All Questions
2,495 questions
14
votes
0
answers
933
views
Relation between Igusa tower and $p$-adic modular forms
As the title suggests, my question is devoted to understand (and maybe get some good references) the relation between the Igusa tower for a modular curves and $p$, or maybe $T$-adic modular forms. I ...
- 28.2k
2
votes
0
answers
184
views
Points on Galois conjugate curves
Let $C$ be a curve defined over a number field $F$ and let $C^{\sigma}$ be its Galois conjugate obtained by applying $\sigma \in Gal(F/\mathbb{Q})$ to the coefficients defining $C$. What can we say ...
- 81
2
votes
1
answer
231
views
Lifting of flat lci maps
Suppose $R$ is a Noetherian ring and $I$ a nontrivial ideal of $R$, and $A_0\to B_0$ a finite faithfully flat lci map of smooth $R_0 := R/I$-algebras.
We fix a smooth $R$-algebra $A$ lifting $A_0$ and ...
- 1,144
8
votes
0
answers
367
views
References for Yoichi Miyaoka's work around Fermat's Last Theorem
Apparently, Yoichi Miyaoka made a serious attempt to prove FLT in 1988. See the following question.
What were the main ideas and gaps in Yoichi Miyaoka's attempted proof (1988) of Fermat's Last ...
8
votes
3
answers
1k
views
Further reading in algebraic geometry
I recently finished reading W. Fulton's "Algebraic Curves" and also attended a lecture series on moduli spaces and am interested in learning about them as well. I looked for a few books to ...
- 739
3
votes
1
answer
121
views
Non-empty closed subsets with empty special fiber
Let $R$ be a dvr and $U\to \text{Spec}(R)$ an affine smooth $R$-scheme with non-empty special fiber $U_0$.
Let $Z\subset U$ be a closed subset. Assume the intersection of $Z$ with $U_0$ is empty.
Is $...
- 54
1
vote
0
answers
194
views
Brauer-Manin obstruction and affine curves
I'm looking for references that can justify to what extent is the following statement true:
Statement. Let $X$ be a smooth geometrically integral curve over a number field $k$. Then the Brauer-Manin ...
- 895
8
votes
0
answers
688
views
An Azumaya algebra from a vector bundle, and a construction of Belov-Kanel and Kontsevich
Let $S/k$ be a scheme over a perfect field $k$ of characteristic $p>0$.
In Automorphisms of the Weyl Algebra, Belov-Kanel and Kontsevich write down the map
$$\alpha: H^0(\Omega^1_{S/k}/d\mathcal O) ...
- 9,650
5
votes
1
answer
534
views
Ordinary abelian varieties and Frobenius eigenvalues
Say $A_0$ is an ordinary abelian variety over ${\mathbf{F}}_q$. Call $\mathcal{A}$ the canonical lift of $A_0$ over $R := W({\mathbf{F}}_q)$. It carries a lift of the $q$-th power map on $A_0$. We ...
- 149k
4
votes
0
answers
112
views
An example of a projective surface of general type where we know all of the rational points
I am looking for an example of a surface $X \subset \mathbb{P}^3$ defined over $\mathbb{Q}$ with the following properties: 1) $X(\mathbb{Q}) \ne \emptyset$; and 2) we know all of the elements $X(\...
- 27k
2
votes
0
answers
180
views
Reference request for an English translation of a book of Tate
In this ongoing program, Professor Mahesh Kakde said that the best reference for learning about Stark and Gross-Stark conjecture is this book of John Tate. But this book is in French. Is there any ...
- 333
4
votes
0
answers
189
views
If the Frobenius endomorphism of a characteristic $p$ ring is epimorphic, is it surjective?
MO question 19282 is about integral epimorphisms of commutative rings, and a counterexample is given to surjectivity. What about the case of the Frobenius endomorphism of a commutative, characteristic ...
- 4,492
4
votes
0
answers
233
views
Structure of $A(L)/A(K)$
Let $L/K$ be an extension of number fields and $A/K$ an abelian variety (or an elliptic curve, or a modular abelian variety).
Then what can we say about the structure of $A(L)/A(K)$ (or of $A(L)_{\...
- 1,364
6
votes
0
answers
233
views
Rational points on varieties whose anticanonical bundle is nef but not ample
Is the following plausible?
"If $X$ is a variety over $\mathbf{Q}$ whose anticanonical bundle $L$ is nef but not ample, there is a number field $K$ such that $X(K)$ contains an infinite set of ...
- 19.2k
1
vote
0
answers
193
views
How to construct a sheaf on the infinitesimal site from a stratified module
Let $X\to S$ be a morphism of schemes.
Proposition 2.11 of the book "Notes on crystalline cohomology" by Berthelot and Ogus states that a stratified $\mathcal{O}_X$-module $(E,\{\...
- 218
3
votes
0
answers
426
views
Closed immersion hitting all the $\mathbb{Q}$-points
Let $i:X\to Y$ be a closed immersion of smooth projective varieties over $\mathbb{Q}$.
Assume that $Y(\mathbb{Q})$ is infinite and $X(\mathbb{Q})\to Y(\mathbb{Q})$ is surjective. Also assume that $X$ ...
CommunityBot
- 1
21
votes
1
answer
2k
views
Cohomology of Shimura varieties and coherent sheaves on the stack of Langlands parameters
In Zhu's Coherent sheaves on the stack of Langlands parameters theorem 4.7.1 relates the cohomology of the moduli stack of shtukas to global sections of a certain sheaf on the stack of global ...
- 103
9
votes
1
answer
387
views
Cotangent spaces of finite flat group schemes in short exact sequences
Fix $(R,m)$ a complete DVR of mixed characteristic $(0,p)$ with perfect residue field, and consider finite flat commutative group schemes $G = Spec(A)$ over $R$. One can associate a differential ...
- 23.8k
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 ...
CommunityBot
- 1
5
votes
1
answer
430
views
Comparison of weight filtration on cohomology of complex manifold
Let $X$ be a smooth scheme of finite type over $\mathbb{Z}$ (or let's say a finitely generated $\mathbb{Z}$ algebra). To each prime $p \in \mathbb{Z}$ we can consider the $\mathbb{F}_p$ variety $$X_{\...
- 35.2k
1
vote
0
answers
143
views
A specific Diophantine equation related to the congruent number question
Let $n$ be an odd square free natural number. J.B. Tunnel in his 1983 paper, showed that a number $n$ is congruent, if and only if the number of triples of integers satisfying $2x^2+y^2+8z^2=n$ is ...
- 159
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 ...
- 10.9k
2
votes
0
answers
187
views
Does the map $\theta[1/p]: A_{\mathrm{inf}} \otimes \mathbb Q_p \to \mathbb C_p$ split?
This question might be very elementary to someone who knows p-adic hodge theory/perfectoid stuff etc.
Recall that $\mathbb C_p = \hat{\overline{\mathbb Q_p}}$ and $\mathbb C_p^\flat$ is it's tilt. We ...
- 7,746
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{...
- 37k
5
votes
1
answer
408
views
On universally closed morphisms of reduced schemes
In this question I'd like to examine some properties of universally closed morphisms.
The question is self-contained. It can also be seen as a follow-up to this question.
Let $R$ be a discrete ...
CommunityBot
- 1
2
votes
1
answer
324
views
Computing $H^1$ with coefficients in a torsion-free abelian group
Let $k$ be a number field and denote by $H^i(k,-)$ the Galois cohomology functor $H^i(\mathrm{Gal}(\bar{k}/k),-)$. Let $X$ be a smooth geometrically integral curve over $k$. One can easily show that ...
- 21.4k
4
votes
1
answer
327
views
Detecting closed immersions on fibers
Let $R$ be a dvr and $f : X\to S$ a universally closed morphism of $R$-schemes.
Assume $X$ and $S$ are $R$-flat and universally closed.
If the special fiber of $X\to S$ is a closed immersion, is $X\...
CommunityBot
- 1
2
votes
1
answer
308
views
An example of Serre on the cohomology of some CM elliptic curves
Let $E$ be the elliptic curve over $\mathbf{Q}_3$ with Weierstrass equation $y^2 = x^3-x$.
It has complex multiplication by $\mathbf{Z}[\sqrt{-1}]$, with $\sqrt{-1}$ acting as $(x,y)\mapsto(-x,iy)$.
...
- 149k
8
votes
1
answer
441
views
Minimal vs characteristic polynomial of geometric Frobenius
Assume $X$ is a smooth projective variety over $\overline{\mathbf{F}}_p$ and fix a prime $\ell\neq p$.
Let $F_i$ be the geometric Frobenius on $\ell$-adic cohomology
$$H^i_{\rm ét}(X,\overline{\mathbf{...
- 37k
1
vote
0
answers
224
views
Cohomological dimension of continuous étale cohomology of finitely generated fields
Given a finitely generated field $F$ with prime field $k$, we assume $k$ is finite, of characteristic $p$. Fix a prime $\ell$ invertible in $k$.
In the discussion right after [K, Lemma 2.3], the ...
CommunityBot
- 1
3
votes
1
answer
245
views
Can non-geometrically reduced reduced subschemes happen for reductive groups?
The title is meant to be punchy, but also a tongue-in-cheek acknowledgement of the prevalence of ‘reduce’-derived words in this area. (Unfortunately, I overlooked the fact that the question in the ...
- 13k
4
votes
0
answers
144
views
How often is the rank of J_0(p)^- zero
As mentioned in this answer there is a conjecture by
Kimball Martin that, formulated slightly informally, has the following special case.
Conjecture:
On average $J_0(p)$ has 2 simple components when ...
- 2,224
0
votes
1
answer
224
views
Notation on a Mumford's paper
I am reading the paper " D. Mumford. Rational equivalence of $0$-cycles on surfaces. J. Math. Kyoto Univ. 9 (1969) 195 - 204" and I do not understand a notation in bottom of page 197. It ...
CommunityBot
- 1
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 ...
CommunityBot
- 1
4
votes
0
answers
262
views
de Rham Bloch-Ogus theory in positive characteristic
In their famous paper Gersten's conjecture and the homology of schemes, one of the results that Bloch and Ogus prove is that the second page of the coniveau spectral sequence for $X$ smooth over a ...
- 2,054
1
vote
0
answers
73
views
Would the iterated finite abelian descent obstruction equality hold for curves?
Let $X$ be a smooth projective geometrically integral variety over a number field $k$. We begin with some established notions in the theory of descent obstruction to the local-global principle, ...
- 895
3
votes
3
answers
1k
views
Topology on $p$-adic period ring in an article by Fontaine
Fix a $p$-adic field $K$ with perfect residue field $k.$ Let $\mathcal{C}$ be the completion of the algebraic closure of $K,$ and let $$R = \varprojlim \mathcal{C}/p,$$ where the transition maps in ...
- 63.9k
6
votes
2
answers
1k
views
Topology on $p$-adic period rings in an article by Fontaine, part II
This is a follow-up to this question. See that question for background and relevant notation.
In the answer to that question, it is claimed, if I understand the answer correctly, that a basis of ...
- 9,283
4
votes
0
answers
117
views
Laurent polynomials of the form $p(x)\cdot p(x^{-1})$
Let $R$ be a commutative, associative ring with $1$ and let $\tau: R[x^{\pm 1}]\to R[x^{\pm 1}]$ be the $R$-algebra involution $\tau(x)=x^{-1}.$ Then it is natural to ask which elements of the ...
- 2,390
3
votes
1
answer
200
views
Interpolation of scheme-theoretic endomorphisms of closed fibers
Let $S$ be a scheme and $f : X\to S$ be an $S$-scheme. This question asks for examples of maps of sets $X(S) \to X(S)$ that do not come from an $S$-scheme endomorphism of $X$, but that, roughly, ...
- 9,061
3
votes
1
answer
332
views
Generalization of torsion points on Jacobian of genus 2 over finite fields (with respect to the theta divisor)
Let $J(C)$ be the jacobian of a hyperelliptic curve $C$ of genus 2 defined over finite field $\mathbb{F}_q$. Let $\Theta$ be the image of the curve on the Jacobian under the embedding $P \mapsto P - \...
- 984
1
vote
0
answers
377
views
Bad primes for algebraic curves
I am confused with general notion of integral models of algebraic varieties. Let us focus on, say, algebraic curves.
If $X$ is a not necessarily projective algebraic curve over a number field $K$, is ...
- 1,428
10
votes
0
answers
217
views
Are the nonnegative rationals diophantine with only two quantifiers?
Definition: A subset $D\subseteq \mathbb{Q}$ is diophantine if it is the projection of the zero set of a polynomial, i.e. there exists a polynomial $f\in\mathbb{Q}[X,Y_1,\dots,Y_n]$ for some $n$ such ...
3
votes
0
answers
106
views
A uniform version of Bashmakov's theorem for elliptic curves
Let $E/\mathbb Q$ be an elliptic curve. Serre's open image theorem is the statement that the image of the Galois group $G_{\mathbb Q}$ into $GL_2(\mathbb Z/n\mathbb Z)$ by it's action on the torsion ...
- 7,746
4
votes
0
answers
1k
views
Variational Hodge Conjecture vs Hodge Conjecture
Motivation.
Let us state the following version of Grothendieck's variational Hodge Conjecture:
Conjecture (VHC). Let $\mathcal{X}\to S$ be a proper smooth map of smooth algebraic varieties over $\...
2
votes
1
answer
359
views
Flatness of $\Omega^1_{X/S}$
Let $f : X\to Y$ be a syntomic morphism of locally Noetherian $S$-schemes (i.e. flat and lci) and assume $X$ and $Y$ are smooth over a locally Noetherian scheme $S$.
Q1: is $\Omega^1_{X/Y}$ a flat $\...
- 16.2k
2
votes
1
answer
121
views
Checking formality of a perfect complex Zariski-locally
Let $X$ be a locally Noetherian scheme and $K^{\bullet}$ a perfect complex of $\mathcal{O}_X$-modules. We say $K^{\bullet}$ is "formal" if it is quasi-isomorphic to the complex $\bigoplus_{n}...
- 1,347
3
votes
1
answer
312
views
Geometric line bundles on the Tate curve
Let $E_q$ be the rigid analytic Tate elliptic curve over a complete algebraically closed non-archimedean field $K$ of mixed characteristic $(0,p)$, with parameter $q\in K^{\times}$ with $|q|<1$.
...
- 16.2k
1
vote
0
answers
192
views
p-adically completed Hodge-completed de Rham algebra
Let's look at this paper on page 40.
Let $K$ be a finite extension of $\mathbb Q_p$. One can define the derived de Rham algebra $L\Omega_{\mathcal{O}_{\overline{K}}/\mathcal{O}_K}^{\bullet}$. There is ...
- 275
3
votes
1
answer
188
views
Maximal closed subscheme stable under the action of a finite connected group scheme
Let $k$ be a field of characteristic $p>0$, $X$ a smooth projective $k$-variety and $Y\subseteq X$ a closed irreducible subvariety. Let $G$ be a connected finite $k$-group scheme acting on $X$.
...
- 7,377