All Questions
2,494 questions
12
votes
1
answer
944
views
Comparing singular cohomology with algebraic de Rham cohomology
Let $X$ be a smooth projective variety over a number field $K$. Then there are two cohomology groups we can attach to $X$: the algebraic de Rham cohomology group
$H^k_{\text{dR}}(X/K), $
which is a ...
- 1,949
5
votes
1
answer
160
views
Derived subalgebra of a restricted Lie algebra
Let $L$ be a restricted Lie algebra over a field of characteristic $p>0$. It is well known that the commutator subalgebra $[L,L]$ is not necessarily restricted (that is, closed under the $p$-map).
...
- 630
47
votes
3
answers
5k
views
"Cute" applications of the étale fundamental group
When I was an undergrad student, the first application that was given to me of the construction of the fundamental group was the non-retraction lemma : there is no continuous map from the disk to the ...
- 7,320
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 ...
- 649
6
votes
1
answer
786
views
B. W. Jordan's thesis on arithmetic of Shimura curves
I'm looking for Bruce W. Jordan's thesis: On the diophantine arithmetic of Shimura curves. Thesis, Harvard University, 1981.
I could not find the pdf at the following site.
https://www.math.harvard....
- 1,364
1
vote
1
answer
115
views
About the power of numbers primes distribution
Let $r>0$, $p\neq q$ two primes numbers and $A=\{(m,n)\in\mathbb N^2; |p^m-q^n|\leq r\}$.
Is it true that $A$ is a finite set?
- 4,073
1
vote
0
answers
88
views
Extensions in the category $F\text{-Isoc}(X)$
Let $X$ be a smooth affine scheme over a finite field $k$, let $W(k)$ denote its Witt ring, and by $K$ its fraction field.
Let $F\text{-Isoc}(X/K)$ denote the category of convergent $F$-isocrystals on ...
- 2,907
3
votes
0
answers
215
views
Global class field theory and closure of unit groups
I'm looking for a reference for the following facts from global class field theory that I found without proofs. I will state them as questions, just in case I get the statement wrong. We fix $K$ a ...
- 85
2
votes
0
answers
354
views
Square-zero extensions mod $p^n$
$\DeclareMathOperator\LMod{LMod}\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Sp{Sp}$A square-zero extensions of rings is, conceptually, a map of rings $R \to A$ such that any two elements in the ...
- 68
2
votes
0
answers
250
views
Maximal p-extension and pro-p extension
I’m studying Iwasawa theory and I meet some questions Thanks a lot for your help.
Q_1: About terminology $p$-extension.
I find many reference use maximal $p$-extension or maximal abelian p-extension ...
- 319
1
vote
1
answer
218
views
Dimension of Zariski closure of a closed point of generic fiber
Let $S= \operatorname{Spec} A$ be a local Dedekind scheme of dimension $1$, (eg spectrum of localization at a prime of the ring of integers of a number field). Let $s \in S$ it's unique closed point ...
- 6,006
4
votes
1
answer
183
views
Primes of bad reductions for quotients of elliptic curves
Let $E$ be an elliptic curve over a number field $K$ and $p$ a prime. Suppose that $E$ has a $K$-rational $p$-torsion, which gives the short exact sequence $0\to\mathbb{Z}/p\to E[p]\to\mu_p\to0$ of ...
- 1,428
0
votes
1
answer
160
views
Another generalisation of euclidean division on integers
Let $n \in\mathbb N^*$.
What are all the surjective functions $f: \mathbb N \rightarrow \{0,...,n-1\}=E $ such that there exist functions $g,h$ from $E^2$ to $E$ with:
$\forall (m,k) \in\mathbb N^2,f(...
- 4,073
3
votes
0
answers
171
views
Large 2-part of Tate–Shafarevich group over $\Bbb{Q}$ with small number of prime factor of discriminants
$\newcommand{\Sha}{\operatorname{Sha}}$Let $E/\mathbb{Q}$ be an elliptic curve, and let $\Sha(E/\mathbb{Q})$ denote the Tate–Shafarevich group of $E/\mathbb{Q}$. It is known that the 2-primary ...
- 1,541
11
votes
1
answer
580
views
Do people prefer working on $\mathrm{GSp}$ and $\mathrm{GU}$ rather than $\mathrm{Sp}$ and $\mathrm{U}$, and why?
I am a new learner of Iwasawa theory and currently reading the famous paper by Skinner-Urban in 2014, and the following-up works by many other people. When reading these papers, I found that some ...
- 639
3
votes
2
answers
412
views
Infinitely many elliptic curve with twist rank more than $1$ in specific case
Let $E/\Bbb{Q}$ be an elliptic curve. Let $D$ be a square free negative integer.
It is conjectured that 50％ of twist of elliptic curve $E_D$ has rank $0$ and $50％$ has rank $1$.
But is some particular ...
- 1,541
2
votes
1
answer
277
views
Understanding an example of abelian-type Shimura varieties
I'd like some help understanding the idea of abelian-type Shimura varieties. In paricular, I understand an abelian-type Shimura datum $(G,X)$ generally parameterizes non-rational Hodge structures ...
- 2,054
1
vote
0
answers
145
views
Multiplicity and the perfect projective line
Let $\mathbf{F}_p$ be the field with $p$ elements, and $X = (\mathbf{P}^1_{\overline{\mathbf{F}}_p})^\text{perf}$ the inverse perfection of the projective line over $\mathbf{F}_p$.
Let $\Gamma$ be the ...
- 85
6
votes
1
answer
305
views
Definition of modular curve associated to $\Gamma(N)$
For a positive integer $N$, we define $$\Gamma(N)=\big \{ \begin{bmatrix} a & b \newline c & d\end{bmatrix}\in \operatorname{SL}_2(\mathbb{Z}): \begin{bmatrix} a & b \newline c & d\end{...
- 831
2
votes
0
answers
275
views
Is there any relation between Berkovich spaces over $\Bbb Z$ and Arakelov theory?
As I understand it, both Arakelov geometry and Berkovich geometry over $\Bbb Z$ (or $\mathcal O_K$) consider geometric objects that contain in some sense information about both archimdean and ...
- 804
6
votes
1
answer
167
views
$\mathbb{Q}$-rank of the space of angles of pythagorean triples
A pythagorean triple is a triple of integers $(a,b,c)$ with $a^2 + b^2 = c^2$. Given a triple, $(a/c, b/c)$ is a point on the unit circle, so we may associate to it the normalized angle
$$\theta_{a,b} ...
- 7,547
2
votes
0
answers
127
views
Classification of restricted Lie algebras of reductive groups
$\DeclareMathOperator\Lie{Lie}$Let $G/K$ be a reductive group over a field $K$. In characteristic $0$ the Lie algebra is invariant under base change of fields, so to understand $\Lie(G)$ it is enough ...
- 461
5
votes
0
answers
261
views
Equations for conic del Pezzo surfaces of degree one
Let $X$ be a del Pezzo surface of degree one over a field $k$ of characteristic not $2$ equipped with a conic bundle $\pi: X \rightarrow \mathbb{P}^1$. By Theorem 5.6 of this paper, $X$ admits a ...
- 241
0
votes
1
answer
152
views
Almost Pell type equation
Consider the following Diophantine equation
$$
2x^2-Ny^2 = -1.
$$
where $N$ is an integer. Is there any result expressing the values of $N$ for which the above equation admits an integral solution?
- 8,998
7
votes
1
answer
310
views
Faithful representations of integral models
I am reposting a question that I had asked on stackexachage three weeks ago.
Let $G/\mathbb{Q}$ be a connected reductive group, and $\mathcal{G}/\mathbb{Z}$ be an integral model (i.e. flat affine ...
- 831
7
votes
0
answers
174
views
Failure of injectiveness of maps between cotangent spaces of abelian varieties
Let $p$ be a prime and $K$ a finite extension of $\mathbb Q_p$ with ramification index $e$. Let $\mathcal O_K$ be the ring of integers of $K$ and $k$ its residue field and the unique maximal ideal. ...
- 2,224
2
votes
1
answer
326
views
Geometric Interpretation of absolute Hodge cohomology
$\quad$Let $\mathcal{Sch}/\mathbb C$ denote the category of schemes over $\mathbb C$. For an arbitrary $X\in\mathcal{Ob}(\mathcal{Sch}/\mathbb C)$, Deligne in his Article defined a polarizable Hodge ...
user508433
2
votes
1
answer
252
views
Images of smooth schemes under lci morphisms
Let $S$ be a Noetherian scheme, $f : X\to S$ a smooth quasi-projective morphism, $g : X\to Y$ a morphism of finite type, and $h : Y\to S$ a smooth projective morphism with $h\circ g =f$.
Can we say ...
user505967
4
votes
1
answer
435
views
Étale group schemes and specialization
If $A$ is an abelian variety over a finite field $\mathbf{F}_q$, then $A(\mathbf{F}_q)$ (resp. $A(\overline{\mathbf{F}}_q)$) is a finite (resp. infinite torsion) group, but $A(\mathbf{F}_q(t))$ is a ...
user501361
0
votes
0
answers
129
views
Extend line bundle on regular curve to it's regular model
Let $S$ be the spectrum of an excellent discrete valuation ring with field of fractions $K$ and $C$ be a proper integral regular curve over $K$.
Assume, $C$ admits a proper regular flat model $\...
- 6,006
14
votes
2
answers
2k
views
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 ...
- 847
0
votes
1
answer
205
views
Rational points on genus 3 curves defined by short equations
(a) Find all pairs of rational numbers $(x,y)$ such that
$$
y^3-y=x^4-x.
$$
(b) Find all pairs of rational numbers $(x,y)$ such that
$$
y^3+y=x^4+x.
$$
If not a complete answer, I would be happy to ...
- 7,182
2
votes
0
answers
259
views
The group of the modular automorphisms of the Shimura curves
Let $B$ be a rational indefinite division quaternion algebra, $(X,G)$ the Shimura datum associated with $B$ (i.e., $X$ is the upper half plane and $G(R) = (B \otimes_\mathbb{Q} R)^*$ for a ring $R/\...
- 1,364
20
votes
0
answers
408
views
Ado's theorem and the reduction to positive characteristic
The synopsis: proofs of Ado theorem in positive characteristic are simple, and in characteristic $0$ are difficult. Can one infer the characteristic $0$ case from the positive characteristic case?
The ...
- 2,934
1
vote
0
answers
162
views
Motivic complex on arithmetic schemes
If we believe the finite generation of motivic cohomology for regular arithmetic schemes like $X$ then we can see that (using Quillen-Lichtenbaum)for infinitely many primes $l$ we have an isomorphism ...
- 5,901
1
vote
1
answer
238
views
When $E_D:y^2=x^3+17D^2x$ has even rank?
Let $E:y^2=x^3+17x$ be an elliptic curve.
In this MO page(Infinitely many elliptic curve with twist rank more than $1$ in specific case), Nulhomologous's and other's comment reads from parity ...
- 1,541
2
votes
0
answers
134
views
Isom-functor for generalized elliptic curves is representable
I am studying Deligne-Rapoport's 'Les Schémas de Modules de Courbes Elliptiques'. The following excerpt is from the proof of Theorem 2.5, Chapter III, page DeRa-61,
(page DeRa-61) (*) For $C_i$, ...
- 21
4
votes
1
answer
292
views
Discrepancy in the calculation of $2$-Selmer group by Magma and LMFDB
The result of LMFDB claims (https://www.lmfdb.org/EllipticCurve/Q/1640/c/1 )
that (2-part of) Tate-Shafarevich group $\mathrm{Sha}(E/\Bbb{Q})$ of elliptic curve $y^2=x^3-8747x-314874$ has order $16$. ...
- 1,541
3
votes
1
answer
303
views
Leading constant in Batyrev-Tschinkel's refinement of Manin conjecture
Background: Let $X$ be a Fano variety over number field $K$, where its anticanonical bundle $K_X^{-1}$ is ample. Let $i: X \to \mathbb{P}^n$ be the anticanonical embedding, where $K_X^{-m} \cong i_*O(...
- 267
2
votes
0
answers
63
views
Fibre functors of the category $F\text{-Isoc}(X)$
Let $X$ be a smooth affine scheme over a finite field $k$. Denote its Witt ring by $W(k)$, and the fraction field of its Witt ring by $K$. Let $F\text{-Isoc}(X)$ denote the category of convergent $F$-...
- 2,907
2
votes
1
answer
214
views
Cohomology of $\mathcal{O}_{F^S}[\frac{1}{S}]^\times$
$\quad$Let $F$ be a number field, $\ell$ a prime, and $S$ a finite set of places of $F$ including all Archimedean places and places over $\ell$.
$\quad$Then we have
$$\mathrm{H}^1\left(G_{F,S},\...
user508433
3
votes
1
answer
245
views
$\mathbf{Z}$-points of quasi-projective schemes
Let $U\subset\mathbf{P}^n_{\mathbf{Z}}$ be an open subscheme such that the smooth morphism $U\to\text{Spec}(\mathbf{Z})$ is surjective. Suppose $U(\mathbf{Q})\neq\varnothing$ and $U(\mathbf{Z}_p)\neq\...
- 85
79
votes
12
answers
13k
views
Is there a high-concept explanation for why characteristic 2 is special?
The structure of the multiplicative groups of $\mathbb{Z}/p\mathbb{Z}$ or of $\mathbb{Z}_p$ is the same for odd primes, but not for $2.$ Quadratic reciprocity has a uniform statement for odd primes, ...
- 118k
1
vote
1
answer
339
views
Cohomology of singular curves
Suppose $X$ is a singular quasi-projective curve over the complex numbers, and $X'$ is a good nonsingular compactification of a resolution of singularities $Y\to X$. Let $D$ be the complement of $Y$ ...
user505967
4
votes
1
answer
409
views
The notion of morphisms between two moduli problems in Katz-Mazur
I am reading Katz-Mazur Arithmetic Moduli of Elliptic Curves, and have some questions about the notion of morphisms between two moduli problems.
What is the proper definition of morphisms between two ...
- 161
28
votes
3
answers
5k
views
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 ...
user140765
4
votes
1
answer
201
views
Modular interpretation of the stalks of modular curves
One may see the modular interpretation of (points of) modular curves in the very first course on modular forms and modular curves. I am wondering if it is well-known that modular interpretation of the ...
- 1,428
2
votes
1
answer
170
views
Automorphism of positive characteristic field
Suppose $K$ is a field with $\text{char}(K) \geq 0$. Let $L$ be a cyclic extension of $K$ with degree $2n$. We consider the generator $\sigma$ of the Galois group $\text{Gal}(L/K)$.
I am interested in ...
- 923
6
votes
0
answers
271
views
What, if anything, do we hope and expect to understand about (classical) Galois groups?
I was reading Franz Lemmermeyer's introduction to Fermat's Last and Wiles' Theorem, where he states
Galois representations $\rho_p : G_\mathbb Q\rightarrow GL_2(\mathbb Z_p)$ are used for studying ...
- 982
3
votes
0
answers
227
views
Tate isogeny theorem over varieties?
Let $X$ be a nice scheme, $\pi:E\to X$ an elliptic curve, and $\ell$ a prime invertible on $K$. Then we can consider the "Tate module" $(R^1\pi_*\mathbb{Z}_{\ell})^\vee=\hbox{''}\varprojlim\...
- 371