All Questions
2,494 questions
1
vote
0
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180
views
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 ...
- 6,016
25
votes
8
answers
3k
views
Relatively concise English expositions of the proofs of the various Weil conjectures
Where can I find relatively concise (i.e. not excessively wordy and waxing poetic about history and intuitions and such, doesn't spend an eternity carefully developing various parts of the theory of ...
user97565
16
votes
2
answers
687
views
Deligne's theorem on finite flat group schemes and generalizations
Recall Deligne's theorem that for a finite flat commutative group scheme $G$ of order $n$, the multiplication by $n$ map $[n]: G \to G$ is the zero map.
I have seen the proof a few times but I can't ...
- 7,746
3
votes
1
answer
309
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
1
answer
230
views
Special value of Hecke $L$ function
Let $E:y^2＝x^3-x/ \Bbb{Q}(i)$ be elliptic curve and $L(E,1)$ be a special value of $L$ function of $E$ at $1$.
Let $L(ψ,1)$ be value at $1$ of Hecke $L$ function with respect to Hecke character $ψ$, ...
- 1,541
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
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{\...
- 254
6
votes
1
answer
507
views
Degree-2 étale covers of curves in characteristic 2 vs torsion points on the Jacobian
It can be found, at Hartshorne exercise 4.2.7 for example, that in the case where $\operatorname{char}(k) \neq 2$ we have a nice correspondence between etale degree 2 covers of a curve $C$ and 2-...
- 679
1
vote
1
answer
237
views
A question involving the three-dimensional Kloosterman sum
Sorry to disturb. I have a question involving the three-dimensional Kloosterman sum, which needs some help from the experts here.
For any $\alpha, \beta, \gamma \in \mathbb{Z}$ and $q\in \mathbb{N}^+$,...
- 563
1
vote
1
answer
94
views
Reference for auto-duality of nearly ordinary deformations associated to Hida families
suppose we have a $p$-stabilized newform $f$ of classical level $\Gamma_0(p)$; then there is a unique ordinary deformation into a Hida family $\mathbf{f}$ defined over a finite flat extension $R/\...
- 2,054
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
5
votes
1
answer
343
views
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 ...
- 1,906
2
votes
1
answer
261
views
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{...
- 8,998
-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-\...
- 13.9k
6
votes
1
answer
351
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
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 ...
- 1,364
3
votes
0
answers
189
views
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 $...
- 7,746
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 $...
- 247
3
votes
1
answer
385
views
Overconvergent modular forms and the level at $p$
I am a little bit confused about the basic theory of overconvergent modular forms, so here is a question that I think will be straightforward for those who know the theory but would help me a lot.
The ...
- 241
9
votes
1
answer
473
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}(...
- 91
1
vote
0
answers
96
views
Are the irreducible components appearing in the resolution of singularities of a Hilbert modular surface defined over $\mathbb{Q}$?
It seems to me that this is claimed in van der Geer's "Hilbert modular surfaces" on p. 245 at the beginning of XI.2 (without justification).
My current state of belief/knowledge:
The ...
- 93
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
2
votes
1
answer
515
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
3
votes
0
answers
127
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 ...
- 1,541
2
votes
0
answers
177
views
How do characters of representations in cohomology depend on the (positive-characteristic) field?
The following sentence appears in Jantzen - Representations of algebraic groups, 2nd edition, p. x, where $G$ is a reductive group over an algebraically closed field $k$, $B$ is a Borel subgroup, $T$ ...
- 13k
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?
- 705
8
votes
2
answers
704
views
Isomorphic endomorphism algebras implies isogenous (for abelian varieties over finite fields)?
$\newcommand{\F}{\mathbb{F}}
\newcommand{\End}{\mathrm{End}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}$
I would like to know if the following is true:
Proposition A : Let $A_1, A_2$ ...
- 1,742
52
votes
8
answers
28k
views
Roadmap for studying arithmetic geometry
I have read Hartshorne's Algebraic Geometry from chapter 1 to chapter 4, so I'd like to find some suggestions about the next step to study arithmetic geometry.
I want to know how to use scheme ...
3
votes
1
answer
328
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 ...
- 71
4
votes
1
answer
370
views
On a consequence of the Gerritzen-Grauert Theorem
Let $K$ be a local field of characteristic zero and $X$ an affinoid rigid space over $K$. Let $U\subset X$ be an affinoid subdomain, and consider a finite family of points $\{p_{1},\cdots, p_{n}\}\...
- 541
2
votes
0
answers
147
views
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^...
- 930
47
votes
1
answer
1k
views
Summing infinitely many infinitesimally small variables makes sense in algebra
There is an identity $e^x=\lim_{n\to \infty} (1+x/n)^n$, and I always thought it is a purely analytic statement. But then I discovered its curious interpretation in pure algebra:
Consider the ring of ...
- 3,772
2
votes
2
answers
358
views
Section of conic bundle
Suppose $X$ is a smooth projective surface with a dominant morphism $\pi:X \rightarrow \mathbb{P}^{1}$ over a field $k$, where all the fibres of $\pi$ are conics (i.e. a conic bundle). If $\pi$ admits ...
- 481
5
votes
1
answer
361
views
diagonal cubic hypersurfaces
At the end of
https://encyclopediaofmath.org/index.php?title=Cubic_hypersurface#References
it is stated that the diagonal cubic hypersurface
$$
\sum_{i=0}^{2m+1} a_i x_i^3 = 0, m\ge 2
$$
(and ...
- 519
6
votes
1
answer
643
views
Classification of simple Lie algebras over finite fields
Classification of simple (or simple-restricted) Lie algebras over algebraically closed fields in positive characteristic is studied for a long time. Today, we know all finite-dimensional simple (or ...
- 145
19
votes
1
answer
2k
views
Perfectoid approach to resolution of singularities in char $p$
Since perfectoid techniques have built a bridge between char $0$ and char $p$ worlds, it is conceivable that they can be applied to resolution of singularities in char $p$ using their successful ...
- 199
4
votes
0
answers
174
views
Spencer complex and de Rham Complex
in those lectures notes written by Claude Sabbbah: https://perso.pages.math.cnrs.fr/users/claude.sabbah/livres/sabbah_nankai110705.pdf
there is the proposition 1.4.4 where he says that there is a ...
- 385
4
votes
1
answer
252
views
Finding the $K=\mathbb{Q}(\sqrt{6})$-rational points on the twist of $X_{0}(26)$
Let $K=\mathbb{Q}(\sqrt{6})$. I am looking to determine all $K$-rational points on the curve
$$C: y^{2}=3x^6-24x^5+24x^4-54x^3+24x^2-24x+3.$$
More precisely, $C$ is a twist of the modular curve $X_{0}(...
- 83
0
votes
1
answer
125
views
Why does $[I](P)＝0$ ($P\in E$) imply $[\psi(I)](P)＝0$ ? ($\psi$ is Hecke character of elliptic curve)
Let $K$ be a imaginary quadratic field, $R_K$ be ring of integers of $K$, and $E/K$ be elliptic curve which has CM over $K$.
Let $\psi_E$ be Hecke (Grössencharakter) character of $E/K$.
Let fix prime ...
- 1,541
2
votes
1
answer
324
views
Rank of $\mathbb{Z}_{p}$-module $H_{et}^{i}(X,\mathbb{Z}_{p}(r))$
I want to ask the following question.
Let $X$ be a smooth projective variety of dimension $d$ over $p$-adic field $k$ ( i.e. finite extension of $\mathbb{Q}_{p}$). Is it true that etale cohomology $H_{...
- 629
11
votes
4
answers
1k
views
Explicit large finite fields in characteristic $2$
Every finite field of characteristic $2$ ist given by $\mathbb{F}_2[x]/P(x)$ for some irreducible polynomial $P\in \mathbb{F}_2[x]$.
For small degree, a simple algorithm gives a way to find $P$. Is ...
- 7,722
3
votes
1
answer
167
views
How to compute Selmer set?
Let $X$ be an affine variety and $G$ an affine algebraic group (for example $\operatorname{PGL}_n$). How do I compute the Selmer set
$$ \operatorname{Sel}_\zeta(\mathbb{Q},G) = \{\tau \in H^1(\mathbb{...
3
votes
0
answers
424
views
Generalization of the Castelnuovo-Severi inequality
The Castelnuovo-Severi says that given curves $X,Y,Z$ over a perfect field $k$ and nonconstant morphisms $f:X\rightarrow Y$ and $g:X \rightarrow Z$ which do not both factor through any morphism $X\...
- 221
2
votes
1
answer
361
views
Lie algebroid in algebraic geometry
When I did net-surfing at home, I met some geometric backgrounds of Lie algebras and encountered the concept of Lie algebroids. In differential geometry, a Lie algebroid seems to be defined as ...
- 145
4
votes
1
answer
322
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 = \{...
- 8,998
12
votes
1
answer
440
views
Arithmetic groups and integral points of integral structures
If $\mathbf{G}$ is an algebraic group defined over $\mathbb{Q}$, a subgroup of $\mathbf{G}(\mathbb{Q})$ is arithmetic if it is commensurable to $\mathbf{G}(\mathbb{Q}) \cap \operatorname{GL}_n(\mathbb{...
- 2,050
8
votes
2
answers
632
views
Motivation of the construction of $p$-adic period rings
Let $B$ be either $B_{\text{dR}}$ or $B_{\text{crys}}$. For a $\mathbb{Q}_p$-representation $V$ of the absolute Galois group $\mathrm{Gal}(\overline{K}/K)$ of a $p$-adic field $K$ (a finite extension ...
- 1,428
3
votes
0
answers
120
views
Resolving the "wild" singularities of $\mathbb A^n/C_n$
Let the cyclic group on $n$ elements, $C_n$, act on $\mathbb A^n$ by permuting the co-ordinates (over a field $k$). If $n \neq 0 \in k$, we can resolve the singularities of $X = \mathbb A^n/C_n$ by ...
- 7,746
3
votes
0
answers
123
views
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
3
votes
0
answers
96
views
Descent obstruction of an open curve in an elliptic curve
Let $E$ be an elliptic curve over a number field $k$, and for an extension $K/k$ we denote by $E_K$ the base change $E \times_k K$. By fixing an embedding $k \hookrightarrow \mathbb{C}$, the etale ...
- 895