Questions tagged [arithmetic-geometry]
Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.
2,044
questions
3
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points on non-hyperelliptic curves of genus 3
I have a non-hyperelliptic curve $C$ of genus 3 and I'm interested in finding the $K$-rational points on the curve with $K$ a fixed imaginary quadratic number field.
As $C$ is a non-hyperelliptic ...
- 31
1
vote
0
answers
184
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Vanishing of the local étale cohomology sheaf (?)
Let $X$ be a locally noetherian regular scheme, and let $Z$ be a closed subscheme of $X$ whose codimension $d > 0$ at every point.
Let $U$ be the complement of $Z$ in $X$.
For a sheaf $\mathscr{F}$ ...
- 185
2
votes
0
answers
209
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Is the ring of power series with $p$-adic coefficients Huber?
I have been reading the Berkeley lectures and got stuck with this question. Let $\mathbb{Q}_p [[t]]$ denote the ring of power series with $p$-adic coefficients. Is there a natural topology (e.g. the ...
- 103
3
votes
1
answer
271
views
Projective dimension of group ring
Assume that $G$ is a group and $R$ is a p.i.d. What can we say about the projective dimension of $R[G]$? For example can we say that this dimension is at most $1$ for reductive groups? (I think if $...
- 1,043
4
votes
0
answers
132
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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
4
votes
1
answer
489
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On the local properties of rigid analytic varieties
Let $K$ be a non-archimedean field complete with respect to a discrete valuation with ring of integers $\mathcal{R}$, uniformizer $\pi$ and residue field $k$. Consider an affinoid analytic $K$-variety ...
1
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0
answers
81
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Are there known situations where this weaker form of the section conjecture holds?
Let $k$ be a number field. The section conjecture predicts that for a (smooth geometrically connected) hyperbolic curve over $k$, the profinite Kummer map $\kappa :X(k) \rightarrow \mathscr{J}_{\pi_1(...
- 865
5
votes
2
answers
288
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Generalization of $j(E) \in \overline { \Bbb{Z}}$ to abelian varieties of arbitrary dimension
Let $E/ \Bbb{C}$ be an elliptic curve which has complex multiplication over a number field $K$.
Then it is widely known that $j(E) \in \overline { \Bbb{Z}}$.
What is the known generalization of this ...
- 1,405
16
votes
0
answers
913
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Finiteness for motivic local systems
Let $X$ be a smooth proper algebraic curve over $\mathbb{C}$. Say a complex local system $\mathbb{V}$ on $X$ is motivic if there exists a dense Zariski-open subset $U\subset X$, and a smooth proper ...
- 22.2k
2
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answers
90
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Reconciling two notions of finite descent obstructions
Let $k$ be a number field and $X$ a smooth geometrically connected variety over $k$. We denote by $H(k,X)$ the set of sections $G_k \rightarrow \pi_1(X)$, where $G_k$ is the absolute Galois group of $...
- 865
1
vote
0
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158
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Maximal unramified extension and algebraic closure of $\operatorname{Frac}(\widehat{A_{\mathfrak{m}_A}})$
$\DeclareMathOperator\trdeg{trdeg} \DeclareMathOperator\Frac{Frac} $
Let $k$ be an algebraically closed field of characteristic $0$ and $K$ a function field over $k$. let $(A, \mathfrak{m}_A)$ a ...
- 6,000
3
votes
0
answers
120
views
Isogeny of elliptic curve over positive characteristic $p$ which does not come from characteristic $0$
Let $K$ be quadratic imaginary field. Let $E$ be an elliptic curve which has CM over $R_K$
($R_K$ is ring of integers of $K$).
According to SIlverman's ''ADvanced topics in the arithmetic of elliptic ...
- 1,405
3
votes
1
answer
163
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On the stability of having a normal formal model under finite extensions of the base field
Let $K$ be a finite extension of the $p$-adic numbers with valuation ring $\mathcal{R}$ and uniformizer $\pi$. Consider a smooth and connected rigid $K$-variety $X=Sp(A)$ and assume that the affine ...
8
votes
1
answer
314
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On actions of finite groups on adic spaces
Let $K$ be an algebraically closed complete non-archimedean field and consider the unit ball $\mathbb{B}^{1}_{K}=Sp(K\langle t\rangle)$. We have an action of $\mathbb{Z}/2\mathbb{Z}$ on $\mathbb{B}^{1}...
5
votes
0
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483
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Perfect algebraic spaces on a paper of Xinwen Zhu
I have problem reading Xinwen Zhu's paper Affine Grassmannians and the geometric Satake in mixed characteristic about perfect algebraic spaces in Section A.1.
Let $k$ be a perfect field of ...
- 151
0
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0
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98
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Identity component of $\mathrm{Ker}(E^n→E^m)$ in the advanced topics in the arithmetic of elliptic curves
$\DeclareMathOperator\Ker{Ker}$Silverman's "Advanced topics in the arithmetic of elliptic curves", p.115 reads $$0\to\mathfrak{a}^{-1}Λ\to\Bbb{C}\to\Ker(E^n\to E^m)\to Λ^n/A^tΛ^m\quad (1)$$ ...
- 1,405
17
votes
1
answer
2k
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How does the cohomology of the Lubin-Tate/Drinfeld tower fit into categorical p-adic local Langlands?
In conjecture 6.1.14 of this article, Emerton-Gee-Hellmann formulate the p-adic local Langlands conjecture, which posits the existence of a fully faithful functor from (the appropriate derived ...
- 3,269
3
votes
0
answers
177
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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 $...
- 7,646
2
votes
0
answers
141
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Can we say anything about the zeros and Galois group of the polynomial $(x^p-a)^{p^2}-p^{p^2+1}x+p^{p^2} a=0$?
Let $p$ be an odd prime number and $\mathbb Q_p$ be the $p$-adic number field. Let $K=\mathbb Q_p(a)$ be the extension by $a=p^{\frac{p^2+1}{p^3-1}}$.
Consider the polynomial $f(x)=(x^p-a)^{p^2}-p^{p^...
- 870
4
votes
0
answers
229
views
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 $\...
- 153
6
votes
1
answer
421
views
Why do Chern forms show up in Arakelov geometry?
Let $X$ be a regular, projective flat scheme over $\Bbb{Z}$, let $\bar{L}$ be a hermitian line bundle on $X$. In order to define the height of an integral closed subset $Y$ we define it on closed ...
- 183
5
votes
1
answer
347
views
On the noetherianess of some subalgebras of an affinoid algebra
$\DeclareMathOperator\Sp{Sp}$Let $X=\Sp(A)$ be a connected smooth affinoid rigid space over a discretely valued non-archimedean field $K$. Let $\mathcal{R}$ be a valuation ring of $K$, and fix a ...
5
votes
0
answers
424
views
What does Colmez's conjecture tell us?
There is the well known conjecture of Colmez, which describes the logarithmic derivative of the $L$-function of a character via the periods of CM-abelian varieties. Equivalently it describes the ...
- 4,246
0
votes
2
answers
471
views
Is $S^1=\mathbb{R}/\mathbb{Z}$ a ring? [closed]
Is there a ring structure on $S^1=\mathbb{R}/\mathbb{Z}$?
- 625
2
votes
0
answers
175
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Why are they called reductive groups? [duplicate]
The reductive groups play a central role in the Langlands correspondence. Why are these groups called reductive? Does this name suggest something conceptual about these groups?
- 625
3
votes
1
answer
142
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Selmer groups and fppf cohomology
Let $\mathcal{O}$ be a Dedekind domain and $K = \mathrm{Frac}(\mathcal{O})$ its field of fractions. Let $E / K$ be an elliptic curve and $\mathcal{E} / \mathcal{O}$ its Neron model and $\mathcal{E}^\...
- 3,301
2
votes
0
answers
97
views
Selmer ranks unbounded?
Is it known if the Selmer ranks of quadratic twist families are unbounded?
Suppose that $E/K$ is an elliptic curve defined over a number field. For each quadratic extension $F/K$ I can form the twist $...
- 3,301
1
vote
0
answers
197
views
Projective scheme over the integers
Let $X$ be a projective scheme over $Spec(\mathbb{Z})$. Let $X_{p}$ be the reduction at $p$ of $X$. If for any prime $p$, $X_{p}$ is normal, can we deduce $X$ is normal? Or any counterexamples?
- 541
1
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0
answers
143
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Moduli interpretation for integral models of PEL Shimura variety at parahoric level?
Kottwitz has built canonical integral models for a large family of PEL Shimura varieties, associated to a certain reductive group $G$ over $\mathbb Q$, when the structure level has the form $K = K_pK^...
- 687
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0
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96
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$K$-ranks of some algebraic groups in Lubotzky's "Discrete groups, expanding graphs and invariant measures"
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$Let $G$ be a semisimple algebraic group and $K$ any field. Then
the $K$-rank of $G$ is the maximal rank $r$ of a $K$-splitting
torus $T \cong (K^...
- 6,000
-1
votes
1
answer
129
views
Does this quadratic system admit an integral or a rational solution?
Let $a,b$ be coprime and say $0<a<b<2a$.
Consider the quadratic system:
$$\alpha\delta-\beta\gamma=1$$
$$(\alpha^2-(\alpha\delta+\beta\gamma))a^2b+\beta^2b^3+(2\alpha\beta-\beta\delta)ab^2-\...
- 13.7k
2
votes
0
answers
127
views
Is there the specialisation map of etale K theory?
Take a smooth proper morphism of schemes $X\to S$. Fix a point $t\in S$ and a point $s\in \overline\{s\}$. For a prime $l$ which is invertible in $S$, is there the natural specialization map of etale ...
- 489
5
votes
1
answer
662
views
A regular, geometrically reduced but non-smooth curve
Can anyone give an example of a projective, regular, geometrically reduced but non-smooth curve ?
Of course, the base field should be imperfect.
In Exercise 4.3.22 of Qing Liu's book Algebraic ...
- 600
0
votes
0
answers
253
views
Geometric meaning of cusps/component labels in Katz-Mazur book
In Katz-Mazur book "Arithmetic moduli of elliptic curves" there is a short section (see image below) regarding the cusp-labels and component-labels.
The set of cusps labels intuitively ...
- 299
4
votes
1
answer
208
views
Does a smooth relative curve $X/S$ embed into $\mathbb{P}^3_S$?
Suppose that $\pi:X\to S$ is a smooth projective morphism of relative dimension 1. If $S$ is the spectrum of an algebraically closed field, then it is known that $X$ embeds into $\mathbb{P}^3_S$.
...
- 1,102
-4
votes
2
answers
394
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Do these irrationals exist?
An irrational $a$ verifies : $\{a\times n+k;(n,k)\in\mathbb Z^2 \}$ is dense in $\mathbb R$.
If you take $a$ universe then : $\forall b\in \mathbb N^*, \{a\times n^{b}+k;(n,k)\in\mathbb Z^2\}=A(a,b)$ ...
- 3,795
1
vote
1
answer
141
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Power series corresponding to $[a]\in \operatorname{End}(E)$ ($a \in R_K$) can be expressed as $[a](t)=at+\text{(term higher than degree $2$)}$?
Let $K$ be an imaginary quadratic field and $E/K$ be an elliptic curve which has complex multiplication on $K$.
Let $R_K$ be ring of integers of $K$.
Let $ \hat{E}$ be its formal group of $E$.
Take $...
- 1,405
15
votes
1
answer
854
views
A possible gap in Faltings note to prove the Tate conjecture for finitely generated field over $\mathbb{Q}$
Qeustion:
Given a Lie algebra $\mathfrak{g}$ over $\mathbb{Q}_\ell$ with an ideal $\mathfrak{g}^O$ and a subalgebra $\mathfrak{h}$,
such that $\mathfrak{g}=\mathfrak{g}^O+\mathfrak{h}$.
Now given a ...
- 178
18
votes
0
answers
2k
views
Cycles in algebraic de Rham cohomology
Let $F$ be a number field, $S$ a finite set of places, and $X$ a smooth projective $\mathscr{O}_{F,S}$-scheme with geometrically connected fibers. For each point $t\in \text{Spec}(\mathscr{O}_{F,S})$, ...
- 22.2k
0
votes
1
answer
124
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Why does $[I](P)＝0$ ($P\in E$) imply $[\psi(I)](P)＝0$ ? ($\psi$ is Hecke character of elliptic curve)
Let $K$ be a imaginary quadratic field, $R_K$ be ring of integers of $K$, and $E/K$ be elliptic curve which has CM over $K$.
Let $\psi_E$ be Hecke (Grössencharakter) character of $E/K$.
Let fix prime ...
- 1,405
2
votes
0
answers
97
views
Number of points of parabolic Springer fibres for general reductive groups
My question is the same as this post but for an arbitrary reductive $G$ instead of just $\mathrm{GL}_n$.
Let $G$ be a connected split reductive group over a finite field $k$.
Let $P$ be a parabolic ...
- 2,681
2
votes
1
answer
179
views
Generation of trace fields of Frobenii on local systems
Let $\overline{X}$ be a smooth proper curve over $\mathbb{F}_q$, for some $q$, $S$ a collection of $\mathbb{F}_q$ points of $\overline{X}$, and set $X=\overline{X}-S$.
For a rank $n$ $\overline{\...
- 222
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votes
0
answers
138
views
Why is image of prime ideal under Hecke (Grossencharacter) character is prime element of the local field?
Let $K$ be a imaginary quadratic field, and $E/K$ be elliptic curve which has CM over $K$.
Let $ψ_E$ be Hecke(Grossencharacter) character of $E/K$.
Let fix prime ideal $I$ of $K$.
Then, why $ψ_E(I)$ ...
- 1,405
7
votes
0
answers
117
views
Upper bound on $\#\{p \leq B :\#E(\mathbb{F}_p) \equiv a \mod b\}$ for large $b$
Let $E$ be a fixed elliptic curve over $\mathbb{Q}$. Is there a good upper bound on $\#\{p \leq B :\#E(\mathbb{F}_p) \equiv a \mod b\}$ when $b$ is large (maybe around $\sqrt{B}$)? I don't mind ...
- 270
1
vote
0
answers
171
views
Homomorphism of formal group of elliptic curve corresponding to its endomorphism
Let $E$ be an elliptic curve and $ \hat{E}$ be its formal group.
Rubin's lemma $3.7$ in 'Elliptic curves with complex multiplication' reads
For arbitrary $φ∈End(E)$, there exists unique $φ(t)∈End( \...
- 1,405
4
votes
0
answers
125
views
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 ...
- 1,744
3
votes
1
answer
351
views
The Weil restriction of a simple algebraic group
Let $F$ be a number field, $G$ an $F$-simple affine algebraic group.
Then is the Weil restriction $\operatorname{Res}_{F/\mathbb{Q}} G$ $\mathbb{Q}$-simple?
I couldn’t find any references.
- 185
3
votes
1
answer
263
views
$p$-power torsion of semiabelian variety
Let $K$ be a finite extension field of $\mathbb{Q}_p$. Let us consider a semiabelian variety $G$ defined over $K$, i.e there exists an extension of an abelian variety $B$ and a torus $T$ defined over $...
- 247
10
votes
3
answers
2k
views
Simple motivation to study arithmetic geometry
Is there a simple-to-understand diophantine equation (in the sense that it's easy to explain to a child) that has a positive integer solution, but to prove that such a solution exists and to find it ...
- 473
2
votes
0
answers
188
views
Intermediate extensions of pure perverse sheaves (BBD 5.4.3)
I am working my way through "Faisceaux pervers" by Beilinson, Bernstein and Deligne and can't wrap my head around Corollary 5.4.3. To me it seems that one of the hypotheses is extraneous, ...
- 1,021