All Questions
2,494 questions
56
votes
8
answers
8k
views
Questions about analogy between Spec Z and 3-manifolds
I'm not sure if the questions make sense:
Conc. primes as knots and Spec Z as 3-manifold - fits that to the Poincare conjecture? Topologists view 3-manifolds as Kirby-equivalence classes of framed ...
6
votes
1
answer
268
views
Hochschild cohomology of an Azumaya algebra
Let $k$ be a field. Given a commutative $k$-algebra $Z$ and an associative algebra $A$ that is Azumaya over $Z$, do we have an isomorphism of Hochschild cohomologies: $HH^*(A) \cong HH^*(Z)$?
This is ...
- 410
2
votes
1
answer
312
views
Pairwise intersecting circles in the plane
If I am looking at a collection $\mathcal{C}$ of circles $\{C_1,...,C_n\}$ all of which have some radii $\{r_1,...,r_n\}$ where $r_i\in\mathbb{R}^{+}$ for each $i \in[n]$. In $\mathcal{C}$, all the ...
- 135
5
votes
1
answer
576
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 $Q$?
- 8,998
2
votes
0
answers
458
views
Connected-étale sequence $ 0 \to G^0 \to G \to G^{\text{ét}} \to 0$ for affine finite group scheme $G$
Let $G$ be an affine finite commutative group scheme over a complete (or at least Henselian; thanks to Jason Starr for calling attention to it) local ring $R$, and assume the residue field $\kappa=R/m$...
- 6,016
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
3
votes
1
answer
259
views
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 ...
- 3,362
4
votes
1
answer
419
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
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 ...
- 333
35
votes
2
answers
2k
views
Durov approach to Arakelov geometry and $\mathbb{F}_1$
Durov's thesis on algebraic geometry over generalized rings looks extremely intriguing: it promises to unify scheme based and Arakelov geometry, even in singular cases, as well as including geometry ...
- 14.8k
4
votes
1
answer
273
views
Relation between rational Tate module and universal cover of a p-divisible group
We can associate two $\mathbb Q_p$ vector spaces to a $p$-divisible group, and I'm a little confused about the relation between these two groups. First of all, I think part of my problem is that when ...
- 1,093
0
votes
0
answers
101
views
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 $$0\to\mathfrak{a}^{-1}Λ\to\Bbb{C}\to\Ker(E^n\to E^m)\to Λ^n/A^tΛ^m\quad (1)$$ ...
- 1,541
4
votes
1
answer
234
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 \...
- 847
2
votes
0
answers
90
views
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 $...
- 895
3
votes
1
answer
552
views
Subrings of Chow rings
Let $X$ be a smooth projective variety over $\mathbf{F}_p$, call $\overline{X}$ the base change to $\overline{\mathbf{F}}_p$, and denote by $F$ the base change to $\overline{X}$ of the absolute ...
user178246
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
9
votes
2
answers
520
views
Chevalley-Warning-Ax for double covers
Let $f(x_1,\ldots,x_n)$ be a polynomial of degree $d$ with coefficients in the finite field $\mathbb F_q$ and let $V(f)\subseteq\mathbb F_q^n$ be its set of zeroes. Assume $d<n$. Then Chevalley ...
- 15.5k
5
votes
1
answer
223
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 ...
- 9,501
3
votes
0
answers
202
views
Pilloni's cohomological corrispondence factorization
I'm trying to understand the proof of Lemma 7.1.1 at page 39 of Pilloni's paper on Higher Hida and Coleman Theory for $GSp_4$. In particular, what is not clear to me is the diagram relating Serre-Tate'...
- 91
1
vote
0
answers
83
views
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(...
- 895
48
votes
4
answers
4k
views
Fermat's last theorem over larger fields
Fermat's last theorem implies that the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}$ is finite.
Is the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}^{\text{ab}}$ finite?
Here $\...
- 11.3k
3
votes
1
answer
401
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\...
- 8,998
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
2
votes
0
answers
146
views
How to compute the character of the Steinberg module for the group $\mathrm{SL}_n$ over a field of characteristic $p$?
It is known that the Steinberg representation $V$ of the group $\mathrm{SL}_n$ over a field $k$ of characteristic $p$ (maybe one needs to assume that $k$ is perfect, I am not sure) is the irreducible ...
3
votes
1
answer
769
views
Explicit defining equations for del Pezzo surfaces
Are there known explicit parametrizations of smooth del Pezzo surfaces, say of degree $5$ or higher, as surfaces cut out by equations in some projective space?
The closest I've been able to find is on ...
- 131
18
votes
7
answers
3k
views
SAT and Arithmetic Geometry
This is an agglomeration of several questions, linked by a single observation: SAT is equivalent to determining the existence of roots for a system of polynomial equations over $\mathbb{F}_2$ (note ...
- 1,368
17
votes
4
answers
2k
views
Usage of étale cohomology in algebraic geometry
I'm a student interested in arithmetic geometry, and this implies I use étale cohomology a lot. Regarding its definition, étale cohomology is a purely algebro-geometric object. However, almost every ...
- 603
3
votes
1
answer
446
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
0
answers
284
views
modularity of elliptic curves over function fields in positive characteristic
Let $F$ be a global function field over a finite field, and $E$ be an elliptic curve over $F$. Much like the number field case, it is natural to study the Galois representation on the Tate module of $...
- 685
1
vote
1
answer
208
views
$\sigma$-compactness of some locally compact Hausdorff topological groups
Is the topological group $(\mathbf{Q}_p/\mathbf{Z}_p)^{\oplus k}$, $k\ge 1$, a $\sigma$-compact topological group when endowed with its natural $p$-adic topology?
More generally, I'm looking for a ...
user480741
15
votes
2
answers
2k
views
Biggest Field Of Characteristic $p$
The Surreal nummbers, $\boldsymbol{No}$, are according to Wikipidia the biggest ordered field, and the Surrcomplex numbers are the biggest field of characteristic 0. Biggest in the sense that every ...
- 943
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
4
votes
0
answers
64
views
An analog of a BGG resolution in subregular case in positive characteristic
Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$. For every regular weight $\lambda$ of $\mathfrak{sl}_n$, we have the ...
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
7
votes
1
answer
400
views
Does perfect fraction field imply perfect residue field?
Let $A$ be a local integral domain of characteristic $p$. Let $K$ be the fraction field and let $k$ be the residue field of $A$. If $K$ is perfect, is $k$ necessarily perfect?
Thoughts:
If $A$ is ...
- 428
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^...
- 6,016
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
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
1
answer
268
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 $\ell$ ...
- 343
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
4
votes
0
answers
204
views
Explicit description of wonderful compactification for PGL_3
Let $k$ be an algebraically closed field of positive characteristics. Let $X$ be the wonderful compactification of $PGL_3$ (see for example section 6 of "Frobenius Splitting Methods in Geometry ...
- 163
2
votes
2
answers
417
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
4
votes
0
answers
296
views
de Rham Witt complex vs. de Rham complex of the Witt ring
I am reading the paper "Revisiting the de Rham-Witt complex" by Bhatt-Lurie-Mathew and I am a bit confused about the difference between $W\Omega_R^*$ and $\hat{\Omega}^*_{W(R)}$.
Let $\...
- 218
19
votes
1
answer
419
views
Varieties with the same number of $\mathbb{F}_p$-points for all but finitely many primes
If two varieties over $\mathbb{Q}$ have the same number of $\mathbb{F}_p$-points for all but finitely many primes do they have the same number of $\mathbb{F}_{p^n}$-points for all $n>1$ and for all ...
- 325
2
votes
2
answers
270
views
Why is $P^n(K)$ compact, when $K$ is a local field?
In Silverman's book AEC, question 7.6 asks to prove $E_0(K)$ has finite index in $E(K)$ for $K$ a local field. For part (a), I know the topology on $P^{n}(K)$ is the quotient topology on $K^{n+1}$, ...
- 455
6
votes
0
answers
651
views
Are crystalline cohomology obsolete?
I'm interested in $p$-adic cohomology theories now. I have learned that since de Rham cohomology behaves badly in char $p$, people invented crystalline cohomology in smooth cases and later rigid ...
- 375
4
votes
0
answers
77
views
Conjugacy of cocharacters from conjugacy of labelled diagrams
Everything to follow is over some fixed algebraically closed field $k$. Although all the definitions make sense regardless of characteristic, the meat of the question is about small positive ...
- 13k
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
9
votes
2
answers
910
views
Example of a smooth projective family of varieties in characteristic $p$ where the Hodge numbers jump
Let $\mathbb F$ be an algebraic closure of the field of order $p$. Let $S=\textrm{Spec}(\mathbb F[[z]])$ with special point $s$ and generic point $\eta$. I'm looking for an example of a smooth ...
- 790
11
votes
1
answer
2k
views
Effective weight-monodromy conjecture
$\DeclareMathOperator\Gr{Gr}$Let $G$ be the absolute Galois group of a finite extension of $\mathbb{Q}_p$ with inertia subgroup $I$, and let $V$ be an $\ell$-adic representation of $G$. Grothendieck's ...
- 15.4k