All Questions
2,494 questions
5
votes
1
answer
362
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 ...
- 541
4
votes
0
answers
252
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 $\...
- 305
3
votes
1
answer
443
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.
- 141
5
votes
0
answers
481
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 ...
- 4,428
4
votes
1
answer
572
views
Relation between stacky curves and "M-curves"
A tame stacky curve over a field $k$ is a geometrically connected proper smooth DM stack of dimension 1 which has a dense open substack which is a scheme, and whose automorphism group of each ...
- 9,580
0
votes
2
answers
545
views
Is $S^1=\mathbb{R}/\mathbb{Z}$ a ring? [closed]
Is there a ring structure on $S^1=\mathbb{R}/\mathbb{Z}$?
- 63.9k
2
votes
0
answers
177
views
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?
- 5,429
3
votes
1
answer
150
views
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}^\...
- 984
2
votes
0
answers
103
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 $...
- 3,730
10
votes
1
answer
1k
views
$\ell$-adic Weil cohomology theory
I have a reference or counterexample request. Suppose $k$ is a field and $\ell\neq char(k)$. There are several common references that show that $H^i_{et}(-, \mathbb{Q}_\ell )$ is a Weil cohomology ...
- 2,821
4
votes
1
answer
227
views
Compute de Rham-Witt sheaves
I am really new to this, but I am having a hard time understanding all the de Rham-Witt construction.
It seems to be really difficult to compute anything with those beasts: like I cannot find any ...
1
vote
0
answers
205
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?
- 653
1
vote
0
answers
186
views
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^...
- 769
1
vote
0
answers
98
views
$K$-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^...
- 6,006
-1
votes
1
answer
131
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-\...
- 37.7k
9
votes
0
answers
1k
views
Ample vector bundles, $H^1=0$ and global generation in characteristic $p$
This is a follow up from Ample vector bundles on curves question, in characteristic $p>0$. First, some background. It is known that in characteristic $0$, a vector bundle $E$ on say a projective ...
- 12.9k
2
votes
0
answers
136
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 ...
- 519
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 $t\in \text{Spec}(\mathscr{O}_{F,S})$, ...
- 23k
3
votes
1
answer
359
views
Description of a Shimura variety
Let $(G, X)$ be a Shimura datum and let $U \subseteq G(\mathbb A_f)$ be an open compact subgroup. By the general theory of Shimura varieties, we get a corresponding algebraic variety $Y(U)$ defined ...
- 12.9k
3
votes
1
answer
275
views
complement of "good reduction" points in p-adic shimura varieties
assume that $X$ is Siegel Shimura variety defined over $\mathbb{Z}_p$, you can take its p-adic formal completion $\mathfrak{X}$,and than take it's adic generic fiber $\mathcal{X}$ and get an adic ...
- 37k
-4
votes
2
answers
405
views
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 : $\forall b\in \mathbb N^*, \{a\times n^{b}+k;(n,k)\in\mathbb Z^2\}=A(a,b)$ ...
- 2,621
11
votes
0
answers
420
views
Good reduction of finite etale covers of abelian varieties
Let $R$ be a dvr (whose residue characteristic is zero if it helps) with fraction field $K$.
Let $A$ be an abelian variety over $K$ with good reduction over $R$. Let $X\to A$ be a finite etale ...
- 81
1
vote
1
answer
141
views
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 $...
6
votes
1
answer
771
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 ...
- 149k
7
votes
1
answer
562
views
Hasse principle for a group
$\DeclareMathOperator\PSL{PSL}$In the paper Ono - "Hasse principle" for $\PSL_2 (\mathbb Z)$ and $\PSL_2(\mathbb F)$ there's a definition of a Hasse principle for a group $G$, but I don't ...
- 12.9k
4
votes
0
answers
197
views
Inequalities between numerical invariants of nonsingular projective Varieties in positive Characteristic
It is well-known that Miyaoka and Yau-type inequalities do not hold in positive characteristic. In "a note on Bogomolov-Gieseker’s inequality in positive characteristic", however, we can ...
- 2,821
15
votes
1
answer
911
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 ...
- 5,050
4
votes
1
answer
219
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 $\mathbb{P}^3_S$.
...
- 4,492
0
votes
1
answer
125
views
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 ...
- 37k
1
vote
0
answers
180
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( \...
- 1,541
6
votes
1
answer
395
views
On the Artin-Rees Lemma for non-commutative rings
Consider a commutative noetherian ring $A$ with an ideal $I\subset A$. The Artin-Rees lemma implies that for f.g. modules $N\subset M$, the $I$-adic topology on $N$ agrees with the subspace topology ...
- 541
2
votes
1
answer
183
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{\...
- 149k
2
votes
0
answers
101
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 ...
- 63.9k
2
votes
0
answers
253
views
Künneth formula for algebraic de Rham cohomology
Let $X$ and $Y$ be finite type schemes over a field $k$, and let $H^i(X/k)$ denote the $i$-th algebraic de Rham cohomology group of $X$ over $k$. I'm interested in the extent to which a Künneth ...
- 63.9k
0
votes
0
answers
142
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 $ψ_E(I)$ ...
- 1,541
7
votes
0
answers
118
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
4
votes
0
answers
129
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 ...
- 1,856
4
votes
2
answers
864
views
When is the period of elliptic curve over the rationals transcendental?
Given an elliptic curve $E/\mathbf{Q}$, when is its period transcendental/algebraic?
- 2,821
3
votes
1
answer
296
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 $...
- 7,377
9
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 ...
- 50.7k
2
votes
0
answers
212
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, ...
- 1,071
10
votes
0
answers
295
views
Relation between Faltings height and height on moduli space
Let $E$ be an elliptic curve over a number field $K$. The difference between the semistable Faltings height $h_F(E)$ of $E$ and the height $h(j_E)$ of the $j$-invariant of $E$ can be bounded in terms ...
- 321
3
votes
1
answer
642
views
Decomposition theorem for polarized abelian varieties in positive characteristic
In characteristic zero we have the following decomposition theorem for polarized abelian varieties: it gives an isomorphism between a PPAV and a product of PPAV's of lower dimension and is valid (as ...
CommunityBot
- 1
1
vote
1
answer
229
views
Purity for proper varieties
Let $X$ be a proper, geometrically connected, geometrically integral variety over $\mathbf{F}_q$. There exists a finite field extension $k/\mathbf{F}_q$ of degree $d$ and an alteration $X'\to X_k$ ...
- 149k
3
votes
1
answer
283
views
Is the theta characteristic attached to an etale double cover of a plane quintic arising from a cubic threefold even in characteristic two?
We work over an algebraically closed field $k$ of characteristic $2$. Let $X$ be a cubic threefold realized as a conic bundle via $f: X \to \mathbb{P}^2$ (after blowing up some line). Let $C \to \...
- 679
0
votes
0
answers
124
views
How extension $\Bbb{Q}_p(\hat{E}[p])/\Bbb{Q}_p$ looks like?
Let $E/ \Bbb{Q}_p$ be an elliptic curve over $ \Bbb{Q}_p$. $\hat{E}$ denote the corresponding formal group of $E$.
I want to know what
$\Bbb{Q}_p(\hat{E}[p])/\Bbb{Q}_p$ is .
At first I tried to prove ...
- 1,541
0
votes
0
answers
135
views
Proof of $[p](x)≡x^p\operatorname{mod}p \Bbb{Z}_p$ for formal group of elliptic curve
Let $E$ be an elliptic curve over $\Bbb{Q}_p$.
Let $ \hat{E}$ be formal group of $E$.
Let $[p](x)＝x＋_\hat{E}＋・・・＋_\hat{E}x$ (add by formal group law $p$ times).
I want to know the proof of $[p](x)≡x^...
- 1,541
3
votes
1
answer
147
views
Bounded torsion of quotients of affine formal models
$\DeclareMathOperator\Sp{Sp}$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 ...
- 541
2
votes
0
answers
190
views
Relation between division point of elliptic curve and formal group of elliptic curve, $\Bbb{Q}_p(E[p])＝\Bbb{Q}_p(\hat{E}[p])$
Let $E/ \Bbb{Q}_p$ be an elliptic curve over $ \Bbb{Q}_p$. $\hat{E}$ denote the corresponding formal group of $E$.
I want to prove
$\Bbb{Q}_p(E[p])＝\Bbb{Q}_p(\hat{E}[p])$.
$ \hat{E}[p]$ denotes $p$ ...
- 1,541
1
vote
1
answer
293
views
What's a right parameter space of abelian varieties over a non algebraically closed fields?
Let $k$ be a field of characteristic not 2 or 3. Then the set of elliptic curves over $k$ can be parametrized by the affine variety $S=D(4a^3+27b^2)\subset\mathbb{A}^2_k$ via the family $E\to S$ where ...
- 149k