Comment by

2011-05-30T06:16:28Z

"Or perhaps not! If so, ask away." Yes, please! I'm interested in!

Comment by

2011-05-30T06:14:57Z

Isn't there a purely cohomological, non geometrical proof?

Comment by

2011-05-15T16:02:12Z

OK, $V_\ell A = \mathbf{Q}_\ell^{2g}$, upon which the Frobenius acts semisimply with eigenvalues Weil numbers of weight $-1$.

Comment by

2011-05-15T15:32:47Z

But this is not all. Let me think about it.

Comment by

2011-05-15T15:28:49Z

I see: The Tate conjecture gives a description of $End_G(T_\ell A)$, in particular of its $G$-automorphisms, in terms of $End(A) \otimes \mathbf{Z}_\ell$.

Comment by

2011-05-15T15:23:50Z

Where can I find the description of its Galois module structure?

Comment by

2011-05-15T13:50:28Z

Does the "only if" follow from fppf descent?

Comment by

2011-02-08T19:29:59Z

The article is here: math.berkeley.edu/~ribet/Articles/motives.pdf page 5.