All Questions
974 questions with no upvoted or accepted answers
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294
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Equivalence of algebraic and topological monodromy representations?
Does anyone know of a reference for the following fact?
Let $M_g$ denote the moduli stack of genus g curves, let $A_g$ denote the moduli stack of abelian varieties, and let $U_g \rightarrow A_g$ ...
- 1,308
5
votes
0
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256
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Symplectic leaves in positive characteristic
I am currently considering a family of filtered algebras over an algebraically closed field of positive characteristic with the property that the associated graded algebra is a finitely generated ...
- 599
5
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0
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143
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Reference request: "effective'' semistable reduction
I am looking for the origin of the following idea: suppose $m$ and $n$ are relatively prime integers $\geq 3$. Let $E$ be an elliptic curve over a number field $K$. Let $L/K$ be a finite extension ...
- 51
5
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486
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Computing intersection number of two arithmetic line bundles
I have some questions in Arithmetic Arakelov geometry
Let $\mathcal X\to Spec(\mathcal O_K)=C$ be an arithmetric projective variety over $C$ , where $\mathcal O_K$, ring of number filed $K$ and $\...
user21574
5
votes
0
answers
307
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Ranks of elliptic curves over Q(t)
I have an elliptic curve $E/\mathbb{Q}(t)$, and I want to compute its rank. Does knowing the rank over $\mathbb{F}_p(t)$ for some prime of good reduction give a bound on the rank over $\mathbb{Q}(t)$?...
- 585
5
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150
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Counting square zero forms over finite fields
Let $p$ be an odd prime and let $R=\Lambda_{\mathbb{F}_p}[x_1,\dots,x_n]$ be the exterior algebra on $n$ generators over the finite field with $p$ elements. This is a graded-commutative ring.
Is ...
- 2,320
5
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241
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Classification of compact Shimura curves
Is there a classification that determines all isomorphism classes of compact Shimura curves at least Shimura curves in $A_g$? I did not find this in the literature and appreciate any helpful ...
- 2,221
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287
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Nori fundamental group and etale fundamental group in positive characteristic
Let $X$ be a smooth projective surface over an algebraically closed field of char $p > 0$. Suppose that $\pi_{1}^{et}(X) = \{1\}$. Can Nori fundamental group scheme of $X$ be non-trivial?
- 163
5
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240
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Semiabelian actions appearing in the toroidal campactification of a degenearting abelian varieties
Given a totally degenerated abelian variety $A_K$ (to make it easier) over a complete discrete valuation field $K$ with $R$, $\pi$ and $k$ the corresponding discrete valuation ring, uniformiser and ...
- 997
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387
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Are these empirical discoveries about the Serre Swinnerton-Dyer ring of prime level modular power series actual theorems?
In this question Joel Bellaiche constructed an algebra, M, of modular forms for
gamma_0 (N) in finite characteristic (which he called p, but I'll call ell) and asked to know its structure. Matt ...
- 5,422
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1k
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Grothendieck monodromy theorem for l-adic sheaves
Hi,
Suppose that $F$ is a local field, $G_F$ its Galois group, $I$ the inertia subgroup, $k$ its residue field.
Let $X$ be a finite type scheme over $k$. Let $C$ be a constructible $l$-adic sheaf on $...
- 2,842
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725
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intersection cohomology and etale cohomology
Hello,
Can someone explain or give a reference on the comparison between intersection cohomology and l-adic etale cohomology of a variety over a field of characteristic zero?
Thanks!
- 2,842
5
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answers
240
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Is the moduli space of genus three smooth quartics affine?
Non-hyperelliptic curves of genus three are smooth quartics. Is the moduli space of such curves affine?
I think this follows from a more general result on smooth complete intersections, but I'm ...
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397
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false elliptic curves and principal polarizations
Hi,
Let $\Delta$ be a quaternion algebra over $\mathbf Q$ and let $\mathcal O_\Delta$ be a maximal order in $\Delta$.
Recall that a false elliptic curve over a field $K$ is a pair $(A/K,i)$ ...
- 2,842
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393
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What is the shape of the zeta function of a singular hypersurface?
So let $X$ be a projective hypersurface inside $\mathbb{P}_{\mathbb{Z}}^n$ of degree $d$.
Assume that
(a) $X(\mathbb{C})$ and $X(\overline{\mathbb{F}}_p)$ are irreducible,
(b) and that $X(\mathbb{C}...
- 7,609
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454
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If $p=0$ and $df=0$, is $f$ a $p$th power?
This question is a follow-up to When does the relative differential $df=0$ imply that $f$ comes from the base?. There it was asked, for an $A$-algebra $B$, under what conditions does $df=0$ (in the ...
- 4,487
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834
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Motivic Galois group and Shimura varieties
Hi,
Suppose that one has a Shimura variety $Sh(G,X)$ where $(G,X)$ is the corresponding Shimura datum and suppose that it can be interpreted as a moduli space of motives (e.g. PEL type Shimura ...
- 647
5
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708
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Formal groups in the supersingular reduction case
Let $E/\mathbb{Q}$ be an elliptic curve with potential good supersingular reduction at $p$. Thus, there is a finite extension $K/\mathbb{Q}_p$ such that $E/K$ has good supersingular reduction. Let us ...
- 1,694
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518
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Hochschild-Serre for hypercohomology
I need either a proof or a good reference for the following plausible statement:
Let $S$ be a scheme and let $C$ be a bounded complex of abelian sheaves on $S_{\rm{fppf}}$. Let $S^{\prime}\rightarrow ...
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234
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Modular reduction of exceptional complex reflection groups
I am interested in reducing reflection representations of complex reflection groups modulo a prime $p$. For the infinite family $G(m,r,n)$, it is straightforward to get "good reduction" provided that $...
- 10.7k
5
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672
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choice of local system in Deligne's construction of $l$-adic Galois representations
Hello,
Deligne famously constructed $l$-adic representations of $G_\mathbf Q = Gal(\overline{\mathbf Q}/\mathbf Q)$ starting form cusp modular forms of weight $k$ by looking inside the cohomology ...
- 2,842
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735
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Do all the main properties of constructible and perverse sheaves (in an 'arithmetic' situation) follow from results of Gabber?
This question is a continuation of Bad behaviour of perverse sheaves over 'general' bases?
Let $S$ (for example) be a finite type separated scheme over $\mathbb{Z}$. I would like: (1) to ...
- 16.9k
5
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530
views
Given two linear operators A and B over a finite field, is there a third operator C whose kernel is the intersection of kernels of A and B?
Let $V$ be a finite dimensional linear space over a finite field $k$. Let $A$ and $B$ be two endomorphisms of $V$.
Question 1. Is there an endomorphism $C$ of $V$, which is expressed in terms of ...
- 3,897
5
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744
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p-divisible groups of superspecial abelian varieties
Let $p$ be a prime and $F$ be an algebraic closure of the field with p elements. I will consider abelian varieties over F up to prime-to-$p$ isogeny. Principal polarizations will be $Q$-homogeneous ...
- 51
4
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answers
110
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Euler factors from bad primes and the Beilinson-Bloch vanishing conjecture
The vanishing part of the Beilinson-Bloch conjecture asserts that for a smooth projective variety $X$ over a number field $K$, $\dim_{\mathbb{Q}} \operatorname{CH}^i(X) \otimes_{\mathbb{Z}} \mathbb{Q} ...
- 531
4
votes
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178
views
Splitting of counit-trace map for $\ell$-adic sheaf $\Bbb Q_{\ell}$
I have a question about following argument on page 3 in paper arxiv.org/abs/1702.04404 by Will Sawin (Proposition 3):
The claim is that for a finite, dominant map $f:X \to Y$ between varieties $X,Y$ ...
- 6,006
4
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100
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Structure of points of elliptic curves in field with restricted ramification
Let $k$ be a finite field of characteristic $p$ and let $C$ be a curve over $k$. Let $E$ be a non-constant elliptic curve over $k(C)$. Taking the Néron model of $E$ and removing the singular fibers ...
- 177
4
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125
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Absolute purity in p-adic case
Suppose that $X$ is a regular proper scheme over $\mathbb{Z}_2$ with smooth generic fiber $X_{\eta}$ and singular special fiber $X_s$. I want to understand $H_{\textrm{et}}^*(X_{\eta},\mathbb{Z}/2)$. ...
- 918
4
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167
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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 ...
- 531
4
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228
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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 ...
- 141
4
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184
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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 ...
- 371
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129
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How does one compute the group action of the automorphism group on integral cohomology?
Suppose I have a curve $X$ (for concreteness, we can take $X$ to be a smooth, projective curve over a finite field $\mathbb F_q$, and even more concretely consider the family of curves described by ...
- 7,746
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249
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Is it always true that the complement of an ample divisor is affine?
Consider a proper and integral scheme $X\rightarrow\operatorname{Spec}(A)$ over a Noetherian ring $A$ and $D\in\operatorname{Div}(X)$ an effective ample Cartier divisor on $X$. Is it true that its ...
- 41
4
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108
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Shafarevich conjecture for Abelian varieties over global function fields
Let $S$ be a finite set of places of a global function field $K$. Are there finitely many Abelian varieties over $K$ with good reduction outside $S$? What if we exclude isotrivial families?
- 679
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135
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Nilpotent orbits in characteristic $0$ vs. positive characteristics
Let $G_\mathbb{C}$ be a connected reductive group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}_{\mathbb{C}}$. For any algebraically closed field $k$, let $G_k$ denote the connected reductive group ...
- 2,751
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183
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Characters of finite fields - Reference request
Let $\mathbf{F}_q$ ($q=p^f$) be a finite field. We are interested in the characters $\chi: \mathbf{F}_q\rightarrow \mathbf{K}$ ($\chi(0)=0$) where the $ \mathbf{K}$ is an alg.closed field of ...
- 41
4
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0
answers
284
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Intermediate arithmetic results in F_1 geometry
Much is made of the search for a proof of the Riemann hypothesis via the field with one element. Are there any lesser classic arithmetic results that have been proved with F_1 geometry, such as the ...
- 447
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103
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Boundary divisor of an equivariant compactification of a non-trivial twist of $\mathbb{G}_a^n$
Let $K$ be a non-perfect field of positive characteristic. Then there exist non-trivial forms of $\mathbb{G}_a^n$ which split over finite purely inseparable extensions of $K$, i.e., algebraic groups $...
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226
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The coarse moduli schemes of the "Shimura stacks" are the canonical models of the corresponding Shimura varieties
Let $F$ be a number field, $B$ a central simple algebra over $F$, $*$ a positive involution on $B$ which fixes $F$, and
$O_B$ a maximal $O_F$-order of $B$ which is stable under $*$.
Assume that $(B, *)...
- 1,364
4
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179
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Reference for Iwahori-Hecke algebras
I recently came across the notion of an Iwahori-Hecke algebra. I would like to learn the basics about this type of algebras (mainly to get an intuition about them, as they seem to be related to some ...
- 541
4
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77
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Conjugacy of cocharacters from conjugacy of labelled diagrams
Everything to follow is over some fixed algebraically closed field $k$. Although all the definitions make sense regardless of characteristic, the meat of the question is about small positive ...
- 12.9k
4
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174
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Spencer complex and de Rham Complex
in those lectures notes written by Claude Sabbbah: https://perso.pages.math.cnrs.fr/users/claude.sabbah/livres/sabbah_nankai110705.pdf
there is the proposition 1.4.4 where he says that there is a ...
- 385
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64
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An analog of a BGG resolution in subregular case in positive characteristic
Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$. For every regular weight $\lambda$ of $\mathfrak{sl}_n$, we have the ...
4
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252
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Lifting the connected-etale sequence of the $p$-torsion of an elliptic curve
Suppose that $R$ is a complete DVR with characteristic 0 fraction field $K$, maximal ideal generated by $p$ and characteristic $p>0$ residue field $k$ which is algebraically closed. Suppose that $\...
- 305
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129
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Statistics about existence of rational points on a curve over $\mathbb{F}_q$
I wish to ask the naive question: if we write down a random curve $C$ over $\mathbb{F}_q$, what can be said about the probability that $C(\mathbb{F}_q)=\emptyset$?
Of course, this depends on the ...
- 1,856
4
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0
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284
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modularity of elliptic curves over function fields in positive characteristic
Let $F$ be a global function field over a finite field, and $E$ be an elliptic curve over $F$. Much like the number field case, it is natural to study the Galois representation on the Tate module of $...
- 685
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147
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Uniqueness of Galois descent
Let $Y,Z$ be $\mathbb{F}_p$-schemes, they are both models of scheme $X$ over $\overline{\mathbb{F}_p}$ . Let $F$ be the absolute Frobenius of $\overline{\mathbb{F}_p}$, If $1_Y\times F$ and $1_Z\times ...
- 653
4
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0
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296
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de Rham Witt complex vs. de Rham complex of the Witt ring
I am reading the paper "Revisiting the de Rham-Witt complex" by Bhatt-Lurie-Mathew and I am a bit confused about the difference between $W\Omega_R^*$ and $\hat{\Omega}^*_{W(R)}$.
Let $\...
- 218
4
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0
answers
204
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Explicit description of wonderful compactification for PGL_3
Let $k$ be an algebraically closed field of positive characteristics. Let $X$ be the wonderful compactification of $PGL_3$ (see for example section 6 of "Frobenius Splitting Methods in Geometry ...
- 163
4
votes
0
answers
244
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Torsionness of the kernel of the pullback map of Picard groups of a normalization map
Let $X$ be a (irreducible) projective variety over a number field $k$, $\pi: \tilde X \to X$ be its normalization, and $\pi^{*}: \mathrm{Pic}(X) \to \mathrm{Pic}(\tilde X)$ be the corresponding map of ...
- 151