All Questions
2,495 questions
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
20
votes
2
answers
2k
views
revisiting $THH(\mathbb{F}_p)$
Reading through Bhatt-Morrow-Scholze's "Topological Hochschild Homology and Integral p-adic Hodge Theory" I encountered the following statement.
We use only “formal” properties of THH throughout ...
- 858
10
votes
4
answers
1k
views
Possible groups of K-rational points for elliptic curves over arbitrary fields
It is known that the group $C(\Bbb R)$ has at most two connected components, and the connected component of the identity is isomorphic to $U(1)$ as a topological group (trivially) and $C(\Bbb Q)$ is ...
- 3,738
40
votes
4
answers
3k
views
Why are Green functions involved in intersection theory?
I've been learning Arakelov geometry on surfaces for a while. Formally I've understood how things work, but I'm still missing a big picture.
Summary:
Let $X$ be an arithmetic surface over $\...
- 1,237
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_{\...
- 1,061
16
votes
3
answers
2k
views
Tower of moduli spaces in Scholze's theory
My question is related to another one I read here in Overflow. I am reading Scholze's papers about moduli spaces of $p$-divisible groups and elliptic curves, and I am very interested in the formal ...
- 161
1
vote
1
answer
262
views
Singularities of arithmetic surface
I have a problem understanding the discussion of example 8.3.54 in Liu's Algebraic Geometry and Arithmetic Curves.
The setting is the following: We have a DVR with uniformiser $t$, characteristic of ...
- 223
14
votes
1
answer
699
views
What numbers are not represented by $5xy+2x+2y$?
What numbers are not represented by $5xy+2x+2y$? Do they have a positive density?
This came up for me while investigating some cases here. Here's what I've found:
All evens are represented with $x=0$...
user44143
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
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
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
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 ...
- 2,751
18
votes
1
answer
3k
views
Conjectures of Peter Scholze about q-de Rham complex: examples
Peter Scholze formulated several conjectures about $q$-de Rham complex in the paper
Canonical $q$-deformations in arithmetic geometry, Ann. Fac. Sci. Toulouse Math. (6) 26 (2017), no. 5, pp 1163–...
- 4,316
8
votes
1
answer
2k
views
Proof of Prop 1.1 in Wiles' "Modular elliptic curves and Fermat's last theorem"
On p. 459 of "Modular elliptic curves and Fermat's last theorem", proof of Prop. 1.1, where it says "Since $H^2(G,\mu_{p^r}) \rightarrow H^2(G,\mu_{p^s})$ is injective for $r \leq s$...", is there any ...
- 2,125
2
votes
0
answers
467
views
Confusion regarding Proposition 1.1 in Wiles's Fermat paper
This is from p. 459 of Wiles's Fermat paper.
Theorem: If $D_{p}$ is a decomposition group at $p$, $A$ is an Artinian local ring with maximal ideal $\mathfrak{m}$ and finite residue field $k$ of ...
- 1,805
4
votes
2
answers
997
views
Pullback of a connection
let $X,Y$ be smooth schemes (or rigid spaces etc..) over a base $S$, let $f:Y \rightarrow X$ be a $S$-morphisn and let $\mathcal{F}$ be a locally free $\mathcal{O}_X$-module with connection $\nabla$. ...
- 43
2
votes
0
answers
218
views
Borel-Weil-Bott theorem for wonderful compactification in characteristic p
Are there any known results for a Borel-Weil-Bott theorem for the wonderful compactifications over characteristic $p$ (i.e., theorems that classify the cohomologies of all line bundles on a wonderful ...
- 41
4
votes
1
answer
198
views
Simple restricted but not restricted simple Lie algebras
Let $F$ be a field which has a positive characteristic $p \ge 2$ and $(\mathfrak{g},[p])$ be a restricted Lie algebras over a field $F$ where $[p]$ is a $p$-th power map on $\mathfrak{g}$. $(\mathfrak{...
- 145
3
votes
0
answers
196
views
Faithfulness of parabolic induction
I've only recently begun to study the representation theory of $p$-adic groups, so the following question might be quite silly.
Let $F$ be a non-archimedean local field of residue characteristic $p$, $...
user484137
15
votes
0
answers
2k
views
A question on Fargues-Scholze
As far as I understand it, the main goal of the recent work of Fargues and Scholze on the geometrization conjecture is to show that the local Langlands conjecture of a local field is equivalent to the ...
- 4,428
4
votes
0
answers
147
views
Uniqueness of Galois descent
Let $Y,Z$ be $\mathbb{F}_p$-schemes, they are both models of scheme $X$ over $\overline{\mathbb{F}_p}$ . Let $F$ be the absolute Frobenius of $\overline{\mathbb{F}_p}$, If $1_Y\times F$ and $1_Z\times ...
- 653
13
votes
2
answers
1k
views
Finiteness of the Brauer group for flat proper schemes over $\operatorname{Spec} \mathbf{Z}$
One fundamental conjecture on the Brauer group is that $\operatorname{Br}(X)$ is finite for $X/\operatorname{Spec} \mathbf{Z}$ proper. By class field theory (the theorem of Albert–Brauer–Hasse–Noether)...
user19475
2
votes
3
answers
416
views
Density of $d$ for which a generalized Pell equation has a solution
For how many $0 < d \leq D$ is there an integer solution to
$$x^2-dy^2 = -n$$
for $n > 1$? I have circumstantial reason to believe it might be $\sim D^{\frac{1}{2}}$ but I'd be interested in any ...
- 479
6
votes
1
answer
418
views
Number of points of parabolic Springer fibres
Let $P$ be a parabolic subgroup of $\mathrm{GL}_n$ and $u\in P$ a unipotent element. The parabolic Springer fibre associated to $(P,u)$ can be defined by
$$
\mathcal{P}_u:=\{gP\in G/P \mathrel\vert g^{...
- 2,751
3
votes
0
answers
174
views
On continuous seminorms on Fréchet-Stein algebras
Let $K$ be a discretely valued complete non-archimedean field and $U$ be a left Fréchet-Stein algebra as defined in Algebras of p-adic distributions and admissible representations, with a Fréchet-...
- 541
32
votes
1
answer
8k
views
$p$-adic Hodge Theory for rigid spaces, after P. Scholze
I was going over P. Scholze's paper on $p$-adic Hodge Theory for rigid analytic varieties.
This question is around the "Poincaré Lemma" in the paper.
Throughout, let $X$ be a proper smooth rigid ...
user113453
11
votes
2
answers
1k
views
Bounded Torsion, without Mazur’s Theorem
Mazur’s torsion theorem famously tells us exactly which finite groups can occur as the torsion subgroup of $E(\mathbb{Q})$ for an elliptic curve $E$ defined over $\mathbb{Q}$. In particular, it ...
- 584
4
votes
1
answer
309
views
Torsion points on $E/\mathbb{Q}$ with large coordinates
Let $E/\mathbb{Q}$ be an elliptic curve with finitely many rational points.
What are some examples where at least one rational point has large coordinates (compared to the height of $E$)?
- 41
2
votes
0
answers
98
views
Control on the locus of bad reduction for divisors
Let $X$ be a smooth variety over a number field $K$ and let $\mathcal X$ be a normal, projective model of $X$ over the ring of integers $O_K$.
Now assume that $D\subset X$ is an irreducible divisor ...
- 321
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{...
user178246
2
votes
0
answers
147
views
Automorphism groups of "reductive" Lie algebras in positive characteristic
I put "reductive" in quotes because, of course, in positive characteristic one should speak of Lie algebras of reductive groups, not of reductive Lie algebras.
Let $G$ be a reductive group ...
- 13k
4
votes
1
answer
638
views
perfect fields in positive characteristic
Let $k$ be an infinite perfect field in positive characteristic $p$, i.e. every element of $k$ is a $p$th power. I am interested in properties of finite fields that can be extended to $k$. For example:...
- 168
10
votes
1
answer
592
views
Brauer-Manin obstruction on an open subset of an elliptic curve
First a disclaimer. This is an old question that I considered years ago and that I recently remembered. Since I am no longer in active research it may be considered as 'idle curiosity', although I ...
- 4,745
3
votes
0
answers
279
views
Grothendieck trace formula for arbitrary morphisms
The Grothendieck trace formula can be viewed as a generalization of the Lefschetz trace formula in étale cohomology from constant sheaves to constructible $l$-adic sheaves, but restricting to the ...
- 216
4
votes
0
answers
244
views
Torsionness of the kernel of the pullback map of Picard groups of a normalization map
Let $X$ be a (irreducible) projective variety over a number field $k$, $\pi: \tilde X \to X$ be its normalization, and $\pi^{*}: \mathrm{Pic}(X) \to \mathrm{Pic}(\tilde X)$ be the corresponding map of ...
- 151
6
votes
1
answer
242
views
Contraction of some surfaces over a ring of algebraic integers
The situation:
Let $X$ be a 2 dimensional normal quasi-projective $\mathcal{O}_K$-scheme, where $K$ is an algebraic number field. Assume the following conditions on $X$:
$X$ is integral.
$X_K$ is ...
41
votes
2
answers
3k
views
Perfectoid universal covers
It is often said, with varying degrees of rigor or enthusiasm, that every rigid space (say over $\mathbb{C}_p$) has a pro-etale cover which is 'topologically trivial' in some sense. For example, this ...
- 843
6
votes
1
answer
1k
views
Are the Galois actions on automorphisms of twists isomorphic?
This might be a trivial question and I might be overlooking something:
Suppose $k$ is a field with algebraic closure $\overline k$ and absolute Galois group $\Gamma$. Let $X,Y$ be two distinct ...
- 7,746
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 ...
user322153
3
votes
1
answer
263
views
On the exactness of some completed tensor products
Let $X$ be an affinoid variety over a discretely valued non-archimedean field $k$ with valuation ring $\mathcal{R}$. Fix a uniformizer $\omega$. On the section 3.2 of the paper https://arxiv.org/abs/...
- 541
16
votes
5
answers
8k
views
Is the ABC conjecture known to imply the Riemann hypothesis?
I once heard from a graduate student that the ABC conjecture implies the Riemann hypothesis. I can't find a reference for this, but given the department the student is from I tend to believe he might ...
- 3,268
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 ...
- 1,093
13
votes
3
answers
2k
views
Is the map on étale fundamental groups of a quasi-projective variety, upon base change between algebraically closed fields, an isomorphism?
$\DeclareMathOperator\Spec{Spec}$Let $k \subset L$ be two algebraically closed fields of characteristic $0$. Let $U \subset \mathbb P^n_k$ be a smooth quasi-projective variety and let $U_L$ denote the ...
- 1,308
9
votes
1
answer
356
views
Reference request for indecomposable representations of $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$
Is there a good reference on the classification of indecomposable representations of the Lie algebra $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$ (of course, in ...
- 239
9
votes
2
answers
2k
views
Any simple concrete proof of Faltings theorem?
Are there simple proofs of some concrete special cases of Faltings's theorem? Any help would be appreciated.
- 3,104
8
votes
2
answers
2k
views
Can someone suggest a path to study Mordell-Weil theorem for someone studying on their own?
So, I want to read the proof of Mordell-Weil theorem and so, I picked up the book 'Arithmetic of Elliptic Curves' by J. Silverman and J.S. Milne's Elliptic Curves book. But after going through both ...
- 401
41
votes
2
answers
9k
views
What should I read before reading about Arakelov theory?
I tried reading about Arakelov theory before, but I could never get very far. It seems that this theory draws its motivation from geometric ideas that I'm not very familiar with.
What should I read ...
8
votes
1
answer
943
views
Automorphisms over finite field that do not lift to an automorphism in characteristic zero
My main question is the following: is there an automorphism of the affine space $\mathbb{A}^n$ (automorphism of an algebraic variety) defined over a finite field which does not lift to an automorphism ...
- 7,722
4
votes
0
answers
215
views
Reference request: Radicial morphisms & Jacobson-Bourbaki correspondence
My name is Chemy (Przemysław). I am a PhD student at UvA (Amsterdam), and I work on projects related to foliations in algebraic geometry in positive characteristic. Therefore I am avidely reading two ...
- 485
2
votes
0
answers
166
views
Theorem on formal functions when the initial data is a proper map of formal schemes
Let $\pi: X \to S:=\mathrm{Spf}\text{ } A$ be a proper morphism of $\mathbb{Z}_p$-admissible formal schemes and $\mathcal{F}$ be a coherent sheaf on $X$.
Set $S_0=\{x\}$ be a closed point of $S$ and $...
- 1,066