All Questions
974 questions with no upvoted or accepted answers
3
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Variety Isomorphism Problem for Abelian Surfaces
This is a special case of this question, where it is asked whether there exists an algorithm to determine whether two varieties are isomorphic. There, an answer by Bjorn Poonen explains how to solve ...
- 437
3
votes
0
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185
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Jacobians of pointed curves
Let $Y$ be an algebraic curve of genus $g \geq 1$ defined over a number field $K$. If $Y$ has a $K$-point, then one can define the Abel-Jacobi map which embeds $Y$ into its Jacobian variety $\text{Jac}...
- 27k
3
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322
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Tamagawa number of GL(n)
Weil's conjecture, proved by Kottwitz, states that the Tamagawa number of a semisimple, simply connected algebraic group (over a number field) is 1. For example, $SL(n)$ and induced tori. Is the ...
- 3,799
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132
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Is there a way to reduce this problem to two variables through functions coming from arithmetic?
Consider following diophantine equation in $\mathbb Z[x,y,z]$ in three integer variables $x,y,z$
$$x^2+L(y,z)x+L_1(y)L_2(z)=0$$ where $L(y,z)$ is a non-homogeneous linear polynomial in $y,z$ and $L_1(...
- 13.9k
3
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101
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Differential of p-divisible groups
Setting: Let $p$ be a prime number and let $S$ be a scheme such that $p$ is locally nilpotent on $\mathcal O_S$ ($p^N=0$). Let $X$ be a $p$-divisible group over $S$. Let $X[p^n] $ be the kernel of ...
- 309
3
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191
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Automorphy of families of motives
I have a couple of elementary questions regarding automorphy of Galois representations arising from geometric families.
Suppose we have an algebraic family of varieties over a number field, and ...
user127738
3
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238
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Visualization of hidden structures in numbers
[Please allow me a note: The way desribed below allows to depict functions $f:X^2 \rightarrow Y$ completely in two dimensions (without hiding or omitting any information). This allows for depicting ...
- 9,730
3
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174
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Extension of Galois representations from geometry
Let $K$ be a number field. Take two representations $A$, $B$ of $G_K=\mathrm{Gal}(\bar{K}/K)$ over some ring $\Lambda$ (in fact, I only consider the case $\Lambda$ is a field, usually of ...
- 1,428
3
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180
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Lefschetz trace formula over truncated Witt ring
Let $k$ be a finite field, $W_n(k)$ be its $n$-th truncated Witt ring. We have a Frobenius on $W_n(\bar{k})$ whose fixed point is exactly $W_n(k)$. Let $X$ be a finite type separated scheme over $W_n(...
- 6,622
3
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409
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Non algebraizable formal abelian schemes
I'd like to collect a good number of examples of formal schemes over $\text{Spf}(\mathbf{Z}_p)$, whose special fiber is a projective variety over $\mathbf{F}_p$, but that are not algebraizable.
If ...
user97068
3
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78
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Under what conditions are superspecial abelian surfaces isomorphic over a finite field?
Let $E_1$, $E_2$, $E_3$, $E_4$ be supersingular elliptic curves over a finite field $\mathbb{F}_{p^2}$, where $p$ is an odd prime. There is a well known theorem stating that over the algebraic closure ...
- 1,833
3
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123
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Frobenius stratification of imperfect fields
Suppose $k$ is a (non-perfect) field of characteristic $p$, and $Fr$, its Frobenius map. I’d appreciate comments and references on the structure of the fitration/stratification of $k$ given by the ...
user122285
3
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280
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Algebraic vs analytic de Rham cohomology
Let $X$ be a smooth projective variety over $\mathbf{C}$, $\Omega^{\bullet}_X$ its algebraic de Rham cohomology.
Let $p : X_{\rm an}\to X_{\rm Zar}$ the obvious morphism of sites.
We have $p^*\Omega^...
user92332
3
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answers
162
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On what varieties are the conjectures on $L$-functions true
In Peter Schneider's paper "Introduction to the Beilinson conjectures", he lists some hypothesis (conjectures) on the $L$-function associated to the pure motive $H^i(X)$ where $X$ is a smooth ...
- 2,971
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333
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Philosophical question on the role of motivic cohomology
As a physicist who has read almost nothing on the beautiful theories of motives and motivic cohomology, I have some philosophical questions on motivic cohomology. I apologize first for my naive ...
- 2,971
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124
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Characterization of a meromorphic function as arithmetic zeta function
I'd like to know if there is a (conjectural) criterion for a meromorphic function on $\mathbb{C}$ to be the zeta function of an arithmetic scheme, i.e., a statement of the form
"If a meromorphic ...
user122285
3
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answers
114
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Multiplicative structure on Deligne cohomology
Let $X$ be a smooth projective variety over the complex numbers, and $\mathbf{Z}(p)_{\mathcal{D}}$ the Deligne complex on $X$:
$$\mathbf{Z}(p)_{\mathcal{D}} : \ \ \mathbf{Z}(p)\to\mathcal{O}_X\to\...
user92332
3
votes
0
answers
166
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Cycle maps as edge maps
Given a smooth projective algebraic variety over $\mathcal{C}$, let $X$ be its associated complex analytic space.
The exponential sequence on $X$:
$$0\to\mathbf{Z}(1)\to\mathcal{O}_X\to\mathcal{O}_X^...
user119470
3
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answers
131
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Hypercohomology of topological abelian groups
Suppose $X$ is a scheme, and $A, B$ two étale sheaves of locally compact topological abelian groups. Assume there is a map of étale sheaves such that for any $f: U\to X$ étale, $\Gamma(U_{\rm ét}, A)\...
user113452
3
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answers
556
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Chern class map and the exponential sequence
Let $X$ be a smooth projective variety over the complex numbers, and
$$c^1_X : \text{NS}(X)\to H^2_{\rm Betti}(X,\mathbf{Z}(1))$$
the first cycle map to Betti cohomology. The cokernel $\text{coker}(c^...
user95222
3
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239
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A simple question on Periods Conjecture
Suppose $X$ is a variety defined over $\mathbb{Q}$, and $\omega$ is an algebraic form defined on $X$ which induces a nonzero element of the algebraic de Rham cohomology $\mathbb{H}^n(X)$. If $C$ is a ...
- 2,971
3
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221
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Artin $\ell$-adic comparison and Galois action
Let $X_0$ be a smooth projective variety defined over a number field $k$.
Let $\sigma : k\to\mathbf{C}$ be one of the finitely many field embeddings of $k$ into the complex numbers, and call $X := (...
user97068
3
votes
0
answers
517
views
Picard group finitely generated
Let $X$ be a smooth projective variety over a finite field $k$. Is the Picard group finitely generated? Equivalently, is $\text{Pic}^0(X)$ finitely generated?
(I am not assuming $k$ is separably ...
user119470
3
votes
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answers
81
views
Tate Conjecture birational invariant?
Is the Tate Conjecture stable under birational equivalence?
In particular, is the Tate Conjecture for rational varieties known?
user120812
3
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112
views
Indecomposablity in purely inseparable extensions
Let $k$ be a field of characteristic $p$ (e.g. the separable closure of $\mathbb{F}_p(t)$) and consider the extension $k(x)/k$ where $x^p\in k$ but $x$ does not. Consider a (finitely generated) ...
- 31
3
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307
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Semisimplicity conjecture
In this short note Ben Moonen proves that over fields of characteristic zero that are of finite type over their prime field, the Tate conjecture about surjectivity of cycle maps implies the semi-...
user92332
3
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0
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176
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Component groups of commutative group schemes
I'm interested in the following question.
Suppose $P$ is a smooth commutative group scheme over a global field $k$, such that $P$ is separated and locally of finite type.
Suppose, in addition, $P^0$ ...
user119470
3
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421
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Discrete vs. finitely generated subgroups of the adèles
If $U\subseteq\mathbf{R}^n$ is an additive subgroup, discrete with respect to the induced topology, then $U$ is a finitely generated abelian group.
Question.
Given a discrete additive subgroup $U\...
user113453
3
votes
0
answers
255
views
What are the easiest counterexamples to Serre-Lang over non-algebraically closed fields?
Let $k$ be a field of characteristic zero. Let $A$ be an abelian variety over $k$. Let $X\to A$ be a finite etale morphism with $X$ a connected (smooth projective) variety over $k$.
Then, choosing a ...
- 161
3
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171
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Scalar multiplication via the Kummer surface of a genus $2$ curve by $\sqrt{5}$
I hope this is a good question.
Recently I worked with genus two curves $H$ that have multiplication by $[\zeta_5]\in \text{Aut}(H)$, that is, multiplication by $e^{2\pi i/5}$. This automorphism is ...
3
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0
answers
307
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Isotrivial factors of Jacobian
Let $k$ be an algebraically closed field of positive characteristic that it is not the algebraic closure of a finite field. Fix a smooth proper $k$-curve $C$ and write $J_C$ for its Jacobian abelian ...
- 728
3
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113
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Cohomologies of $[V/GL_n]$ in characteristic $p$ for a representation $V$ of $GL_n$
Let $V$ be a representation of $G=GL_n$ (or more generally any reductive group $G$) over an algebraically closed field $\mathbb k$ of characteristic $p$. Let $[V/G]$ be the corresponding quotient ...
- 790
3
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89
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Special and generic points on Shimura varieites with the same special fiber
Let $(G,X)$ be a Shimura datum of Hodge type. We say a point $x\in X$ is generic if the Mumford-Tate group of $x$ is the same as that of $X$ and special if the Mumford-Tate group of $x$ is a torus.
...
- 337
3
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416
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The final step in the proof of Neron-Ogg-Shafarevich as in the paper of Serre-Tate
I have been reading the paper "Good Reduction of Abelian Varieties" of Serre-Tate and in particular the part where they show that the Tate module being unramified implies good reduction.
As in the ...
- 7,746
3
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215
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Why we are interested in p>3 Schoof's algorithm
In the Schoof's algorithm we are particularly interested in $char(K)>3$, where $K$ is the field. I know Schoof's algorithm is mostly used over large prime fields. Also, when we are transforming ...
- 149
3
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answers
131
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Finiteness of rational points on certain open subset of surfaces of general type
Let $X$ be a smooth surface of general type defined over a number field $k$.
The Bombieri-Lang conjecture asserts that there is a proper Zariski-closed set $Z$ in $X$ such that for any finite ...
- 31
3
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135
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How does the fundamental group of $\mathbb G_{m,S}$ depend on the base scheme
Let $S$ be an integral noetherian regular scheme, and let $X =\mathbb{G}_{m,S}$.
How to compute $\pi_1^{et}(X)$?
Note. I am only interested in the part not coming trivially from the finite etale ...
- 39
3
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148
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Given an embedding of $X$ into $\mathbb{P}^n_K$, do you get an induced embedding of any twist of it into $\mathbb{P}^n_K$?
Let $X$ be a projective algebraic curve over some number field $K$, and let $\varphi:X\hookrightarrow \mathbb{P}^n_K$ be an embedding of it (defined over $K$) into some projective space.
Now let $X'$ ...
- 460
3
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0
answers
110
views
G is p-divisible, about the affine rings of G[p]
Let $R$ be a ring and $G$ a $p$-divisible group over $R$. Since $G[p]$ is finite flat over $\text{Spec}(R)$, it is an affine (group) scheme, say $\text{Spec}(B)$. What can be said about $B$ beyond the ...
- 371
3
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97
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CM abelian surfaces (computed locally)
Let $K$ be a CM field such that $[K:\mathbb{Q}] = 4$ and let $K^+\subseteq K$ be the totally real subfield of $K$. For simplicity, assume that $K/\mathbb{Q}$ is Galois and suppose that $p\in \mathbb{Z}...
- 443
3
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128
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On Abhyankar's results cited in a paper of Manin titled "Correspondences, Motifs and Monoidal Transformations"
Consider the following from this paper "Correspondences, Motifs and Monoidal Transformations" of Manin here.
Theorem. Nonsingular three-dimensional projective unirational varieties $V$ over ...
- 113
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132
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Arithmetic version of "Attaching maps" for moduli of curves
I am looking for a reference for attaching maps of moduli of curves with marked points. Especially I would like to know whether they descend over $\mathbb{Z}$. On one hand this seems very hard to ...
- 845
3
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178
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$U_p$ operator is not compact on $p$-adic modular forms
I know that one of the reasons for introducting overconvergent $p$-adic modular forms is that the $U_p$ operator is compact on them.
Is there an easy way to see that $U_p$ is not compact on non-...
- 31
3
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answers
322
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How to Taylor series expand at the prime at infinity
Given a rational number, one can find a Taylor series expansion with respect to any $p$-adic valuation, as covered in Gouvea's introductory text on $p$-adic numbers. My question is how does one do ...
- 2,283
3
votes
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536
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splitting property of etale covering
Theorem (Global Splitting): Let $X$ be an integral separated normal scheme flat and of finite type over $\mathbb Z$. Let $\phi: Y\rightarrow X$ be a connected etale covering which splits completely ...
- 135
3
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180
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Is there a difference between the inertia stack and the universal automorphism group
Let $\mathcal M$ be a stack representing some moduli problem. Let $\mathcal X\to \mathcal M$ be the corresponding universal family.
What is the difference between the inertia stack $I\to \mathcal M$ ...
- 31
3
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0
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211
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Singularities in mixed characteristic
Let $R$ be a regular local ring in mixed characteristic. Moreover, I assume that $R$ is the local ring of a point on a smooth $\mathbb Z_p$-scheme and that $R/pR$ is regular. ($\mathbb Z_p$ is the ...
- 1,590
3
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0
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454
views
What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers $\sqrt{A}$ and the integer $A$
What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers and the integer
As we know,$\sqrt{2} = [1;2,2,2,2,…]$; while $\sqrt{14}= [3;1,2,...
- 3,104
3
votes
0
answers
239
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log structure on Witt ring and Frobenius map
I am a little confused about the log structure of Witt ring and its Frobenius map.
Let $k=\bar{\mathbb{F}}_p$, $W:=W(k)$ the Witt ring.
We know that to deal with the semistable reduction, i.e. scheme ...
- 699
3
votes
0
answers
593
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"Extended" Weil Cohomology Theories
According to Wikipedia, a Weil cohomology theory is a functor from the category of smooth projective varieties over a field $k$, to graded algebras over a field $K$ of characteristic zero, together ...
- 1,838