Questions tagged [arithmetic-geometry]
Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.
1,760
questions
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answers
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Polynomial values are powers of two
The initial question comes from Komal in 1999.
Namely it asks to show that for infinitely many $n$ there is a polynomial $f\in\mathbb{Q}[X]$ of degree $n$ such that $f(0),f(1),\dotsc,f(n+1)$ are ...
- 782
1
vote
0
answers
92
views
Singularities of arithmetic surface
I have a problem understanding the discussion of example 8.3.54 in Liu's Algebraic Geometry and Arithmetic Curves.
The setting is the following: We have a DVR with uniformiser $t$, characteristic of ...
- 121
3
votes
0
answers
70
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(...
- 31
1
vote
0
answers
80
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Image of the Kummer map for abelian varieties over $p$-adic local fields
The following statement might be well-known to the community: let $K$ be a finite extension of $\mathbb{Q}_p$ for some prime $p$. Let $A$ be an abelian variety over $K$. Then the image of the Kummer ...
- 1,180
3
votes
1
answer
407
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If we have a nice formula for number of points on a curve over finite fields, can we get some geometric information of the curve from the formula?
Let $p$ be a prime number and let $q = p^2$. Let $C$ be a separated scheme of finite type over $\mathbb F_q$ of dimension $1$.
If we know that for every $\alpha \in \mathbb Z_{>0}$, "the ...
- 33
4
votes
0
answers
76
views
Newton stratification in the Shimura variety for $\mathrm{GU}(1,n-1)$ over a ramified prime
Consider the PEL Shimura variety $\mathrm{Sh}_{K}$ for $\mathrm{GU}(1,n-1)$, where $K\subset G(\mathbb A_f)$ is an open compact subgroup which is small enough, and $G$ is the group of unitary ...
- 395
2
votes
1
answer
222
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Rational points on a special class of surfaces
Consider a smooth surface of the following form
$$
S = \{f(x,y,t) = p_0(t)x^2+p_1(t)xy+p_2(t)x+p_3(t)y^2+p_4(t)y+p_5(t) = 0\}\subset\mathbb{A}^3
$$
over $\mathbb{Q}$, and set
$$
U_S = \{t' \in \mathbb{...
- 71
7
votes
1
answer
405
views
Weil height vs Moriwaki height
Let $X$ be a projective veriety over a number field. After fixing an embedding into $\mathbb P^n$ (i.e. a very ample line bundle $L$), one can define the Weil height $\hat h_{L}$ by restriction of the ...
- 1,217
4
votes
0
answers
176
views
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
-1
votes
0
answers
93
views
Is there an superpolynomial integral point degree $2$ family satisfying Coppersmith's bounds?
Is there an irreducible degree $2$ bivariate curve (so of genus $0$) which satisfies Coppersmith's bounds but has superpolynomial number of integral points satisfying the bounds (allowed by Falting's ...
- 12.8k
8
votes
1
answer
375
views
Why is the category of motives generated by varieties?
I'm reading Ayoub's paper Motifs des varietes analytiques rigides, but I'm not quite familiar with motives. In this paper, he defines the category of motives to be $\mathbf{RigDM}^{\rm eff}_{\rm Nis}(...
- 81
3
votes
1
answer
155
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Stabilizers in abelian varieties are also abelian? reference request
Let $K$ be a field of characteristic $0$ (number fields is a sufficient generality), $A/K$ an abelian variety, and $X\subseteq A$ a closed reduced subscheme.
I am looking for a reference for the ...
- 3,254
3
votes
0
answers
203
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 ...
- 217
4
votes
1
answer
240
views
Rational points of bounded height on a variety
I would like to ask for some clarification on the following argument which I can not quite understand.
There is a variety $X$ of dimension $n$ over a number field with a degree two map $f:X\...
- 351
5
votes
1
answer
328
views
Does p-adic etale cohomology know the variety has ordinary reduction or not?
For a smooth proper variety $X$ over discrete valuation ring $\mathcal{O}$ of mixed characteristic $(0,p)$, let $X_K$ be the generic fibre over a generic point $\textbf{Spec} K$ and let $X_k$ be the ...
- 159
5
votes
1
answer
243
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About closed points in symmetric product schemes over a finite field
Let $k=\mathbb{F}_q$ be a finite field with $q$ elements and let $X$ be a quasi-projective $k$-scheme. I saw somewhere claims the following results (without explanation):
Let $N$ be a positive ...
- 1,614
3
votes
1
answer
131
views
On the exactness of some completed tensor products
Let $X$ be an affinoid variety over a discretely valued non-archimedean field $k$ with valuation ring $\mathcal{R}$. Fix a uniformizer $\omega$. On the section 3.2 of the paper https://arxiv.org/abs/...
4
votes
1
answer
201
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Finding the $K=\mathbb{Q}(\sqrt{6})$-rational points on the twist of $X_{0}(26)$
Let $K=\mathbb{Q}(\sqrt{6})$. I am looking to determine all $K$-rational points on the curve
$$C: y^{2}=3x^6-24x^5+24x^4-54x^3+24x^2-24x+3.$$
More precisely, $C$ is a twist of the modular curve $X_{0}(...
- 43
4
votes
0
answers
215
views
Number of homomorphisms from a group to $\mathrm{GL}_n(\mathbb{F}_q)$
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\GL{GL}$Fix a group $\Gamma$ and a positive integer $n$. Let $c(q):=\lvert\Hom(\Gamma, \GL_n(\mathbb{F}_q)\rvert$ denote the number of homomorphisms ...
- 2,041
2
votes
0
answers
141
views
Path spaces vs arc spaces
Let $X$ be a smooth projective variety over $\mathbf{C}$ and denote by $\mathcal{L}_m(X)$ the $m$-th jet space, a smooth $\mathbf{C}$-scheme representing the functor on $\mathbf{C}$-algebras
$$A\...
user178246
3
votes
1
answer
215
views
Rank of elliptic curves, parity, finiteness of Sha
$\newcommand{\Sha}{Ш}\newcommand{\alg}{\mathrm{alg}}\DeclareMathOperator\Sel{Sel}\DeclareMathOperator\rank{rank}$
Consider the elliptic curves $E$ of $j$-invariant zero that neither them nor their ...
- 41
5
votes
1
answer
163
views
Computation of the torsion of K-groups related to elliptic curves
Let $E$ be an elliptic curve over $\mathbb Q$. Let $F$ be the rational function field of $E$.
The $K_2$ group of $F$ may be described by elements in $F^\times ⊗_\mathbb{Z} F^\times$ quotiented by the ...
- 7,982
2
votes
1
answer
384
views
Galois invariants and tensor products
Consider a number field $K$ and a finite Galois field extension $L/K$. Let $E$ be an elliptic curve over $K$ and consider the abelian group
$$E(L)\otimes L^{\times}.$$
Every element $g$ in $\text{Gal}(...
user290895
2
votes
2
answers
215
views
Transition maps in trivial direct limit
If $\{X_i, i\in I\}$ is a directed system of abelian groups such that we have
$$\varinjlim_{i\in I}X_i = 0$$
is it true that for every $i$ and large enough $j\ge i$ the transition map $f_{i,j} : X_i\...
user290895
1
vote
0
answers
120
views
On closed subsets in spaces of adèlic points
Consider as in Adèlic points and algebraic closure $\mathcal{X}$ a projective and flat scheme over $\text{Spec}(\mathcal{O}_K)$, with $\mathcal{O}_K$ the ring of integers of a number field $K$.
...
user335418
6
votes
1
answer
326
views
Adèlic points and algebraic closure
Consider $\mathcal{X}$ a projective and flat scheme over $\text{Spec}(\mathcal{O}_K)$, with $\mathcal{O}_K$ the ring of integers of a number field $K$.
Let $F/K$ vary over all finite Galois number ...
user335418
4
votes
1
answer
276
views
Tate-Shafarevich groups under finite Galois field extensions
Suppose $L/F$ is a finite Galois extension of number fields. Let $E$ be an elliptic curve over $F$ and $E_L$ its base change to $L$.
Do we have that $\text{Sha}(E/F)$ is finite if and only if $\text{...
user178246
3
votes
1
answer
266
views
Galois cohomology of abelian varieties
Suppose $A$ is an abelian variety over a number field $K$ and call $M$ the maximal torsion free quotient of $A(\overline{K})$ equipped with its Galois action.
For the first Galois cohomology of $M$, ...
user322153
4
votes
2
answers
437
views
Smoothness of fibers over finite fields
Let $f:X\rightarrow Y$ be a morphism of smooth projective varieties over a finite field of characteristic different from $2$. Is there any result on the existence of a point $y\in Y$ such that $X_y = ...
user58018
4
votes
1
answer
241
views
Number of points of a quadric hypersurface over a finite field
Let $k = \mathbb{F}_q$ be a finite field with $q$ elements and $Q\subset\mathbb{P}^n_k$ a quadric hypersurface defined over $k$.
By the Chevalley-Warning theorem if $n\geq 2$ then $Q$ has a point. Is ...
- 387
3
votes
1
answer
244
views
Smooth surfaces in positive characteristic
Let $K = \mathbb{F}_p$ be a field of positive characteristic $p > 0$. Consider a surface in $\mathbb{A}^3_K$ of the following form
$$
S = \{f_1(x_0)y_0^2+f_2(x_0)y_0y_1+f_3(x_0)y_0+f_4(x_0)y_1^2+...
user114666
2
votes
0
answers
455
views
Confusion regarding Proposition 1.1 in Wiles's Fermat paper
This is from p. 459 of Wiles's Fermat paper.
Theorem: If $D_{p}$ is a decomposition group at $p$, $A$ is an Artinian local ring with maximal ideal $\mathfrak{m}$ and finite residue field $k$ of ...
- 1,489
2
votes
0
answers
67
views
Are tamely ramified representations $\widehat{\mathbb{Q}_p^\text{tr}}$-admissible?
Let $K$ be a finite field extension of $\mathbb{Q}_p$. Let $G_K$ denote the absolute Galois group of $K$, $I_K$ the inertia subgroup and $I_K^{(p)}$ the $p$-Sylow subgroup of $I_K$, i.e. the wild ...
- 413
4
votes
1
answer
237
views
Del Pezzo surfaces of degree four and complete intersections of two quadrics
Let $X = Q_1\cap Q_2$ be a complete intersection of two smooth quadrics, over a field $K$, in $\mathbb{P}^4$ with homogeneous coordinates $y_0,y_1,y_2,y_3,y_4$.
Set $Q_1 = \{F_1 = 0\}$ and $Q_2 = \{...
- 387
3
votes
0
answers
154
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 ...
4
votes
0
answers
135
views
Rational points on ramified coverings of abelian varieties
Let $K$ be a number field $A$ an abelian variety over $K$ and $f: X \to A$ a possibly ramified covering of degree $d$ with $X$ a proper variety. My question is:
Suppose that $f(X(K)) \neq A(K)$, can ...
- 1,775
8
votes
0
answers
201
views
Simultaneous rank jumping of elliptic curves over number fields
Here's something fun that I've been wondering about for a while out of curiosity. It has to do with the rank $\mathrm{rk}(E(K))$ of an elliptic curve $E$ over a number field $K$, and especially how it ...
- 9,042
7
votes
0
answers
169
views
Counting elliptic curves over finite fields with a prescribed number of points
Let $K$ be an imaginary quadratic field with ring of integers $\mathcal{O}_K$, and let $\mathcal{O}$ be an order in $K$ of discriminant $D$ and class number $h(\mathcal{O})$. Then the Hurwitz-...
- 479
4
votes
1
answer
149
views
Is Galois representation induced by semistable elliptic curve semistable?
$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\Aut{Aut}$Let $E$ be a semi stable elliptic curve. Let $\overline{\rho_\ell}: \Gal(\overline{\Bbb Q}/\Bbb Q)\to \Aut (E[l]) $ be mod $\ell$ ...
- 333
0
votes
1
answer
142
views
Maximum dimension of a simultaneous anisotropic subspace of quadratic forms over $ \mathbb{Q} $
Let $(V,q )$ be a quadratic space over $ \mathbb{Q} $. A subspace $ U $ is called totally isotropic if $ q(x) = 0 $ for all $ x \in U $ and a subspace $ U $ is called an anisotropic subspace if $ ...
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votes
1
answer
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views
Is Mazur's analogy between arithmetic and topology formal, in any sense?
I preface my question by admitting I know no algebraic geometry nor algebraic number theory. I do know some algebraic topology. I'm a student.
Recently I learned about sheaf cohomology. Then a little ...
- 223
5
votes
1
answer
186
views
Lines on quadric surfaces
Consider a smooth quadric surface $Q\subset\mathbb{P}^3$ over a field $k$. Are there natural hypotheses one can put on $k$ in order to ensure the existence of a line defined over $k$ on $Q$?
- 387
1
vote
0
answers
96
views
About Definition 2 in Roĭtman's Paper
Let $A(X)$ be the Chow group of $0$-cycles on a nonsingular irreducible projective variety $X$ over an uncountable algebraically closed field of characteristic zero.
In Definition 2 of Roĭtman's paper ...
- 407
4
votes
1
answer
204
views
integral points on elliptic curves in terms of discriminant
I am curios where in the literature was the first time written the following conjecture.
Say we have we have an elliptic curve $E$ given by the Weierstrass equation $y^2=x^3+AX+B$ with $A,B\in \...
- 782
1
vote
0
answers
72
views
Drawing a 3D object in a 3D environment, and converting to math [closed]
So I have been granted a free time and I want to work on a project but first I had to research.
As we know, lines have infinite points, and with lines, we can create infinite shapes. I want to let ...
- 11
5
votes
2
answers
494
views
Birational geometry over finite fields
I apologize in advance since probably my questions are very naive. I would like to understand some central notions in birational geometry, that are clear to me over the complex numbers, for varieties ...
user114666
5
votes
0
answers
210
views
Torsion points of an elliptic curve over number fields. Another proof of Silverman AEC theorem
I am studying the following theorem from Silverman's AEC:
I am wondering whether there exists another proof that doesn't make use of formal groups and is still valid for a number field $K$. Could you ...
- 233
7
votes
0
answers
181
views
Are unramified simple Rapoport-Zink spaces smooth?
I have read on different occasions that unramified simple Rapoport-Zink spaces are formally smooth, eg. stated in Remark 4.13 of Rapoport and Viehman's survey article. These spaces are formal schemes $...
- 395
3
votes
0
answers
118
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 ...
- 6,782
5
votes
2
answers
621
views
Embedding torsors of elliptic curves into projective space
Suppose I have a genus 1 curve $C$ over a field $k$. If $C$ has a point, then we can embed it into the projective plane by a Weierstrass equation. Now let us suppose that $C$ does not have a point (so ...
- 6,782