All Questions
974 questions with no upvoted or accepted answers
3
votes
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120
views
Resolving the "wild" singularities of $\mathbb A^n/C_n$
Let the cyclic group on $n$ elements, $C_n$, act on $\mathbb A^n$ by permuting the co-ordinates (over a field $k$). If $n \neq 0 \in k$, we can resolve the singularities of $X = \mathbb A^n/C_n$ by ...
- 7,746
3
votes
0
answers
189
views
Resolutions of configuration space of the projective line where the complement is of "Tate type"
I would like to find a nice compactification $X_n$ of $F(\mathbb P^1,n)$ (considered as a scheme over $\mathbb Z$), the $n$-fold configuration space of the projective line with the property that the $...
- 7,746
3
votes
0
answers
424
views
Generalization of the Castelnuovo-Severi inequality
The Castelnuovo-Severi says that given curves $X,Y,Z$ over a perfect field $k$ and nonconstant morphisms $f:X\rightarrow Y$ and $g:X \rightarrow Z$ which do not both factor through any morphism $X\...
- 221
3
votes
0
answers
202
views
Pilloni's cohomological corrispondence factorization
I'm trying to understand the proof of Lemma 7.1.1 at page 39 of Pilloni's paper on Higher Hida and Coleman Theory for $GSp_4$. In particular, what is not clear to me is the diagram relating Serre-Tate'...
- 91
3
votes
0
answers
196
views
Faithfulness of parabolic induction
I've only recently begun to study the representation theory of $p$-adic groups, so the following question might be quite silly.
Let $F$ be a non-archimedean local field of residue characteristic $p$, $...
user484137
3
votes
0
answers
104
views
Are supersingular K3 surfaces unirational?
There is a conjecture due to Artin, Rudakov, Shafarevich, Shioda that supersingular K3 surfaces over a finite field are unirational. This paper claims to prove this result but it has had a recent ...
- 7,746
3
votes
0
answers
174
views
On continuous seminorms on Fréchet-Stein algebras
Let $K$ be a discretely valued complete non-archimedean field and $U$ be a left Fréchet-Stein algebra as defined in Algebras of p-adic distributions and admissible representations, with a Fréchet-...
- 541
3
votes
0
answers
102
views
Rationality of plane curves with a certain property
Let $C\subset\mathbb{C}^2$ be an irreducible algebraic curve defined over a number field $F.$ Suppose that for any $(z, w)\in C, z\in \mathbb{\overline{Q}}, w\in\mathbb{\overline{Q}},$
either $z\in F(...
- 53
3
votes
0
answers
279
views
Grothendieck trace formula for arbitrary morphisms
The Grothendieck trace formula can be viewed as a generalization of the Lefschetz trace formula in étale cohomology from constant sheaves to constructible $l$-adic sheaves, but restricting to the ...
- 216
3
votes
0
answers
149
views
What direction does the derivation of an inseparable algebraic variable point in?
I've been thinking about the geometry of inseparable field extensions lately, since I'm studying smoothness in commutative rings in an advanced topics course this semester. I've generally come to the ...
- 1,969
3
votes
0
answers
197
views
How much results in Calabi-Yau manifolds and mirror symmetry depends on the existence of a ricci-flat metric?
An important result of CY manifold is the CY theorem, it talks about the existence of a ricci-flat metric. However, this theorem and its proof are highly analytic.
There are many results about ...
- 1,490
3
votes
0
answers
173
views
Smooth proper varieties over the integers that are not toric
Does there exist a smooth proper variety $X$ over $\operatorname{Spec} \mathbb Z$ that is not toric?
By Fontaine, we know that there is no Abelian scheme over $\operatorname{Spec} \mathbb Z$. Also by ...
- 7,746
3
votes
0
answers
188
views
Does the construction of arithmetic toroidal compactification of $A_{g}$ depend on semistable reduction theorem?
If there is a good theory of arithmetic toroidal compactification over $\mathbb{Z}_{p}$ of the Siegel modular variety with deep enough level structure, then it seems like semistable reduction theorem ...
- 1,024
3
votes
0
answers
377
views
Meaning of "the" general fiber in the paper "La conjecture de Weil : I"
In section 4.1, chapter 4 of Pierre Deligne's paper La conjecture de Weil : I (french version, translation to English) he states:
Let $X$ be a non singular analytic space and purely of dimension $n+1$....
- 519
3
votes
0
answers
206
views
Generalization of conjectures involving Beilinson regulators
I had some questions about the Beilinson conjectures as mentioned in this page. I have to admit I do not know much about Deligne cohomology. The conjectures involve some form of comparison map between ...
- 5,901
3
votes
0
answers
106
views
A uniform version of Bashmakov's theorem for elliptic curves
Let $E/\mathbb Q$ be an elliptic curve. Serre's open image theorem is the statement that the image of the Galois group $G_{\mathbb Q}$ into $GL_2(\mathbb Z/n\mathbb Z)$ by it's action on the torsion ...
- 7,746
3
votes
0
answers
151
views
Computing the group structure of $J(\mathbb{F}_q)$
Let $k$ be a finite field, $X/k$ a smooth curve, $f$ a polynomial of 2 variables which gives an affine model of $X$ and $J$ its Jacobian.
Then how can I compute $J(k)$?
If $X$ is a hyperelliptic curve,...
- 1,364
3
votes
0
answers
426
views
Closed immersion hitting all the $\mathbb{Q}$-points
Let $i:X\to Y$ be a closed immersion of smooth projective varieties over $\mathbb{Q}$.
Assume that $Y(\mathbb{Q})$ is infinite and $X(\mathbb{Q})\to Y(\mathbb{Q})$ is surjective. Also assume that $X$ ...
- 31
3
votes
0
answers
207
views
Schemes with common zeta function
If $S_\zeta$ is the set of all separated schemes of finite type over $\mathbb{Z}$ that have the same arithmetic zeta function $\zeta$, what more can we say about $S_\zeta$ assuming it is non-empty?
- 31
3
votes
0
answers
539
views
A question on the Bombieri-Lang conjecture
Let $X$ be a variety of general type, defined over a number field $K$. Then the Bombieri-Lang conjecture asserts that the set of rational points $X(K)$ (or $X(L)$ for any finite extension $L/K$) is ...
- 27k
3
votes
0
answers
306
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Why the curve $x^2+y^2+y+1=0$ has only one point over $\mathbb{F}_{3^7}$?
According to both sagemath and Magma the curve $x^2+y^2+y+1=0$ has only one point over $\mathbb{F}_{3^7}$.
The projective closure has only one point too.
Q1 What hypothesis are missing to not violate ...
- 25.4k
3
votes
0
answers
292
views
Generalizations of Artin–Verdier duality?
Constructible étale abelian sheafs on $Spec\ O_\mathbb K$, for number fields $\mathbb K$, satisfy Artin-Verdier duality. Are there known any algebraic schemes or algebraic stacks, other than $Spec\ O_\...
- 2,390
3
votes
0
answers
139
views
2-fold linear cover of reductive group of type A
Let $F$ be a nonarchimedean local field of characteristic zero. Let $G=\operatorname{Res}_{E/F}\operatorname{GL}_n$ or $\operatorname{Res}_{E/F}\operatorname{U}_n$, where $\operatorname{U}_n$ is any ...
- 833
3
votes
0
answers
483
views
Uniformization of algebraic curves
Given an irreducible smooth complex-projective curve $X$, I will say that a subgroup $\Gamma< SL(2, {\mathbb R})$ weakly uniformizes $X$ if [corrected] there exists a nonconstant holomorphic map ...
- 12.3k
3
votes
0
answers
197
views
How to write down an explicit equation of given degree yielding a smooth hypersurface in a projective space?
Let F be a field of positive characteristic $p$ and let $d,n$ be two positive integers.
Can we explicitly write down an equation defining a smooth hypersurface $X_d⊂\mathbb P^n_F$ of degree d ?
This ...
- 417
3
votes
0
answers
218
views
Is there a proper smooth variety in characteristic $p$ whose Hodge-to-de Rham spectral sequence does not degenerate at $E_1$?
By Deligne-Illusie, such a variety has no lifting to $W_2(k)$. In their paper they state that they do not know if such an example exists. Has this question been answered since then?
- 4,164
3
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0
answers
162
views
Cohomology of Siegel modular varieties
$\mathcal{A}_g(N)$ is the moduli space of principally polarized abelian varieties with a level $N$ structure.
Set $C_g=\displaystyle{\lim_{\rightarrow}} H^3(\mathcal{A}_g(N), \mathbb{F}_p)$ where the ...
user163668
3
votes
0
answers
246
views
Derived category and L-function
For abelian varieties over $\mathbb{Q}$ $\mathscr{A}$ and $\mathscr{A}'$, if derived categories $D(\mathscr{A})$ and $D(\mathscr{A}')$ are equivalent then L-functions are same $L(s,\mathscr{A})=L(s,\...
- 519
3
votes
0
answers
210
views
Étale homotopy equivalent varieties are deformation equivalent
Let $k$ be an algebraically closed field of characteristic $p>0$.
Let $V_1$ and $V_2$ be étale simply-connected smooth proper varieties over $k$. Assume there is an isomorphism between the prime-to-...
user145520
3
votes
0
answers
112
views
Deformation of p-divisible groups along nilpotent thickening
Let $S_0 \rightarrow S$ be a nilpotent thickening of schemes (no divided power provided) where $p$ is nilpotent, let $G$ be a $p$-divisble group over $S_0$, how to describe all liftings of $G$ to $S$ ...
- 6,622
3
votes
0
answers
232
views
$l$-adic Galois representations factor through a common finite quotient
Let $X$ be a smooth projective geometrically connected variety over $\mathbb{Q}$. Assume that for some $m>0$ we have $h^{i, 2m-i}(X)=0$ unless $i=m$.
Does there exist a number field $E$ such that ...
user158636
3
votes
0
answers
123
views
Commutative group stacks and Galois cohomology
"Classically", if we consider an abelian variety $A$ over some number field $k$, we get a $Gal(\bar{k}/k)$-module $A(\bar{k})$, or equivalently a sheaf of abelian groups on the étale site $\...
- 4,428
3
votes
0
answers
326
views
Kummer theory if $\ell = p$
Background. Let $k$ be a field and let $\ell$ be an integer which is divisible in $k$. Then one has a short exact sequence of abelian étale sheaves
$$ 0 \to \mu_\ell \to \mathbb{G}_m \xrightarrow{(\,\...
- 317
3
votes
0
answers
232
views
Lifting a Frobenius endomorphism under an étale morphism
Let $X$ be a smooth affine scheme over $\mathbb{Z}/{p^2}$ that is a complete intersection, say $X$ is the spectrum of $\mathbb{Z}/{p^2}[x_1,...x_n]/(f_1, ... f_r)$, where $n-r$ is the dimension of $X$....
- 111
3
votes
0
answers
178
views
Finiteness results in the category of schemes up to $\mathbb{A}^1$-homotopy
In algebraic geometry, we know that there exist geometrical conditions on a scheme $X/k$ for having finitely many rational points when $k$ is a number field. Namely for curves there is the Mordell ...
- 4,428
3
votes
0
answers
230
views
Independence of $p$ of Hodge-Tate weights
Let $X$ be a smooth and proper variety over $\mathbb{Q}$. Then for each prime $p$ we have the representation $R_p=H^i_{et}(X\times \overline{\mathbb{Q}_p}, \mathbb{Q}_p)$ of $\mathrm{Gal}(\overline{\...
user145520
3
votes
0
answers
339
views
Integral points on affine varieties
Consider Siegel's theorem. It says that for a smooth affine algebraic curve $C$ over $\mathbb{Q}$ such that $g(C)>0$ any model $\mathcal{C}$ of $C$ over $\mathbb{Z}$ has finitely many $\mathbb{Z}$-...
user145520
3
votes
0
answers
280
views
A complete Tate Huber ring is Banachizable (maybe not)?
I have questions of technical nature.
A complete Tate Huber ring is a complete topological (commutative) ring $A$ admitting an open subring $A_0$ whose topology is the $\varpi A_0$-adic topology, for ...
- 5,562
3
votes
0
answers
86
views
Self-contained reference for projective embedding of moduli of polarized abelian varieties via modular forms
I've been working on reading and understanding Arakelov's '71 paper and he uses the fact that the moduli space of complex abelian varieties of dimension $g$ with polarization of degree $d$ admits an ...
- 892
3
votes
0
answers
201
views
Endomorphisms of elliptic curves, resp formal groups
Let
$E$ be an elliptic curve over a number field $K$,
$\mathcal{E}^w$ a fixed Weierstrass model for $E$ over $R := \mathbf{Z}[a_1,\ldots, a_6]$,
$\mathcal{E}$ the Néron model of $\mathcal{E}$ over ...
user147445
3
votes
0
answers
143
views
What kind of equivalences exist between categories of characteristic $0$ and characteristic $p$?
The tilting equivalence for perfectoid algebras gives an equivalence of categories $$K\text{-perf} \cong K^\flat\text{-perf}$$
where the left-hand-side are algebras in characteristic zero and the ...
- 4,164
3
votes
0
answers
157
views
Field of definition for sheaves
What follows could be formulated for more general extensions than $\mathbb{R}\rightarrow\mathbb{C}$ but I'll stick to this particular case for now. Further, I am somewhat new to this language and I'm ...
user108998
3
votes
0
answers
279
views
Rationality of Eisenstein series g2 and g3 for elliptic curves defined over numberfields
Let $K$ be a number field and let $E/K$ be an elliptic curve. (Fix an embedding of $K$ into the complex numbers $\mathbb{C}$). Let $\eta$ be the invariant differential of $E/K$. Let $\omega_1$ and $\...
- 139
3
votes
0
answers
100
views
Tate modules of Jacobian varieties isomorphic over $\overline{\mathbb{Q}}$ but not over a number field $K$
Let $C_1, C_2$ be two curves defined over a number field $K$. Suppose that $C_1, C_2$ are isomorphic over $\overline{\mathbb{Q}}$ but not over $K$ and that $C_1(K), C_2(K) \ne \emptyset$. Then the ...
- 27k
3
votes
0
answers
73
views
Extending morphisms between semiabelian varieties
In the Storrs volume (Cornell-Silverman), Chapter 2 there is Lemma 1 stating that that if you have two semiabelian varieties over a normal scheme, then a homomorphism defined over an open dense ...
user140765
3
votes
0
answers
127
views
Maximal unramified quotient of $E[p]$ for the action of $G_{\mathbb{Q}_p}$
Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with good and ordinary reduction at an odd prime $p$.
Suppose $E[p]$ denotes the $p$-torsion points of $E$ and $G_{\mathbb{Q}_p} := \text{Gal}(\...
- 303
3
votes
0
answers
113
views
Algorithmically computing Weil cohomology groups
Fix a Weil cohomology theory. If I give you a presentation of a smooth projective scheme over an algebraically closed field, do you have an explicit algorithm for computing its cohomology groups? ...
user138661
3
votes
0
answers
197
views
Existence of regular hypersurface sections
Let $X$ be a irreducible regular projective variety over $Spec(O_K)$ for some number field $K$. Is it known that there exists at least one hypersurface over $Spec(O_K)$ such that cuts $X$ in a regular ...
- 5,901
3
votes
0
answers
144
views
Are there three ordinary elliptic curves $E$, $E_1$, $E_2$ such that $E^2 \cong E_1 \!\times\! E_2$?
Consider the elliptic curve $E\!: y^2 = x^3 + 1$ of $j$-invariant $0$ over an algebraically closed field $k$ of characteristics $p$. Let me remind that $E$ is ordinary (i.e., non-supersingular) iff $p ...
- 1,833
3
votes
0
answers
148
views
Maximum number of integral roots in degree $d$ polynomial?
Given $f(x_1,\dots,x_n)\in\mathbb Z[x_1,\dots,x_n]$ such that
Each coefficient is bound in absolute value by $B$
Degree of each variable in any monomial is bound by $d$
Total degree is $d'$
$f(x_1,\...
- 13.9k