All Questions
2,495 questions
1
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294
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Shimura varieties which are not of abelian type but has a good modular description
Deligne's idea was that Shimura varieties should be understood as moduli space of motives(with extra structures). lot's of Shimura varieties of abelian type can be understood as moduli space of ...
- 1,093
8
votes
0
answers
300
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Which algebra is $\mathbb{F}_p \otimes_{HH^{\cdot} (\mathbb{F}_p)} \mathbb{F}_p$?
Let $HH^{\cdot}(\mathbb{F}_p)$ be the Hochschild cohomology of $\mathbb{F}_p$ over $\mathbb{Z}$, which as a $E_1$ ring is simply $\mathbb{F}_p[x]$ with $x$ in cohomological degree $2$. Then it's clear ...
- 2,356
28
votes
3
answers
5k
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Why to believe the Fargues geometrization conjecture?
In the study of the arithmetic local Langlands correspondence, there is a conjecture that was recently (in this decade) formulated by Fargues.
I can't even concisely state the conjecture so I will ...
- 417
3
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0
answers
292
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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
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139
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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 ...
- 14.2k
4
votes
0
answers
289
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Formal integration (?) in Chabauty’s method
In Mccallum, Poonen’s paper “The method of Chabauty and Coleman”,
the authors define, for the Jacobian $J$ of a geometrically connected smooth proper curve over the rational field and for $\omega \in ...
- 1,364
4
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1
answer
572
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The integral cohomology of the de Rham complex
Consider the usual de Rham CDGA $(\Omega^* Sym^*(V),d)$ for a free $\mathbb{Z}_{(p)}$-module $V$. What is known about its cohomology?
It is easy to compute ranks of primary summands in $H^*(\Omega^* ...
- 149k
4
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0
answers
130
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Castelnuovo–Mumford regularity and wedge powers in positive characterisitc
A vector bundle on $\mathbb{P}^n$ is said to be $r$-regular if
$$H^i(\mathbb{P}^n,F(r-i))=0$$ for all $i>0$.
It is always true that if $F$ is $r$-regular and $G$ is $s$-regular (both vector bundles)...
- 2,356
9
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1
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488
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What is the indecomposable decomposition of holomorphic differentials of an Artin-Schreier curve C as a Z/p-representation?
I am attempting to decompose the holomorphic differentials of an Artin-Schreier-Witt curve as a $\mathbb{Z}/p^n$-representation. This is done in Theorem 1 of Madan-Valentini Automorphisms and ...
- 3,539
41
votes
2
answers
3k
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Perfectoid universal covers
It is often said, with varying degrees of rigor or enthusiasm, that every rigid space (say over $\mathbb{C}_p$) has a pro-etale cover which is 'topologically trivial' in some sense. For example, this ...
- 21.3k
23
votes
1
answer
1k
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Tropicalization of perfectoid spaces and their tilts
Does tropicalization exist in the world of perfectoid spaces? Since it does for Huber's adic spaces, I thought it might for perfectoid spaces too, yet I can't find any explicit references so far.
...
- 21.3k
1
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0
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104
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A subgroup of the $n$-Selmer group
Let $p$ be an odd prime and for the purpose of this question let $n$ be an integer which is a power of $p$.
Let $E$ be an elliptic curve over a number field $F$.
The $n$-Selmer group, denoted by $S_n(...
- 1,283
28
votes
1
answer
984
views
Relation between Schanuel's theorem and class number equation
(Crossposted on math stack exchange: https://math.stackexchange.com/questions/4040249/relation-between-schanuels-theorem-and-class-number-equation)
It was recently brought to my attention that there ...
- 5,434
16
votes
2
answers
2k
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Applications of integral p-adic Hodge theory
What are some geometric applications of integral p-adic Hodge theory (as opposed to rational p-adic Hodge theory)? Answers which understand Hodge theory as the study of Galois stable $\mathbb{Z}_p$-...
- 21.3k
3
votes
3
answers
423
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Are there characteristic-dependent Betti numbers in characteristic not equal to two?
Let $\Delta$ be a finite simplicial complex on $n$ vertices. Let $S = \mathbb{k}[\mathbf{x}]$ be a polynomial ring over a field $\mathbb{k}$ in $n$ variables and $I$ be the Stanley-Reisner ideal of $\...
- 150
150
votes
2
answers
22k
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What is a Frobenioid?
Since there will be a long digression in a moment, let me start by reassuring you that my intention really is to ask the question in the title.
Recently, there has been a flurry of new discussion ...
- 64k
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
4
votes
2
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337
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Formal models of rigid discs of any radii
sorry if this is a too vague. For $K$ some non-archimedean discretely valued field the rigid disc of radius 1 $\mathrm{Sp} \, K\langle T \rangle$ has a formal model $\mathrm{Spf} \, K^{\circ} \{T\}$. ...
- 321
2
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1
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303
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Is the reduction of an absolutely irreducible plane curve still irreducible except for the finite number of cases?
Help me please.
Let $k$ be an algebraically closed field (I am mainly interested in $k = \overline{\mathbb{Q}}, \overline{\mathbb{F}_q}$). Consider a plane curve $C \subset \mathbb{A}^2$ of degree $d$ ...
- 30.5k
2
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0
answers
411
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Equidimensional Morphism
I am reading the paper "Relative Cycles and Chow Sheaves" due to Suslin and Voevodsky. Here we have the following definition:
Definition 2.1.2.
A morphism of schemes $p:X\rightarrow S$ is ...
- 519
2
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0
answers
286
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Moduli interpretation and Ogg's notation for the cusps on modular curves
In Ogg's paper "rational points on certain elliptic modular curves", the author says, using Ogg's notation for cusps,
that for fiexed $d$, if $(y, N) = d$, then for any $x$ satisfying $(x, y,...
- 1,364
10
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0
answers
581
views
Elkies' theorem on supersingular primes and inertness
Suppose $E_{/\mathbb{Q}}$ is an elliptic curve over $\mathbb{Q}$ without CM. By Elkies' theorem, there exist infinitely many primes $p$ for which $E$ has supersingular reduction at $p$.
Question. Is ...
- 271
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0
answers
264
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Can arithmetic geometry accelerate the search for rational points in high dimensions?
There are several ideas in arithmetic geometry that can help in proving the absence of rational points as well exhibiting rational points on algebraic curves.
I am aware there are some obstructions (e....
3
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1
answer
203
views
Density of quadratic points on a hyperelliptic curve
We fix a binary form $F \in \mathbb{Z}[x,y]$ with non-zero discriminant and degree $d = 2g+2$, and consider the hyperelliptic curve
$$C_F: \displaystyle z^2 = F(x,y).$$
We say that a point $(x,y,z)$ ...
- 149k
2
votes
0
answers
198
views
Finding rational points via birational map
Let $C$ be an affine curve given by $p_C(x,y)=0$ where $$ p_C=2x^3y + 2xy^3 +x^3 + y^3 + 5x^2y + 5xy^2 + 2x^2 + 2y^2 + 2x^2y^2 + 2xy $$
and let $\overline{C}$ denote the projective closure of $C$. For ...
- 3,065
7
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444
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Status of the conjectured vanishing of Bloch-Kato H^2
There is a folklore conjecture that $\operatorname{Ext}^2$ vanishes in the category of geometric $p$-adic Galois representations (i.e. representations that are unramified almost everywhere and de Rham ...
- 35.5k
21
votes
1
answer
2k
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Two conjectures by Gabber on Brauer and Picard groups
In a paper I need to make reference to two conjectures by Gabber, from
Ofer Gabber, On purity for the Brauer group, in: Arithmetic Algebraic Geometry, MFO Report No. 37/2004, doi:10.14760/OWR-2004-37
...
- 35.5k
2
votes
0
answers
161
views
Abelian variety corresponding to a vector space
I would like to know what the following statement means:
"Let $B_t$ be the Abelian subvariety in $J_t$ corresponding to the $\mathbb{Q}$-vector subspace $H^1(C_t,\mathbb{Q})_{van}$ in the space $...
- 3,065
5
votes
0
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151
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Reduction theory of higher dimensional algebraic varieties
If $X$ is a nonsigular curve over a number field $K$, one can obtain several arithmetic models of $X$. Namely, we can construct an arithmetic surface $\mathcal X\to\operatorname{spec} O_K$, such that $...
- 1,237
3
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1
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372
views
What is the "geometric height" mentioned by Moriwaki?
Let $K$ be a finitely generated field over $\mathbb{Q}$ of transcendence degree 1, and take a curve $C$ over a number field $k$ such that $k(C)=K$. In "Arithmetic height functions over finitely ...
- 149k
2
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1
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187
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Is every sufficiently general monic quartic rational square infinitely often?
Let $f(x)=x^4+b_3 x^3+ b_2 x^2+b_1 x + b_0$.
Let $g(x)=x^4 f(1/x)$. Let $C : g(x)=y^2$.
$C$ is birationally equivalent to $f(x)=y^2$.
The constant coefficient of $g(x)$ is $1$ since $f$ is monic
and $(...
- 149k
9
votes
0
answers
212
views
Belyi-like results over function fields of characteristic zero
In the paper Unifying themes suggested by Belyi's theorem from 2011, the following question is raised:
Let $X$ be a projective non-singular curve over the function field $K:=\overline{\Bbb{Q}}(t)$. ...
- 3,599
7
votes
1
answer
667
views
Results and conjectures on bounds on degrees of isogenies
given an isogeny between two abelian varieties $\varphi: A\rightarrow B$ (everything definied over a number field $K$), we can factor $\varphi$ through a multiplication-by-$n$-endomorphism on $A$ and ...
- 4,394
8
votes
1
answer
172
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Non-degenerate points on a Jacobian surface
Let $C$ be a (hyperelliptic) genus $2$ curve over a number field $K$ with a $K$-rational Weierstrass point $\infty$. We embed $C$ in its Jacobian $J$ via $\infty$.
Question: Is there a quadratic ...
- 149k
2
votes
0
answers
130
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Iwasawa theory over function fields - How do eigenvalues vary in $\mathbb Z_\ell$ towers?
Consider a tower of curves $\dots \to C_n \to C_{n-1} \dots \to C_1$ over $\mathbb F_q$ where $C_n: f(x^{\ell^n},y) = 0$.We can look at the multiset $S_n$ of eigenvalues of the Frobenius $\sigma_q$ on ...
- 7,746
16
votes
2
answers
503
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Number of height-limited rational points on a circle
Consider origin-centered circles $C(r)$ of radius $r \le 1$.
I am seeking to learn how many rational points might lie on $C(r)$,
where each rational point coordinate has height $\le h$.
For example, ...
- 12.3k
8
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0
answers
196
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Simple abelian varieties of $\mathrm{GL}_2$ type with positive rank and large dimension
$\DeclareMathOperator\GL{GL}$I would like know if there are known constructions of simple abelian varieties of $\GL_2$ type of arbitrarily large dimension and positive Mordell-Weil rank, whose rank is ...
- 63.9k
2
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0
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128
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How do elliptic units generate the module of Euler systems over abelian extensions of imaginary quadratic fields?
I am trying to undesrtand the analogy between the Euler systems over abelian extensions of the rationals and the Euler systems over abelian extensions of imaginary quadratic fields.
As Soogil Seo ...
- 99
5
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0
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154
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Curves of genus 0 over a DVR determined by fibers?
Closely related is this question.
Suppose $R$ is a DVR with fraction field $K$ and residue field $k$ (say finite), and $S = \mathrm{Spec}(R)$.
I am interested in regular, proper, flat schemes $X \to S$...
- 1,338
4
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1
answer
428
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p-torsion in the Picard group of a regular projective curve
Let $K$ be a field of characteristic $p>0$ and $C$ a regular projective geometrically integral curve over $K$.
If $C$ is smooth, then the connected component ${\rm Pic}^0_C$ of the Picard scheme of ...
- 19.6k
5
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184
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Given a point $P$ on a genus-$1$ curve over $\mathbb{Q}_p$, is there an $R$ such that $2 \mid [P - R]$ and $x(\overline{P}) \neq x(\overline{R})$?
Let $p \in \mathbb{Z}$ be prime, and let $f \in \mathbb{Z}_p[x]$ be a quartic polynomial with nonzero discriminant. Let $C/\mathbb{Q}_p$ be the genus-$1$ curve with affine equation $y^2 = f(x)$. Let $\...
- 53
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1
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943
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Automorphisms over finite field that do not lift to an automorphism in characteristic zero
My main question is the following: is there an automorphism of the affine space $\mathbb{A}^n$ (automorphism of an algebraic variety) defined over a finite field which does not lift to an automorphism ...
- 7,722
4
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1
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392
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Unique way to topologise finite algebra over Huber ring
Let me start with the following Lemma.
$\textbf{Lemma}$ Let $A$ be a Tate ring, and let $f\colon A\to B$ be a finite $A$-algebra. Then there is a unique way to topologise $B$ turning it into a ...
- 466
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2
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1k
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Why do the $2$-Selmer ranks of $y^2 = x^3 + p^3 $ and $y^2 = x^3 - p^3 $ agree?
I was playing around with sage, when I found that the Mordell-Weil ranks (over $\mathbb{Q}$) of the elliptic curves $y^2=x^3+p^3$ and $y^2=x^3-p^3 $ almost always agree, for $p$ prime. The first few ...
- 582
6
votes
1
answer
432
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Arithmetic zeta functions of products and fibrations
Suppose $X$ and $Y$ are schemes of finite type over $\mathbb{Z}$.
How is the arithmetic zeta function of their product, $\zeta_{X \times Y} (s)$, related to their individual zeta functions, $\zeta_X(s)...
CommunityBot
- 1
0
votes
0
answers
77
views
On four non-cocyclic integral points on ellipse
Let $(x_i,y_i)\in\mathbb Z^2$ at $i\in\{1,2,3,4\}$ be four (not five and I assume an unique curve exists because of Diophantine constraints and not geometric constraints) non-cocyclic integral points ...
- 13.9k
3
votes
0
answers
197
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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
9
votes
0
answers
607
views
Geometric meaning of twist
It is sometimes the case that a Galois representation or a motive acquires a desirable property only after a twist by a character, usually a Tate twist. The latest example of this I have come across ...
- 1,267
2
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0
answers
190
views
Computing monodromy groups of curves over function fields
Suppose I consider a hyperelliptic curve given by an equation such as $y^2 = x^{n} + tx + 1$ or some variation on this (where $t$ is a parameter on $\mathbb P^1$ and this curve is really a surface ...
- 7,746
0
votes
0
answers
148
views
The scheme of intersection points of algebraic plane curves defined over a number field
This question concerns two related but different notions of "plane curves", where we consider both curves in the projective plane $\mathbb{P}^2$ and the affine plane $\mathbb{A}^2$.
Given ...
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