All Questions
Tagged with ag.algebraic-geometry arithmetic-geometry
1,279
questions
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Companions for positive characteristic arithmetic representations viewed as representations of the topological fundamental group?
Suppose $X / K$ is a variety over a finitely generated field over $\mathbb{Q}$. Fix an embedding $K \subset \mathbb{C}$ and let $\pi := \pi_1(X(\mathbb{C}), x)$ be the topological fundamental group. ...
- 3,393
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votes
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152
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Is the group of homologically trivial cycles in a variety over a finite field torsion?
Let $X$ be a smooth projective variety over $\mathbb{F}_q$. Is any cycle in the Chow group $CH^i(X)$ which is trivial in $\ell$-adic cohomology automatically torsion? For abelian varieties I believe ...
- 371
2
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1
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Why this genus one curve over $\mathbb{F}_5$ appear to violate Hasse-Weil bound?
Working over $\mathbb{F}_5$, the affine curve $x^4+2=y^2$ has no points.
The projective curve $x^4+2y^4=z^2y^2$ has only one point $(0 : 0 : 1)$.
Both curves appear to violate Hasse-Weil bound of $4....
- 78.4k
3
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159
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On the sheaves-functions dictionary
Let $X$ be a variety over a finite field $k$. Let $\pi_{1}(X)$ be the arithmetic etale fundamental group of $X$, and $\rho:\pi_{1}(X)\to k^{\times}$ a continuous character. If $x: \text{Spec}(k)\to X$ ...
- 561
7
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Shouldn't we expect analytic (in the Berkovich sense) étale cohomology of a number field to be the cohomology of the Artin–Verdier site?
Let $K$ be a number field. Consider $X=\mathcal{M}(\mathcal O_K)$ the global Berkovich analytic space associated to $\mathcal O_K$ endowed with the norm $\|\cdot\|=\max\limits_{\sigma:K \...
- 12k
15
votes
2
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How to think of algebraic geometry in characteristic p?
How does a working mathematician usually think about algebraic geometry in characteristic $p$? For the sake of concreteness, and to make things more "geometric" (whatever that means), let's ...
- 1,227
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321
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Closed image of curves under $p$-adic logarithm, Coleman integrals and Bogomolov
Disclaimer: my knowledge of $p$-adic analysis/geometry is minimal.
Consider a smooth, complete curve $C$ of genus $g$ over $\mathbb{C}_{p}$, denote by $J$ its Jacobian and consider the embedding $C\...
- 950
3
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195
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The definition of complex multiplication on K3 surfaces
I am reading this paper on the complex multiplication of K3 surfaces. It seems that this is only defined for complex K3 surfaces, or K3 surfaces over number fields. Is there a more general defintion ...
- 131
8
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290
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Do automorphisms actually prevent the formation of fine moduli spaces?
I have found similar questions littered throughout this site and math.SE (for example [1], [2], [3],…), but I feel like like most of them usually just say that non-trivial automorphisms prevent the ...
- 12k
7
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553
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Are all representations of the geometric étale fundamental group subquotients of representations of the arithmetic étale fundamental group?
Let $X$ be a variety over a field $k$. The étale fundamental group of $X$ fits into the exact sequence:
$$1 \to \pi_1^{\text{geom}}(X) \to \pi_1^{\text{arith}}(X) \to \text{Gal}(\overline{k}/k) \to 1,$...
- 27.9k
1
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99
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Compactifications of product of universal elliptic curves
Let $\mathcal{E}$ be the universal elliptic curve over the moduli stack $\mathcal{M}$ of elliptic curves. As $\mathcal{E}$ is an abelian group scheme over $\mathcal{M}$, we obtain a product-preserving ...
- 12k
9
votes
2
answers
386
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Arithmetic schemes with the same zeta function
Suppose $X$ and $Y$ are $n$-dimensional regular separated schemes of finite type over $\mathbb{Z}$ such that number of $\mathbb{F}$-points of $X$ and $Y$ are equal for all finite fields $\mathbb{F}$.
...
- 1,453
4
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165
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Étale- or fppf-crystalline sites
I have a straightforward question. Let (say) $X/\mathbb{F}_p$ be a smooth proper scheme. On the big crystalline category over $\mathbb{Z}/p^n$ one can take the Zariski or étale topology, and one can ...
- 341
16
votes
2
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654
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Deligne's theorem on finite flat group schemes and generalizations
Recall Deligne's theorem that for a finite flat commutative group scheme $G$ of order $n$, the multiplication by $n$ map $[n]: G \to G$ is the zero map.
I have seen the proof a few times but I can't ...
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454
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Finiteness of the Brauer group for a one-dimensional scheme that is proper over $\mathrm{Spec}(\mathbb{Z})$
Let $X$ be a scheme with $\dim(X)=1$ that is also proper over $\mathrm{Spec}(\mathbb{Z})$. In Milne's Etale Cohomology, he states that the finiteness of the Brauer group $\mathrm{Br}(X)$ follows from ...
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