All Questions
2,494 questions
20
votes
2
answers
2k
views
Frobenius splitting and derived Cartier isomorphism
Let $X$ be a smooth algebraic variety over an algebraically closed field $k$ of characteristic $p>\dim X$. The motivation for my question comes from the following results.
1. If $X$ is Frobenius ...
- 16.2k
2
votes
3
answers
652
views
Sections of morphisms of schemes up to a finite morphism
Let $f:X\longrightarrow S$ be a flat projective morphism of regular integral noetherian schemes such that that the generic fibre $X_\eta\longrightarrow K(S)$ is a smooth projective connected curve ...
- 163
6
votes
1
answer
825
views
More on universal homeomorphisms
I would like to understand this notion better; where could I find some examples? In particular, I am interested in the following questions (and references for the answers).
Is a universal ...
- 16.9k
4
votes
2
answers
339
views
Is the pre-image of a regular subscheme with respect to a universal homeomorphism of regular schemes regular?
Let $f:X\to Y$ be a universal homeomorphism of regular (excellent finite-dimensional) schemes, $Z\subset Y$ be a regular subscheme. Is $f^{-1}(Z)$ necessarily regular?
- 16.9k
4
votes
2
answers
865
views
Extending finite morphisms of curves to finite morphisms of arithmetic surfaces
Let $\pi:Y\longrightarrow X$ be a finite morphism of smooth projective geometrically connected curves over a number field $K$. Let $h:\mathcal{X}\longrightarrow \textrm{Spec} \ O_K$ be a regular ...
- 9,497
3
votes
0
answers
498
views
Wintenberger's mystery
Fontaine describes in §2 of this old survey work by Wintenberger and wonders (on p. 97) that the structures found by Wintenberger are a "complete mystery" and no-one knows a "reasonable geometric ...
- 10.8k
27
votes
1
answer
4k
views
Degeneration of the Hodge spectral sequence
Let $f\colon X \to S$ be a smooth proper morphism of schemes. If $S$ is of characteristic zero (i.e., $S$ is a $\mathbb Q$-scheme), then Deligne has shown:
$R^af_*\Omega^b_{X/S}$ is locally free for ...
- 1,645
14
votes
1
answer
835
views
A remark in Swinnerton-Dyer's paper in Cassels-Frohlich
In Swinnerton-Dyer's charming paper "An application of computing to classfield theory", in Cassels-Frohlich, he discusses the genesis of the Birch/Swinnerton-Dyer conjecture and numerical tests of it ...
- 13.1k
6
votes
1
answer
393
views
finite quotients of fundamental groups in positive characteristic
For affine smooth curves over $k=\bar{k}$ of char. $p,$ Abhyankar's conjecture (proved by Raynaud and Harbater) tells us exactly which finite groups can be realized as quotients of their fundamental ...
- 4,265
2
votes
1
answer
405
views
formal smoothness versus reducedness
Hi,
I have the following situation: $R,H$ schemes (can be assumed noetherian and of finite type) over a field $k$ which we can assume to be algebraically closed, with $H$ reduced, $Y\subset R\times \...
- 141
7
votes
1
answer
519
views
Degree zero zero-cycles on the square of a curve
A well-known mathematician once explained the following conjecture to me, as "an example of how little we know about cycles of codimension $\geq 2$." Let $C$ be a curve defined over a number field $k$...
- 13.1k
8
votes
1
answer
2k
views
About the Serre-Tate theorem
It is somehow a general principle that the (infinitesimal) local behavior of a representable moduli functor $X$ at some point $x$ is closely related to the deformation problem of the structure ...
- 1,305
5
votes
0
answers
735
views
Do all the main properties of constructible and perverse sheaves (in an 'arithmetic' situation) follow from results of Gabber?
This question is a continuation of Bad behaviour of perverse sheaves over 'general' bases?
Let $S$ (for example) be a finite type separated scheme over $\mathbb{Z}$. I would like: (1) to ...
- 16.9k
5
votes
1
answer
571
views
Selmer of an abelian variety versus that of its dual.
What is the precise relationship between the Selmer group of an abelian variety and that of its dual? For instance, does the vanishing of one not imply the same for the other?
To fix ideas, let $A$ ...
- 1,141
12
votes
2
answers
2k
views
Compatibility of Bloch-Kato and Beilinson-Bloch
Suppose $V/K$ is a smooth projective variety. Let $\mathrm{Ch}^{j}(V)_{0}$
be the group of codimension-$j$ homologically trivial $K$-rational cycles on $V$, modulo rational equivalence. A conjecture ...
- 13.1k
16
votes
3
answers
2k
views
Are there any rational solutions to this equation?
I am not sure if this is an appropriate question, but I was asked this by a colleague today and do not know how to answer it.
1) Are there any rational solutions to the following equation:
$$x^3-8x^2+...
- 5,037
1
vote
1
answer
183
views
complete ring as union of finite type algebras
Hi,
why the completion of a local ring $R$ can be written as an increasing union of $R$-algebras of finite type?
- 141
1
vote
2
answers
327
views
Equidistribution in the unit interval of numbers in a real field with bounded Mahler measure
Let $K$ be a real number field, together with a fixed immersion in $\mathbb{R}$, and for each positive real number $M$ consider the set $S_M(K)$ of elements in $K \cap [0,1]$ having Mahler measure ...
- 1,269
2
votes
2
answers
686
views
transcendence of canonical heights
Are there known examples of rational points on elliptic curves/abelian varieties
over number fields with transcendental canonical height? Thanks.
- 3,867
11
votes
2
answers
863
views
Valuations and separable extensions
Let $R$ be a valuation ring containing a field $k$, with residue field $F$ and quotient field $K$. Assume $F/k$ is separable. Is $K/k$ separable?
I have convinced myself that (for a positive answer) ...
- 19.6k
6
votes
2
answers
318
views
formal differences?
Hi,
given a local ring $A$ with maximal ideal $m$ which are differences between $Spec(\hat{A})$ ($\hat{A}$ completion of $A$ along $m$) and $Spf(A)$?
- 141
6
votes
1
answer
731
views
different Shimura data with common underlying group?
A pure Shimura datum is of the form $(G,X)$ with $G$ a connected reductive $\mathbb{Q}$-group, and $X$ a homogeneous space under $G(\mathbb{R})$, subject to Deligne's conditions in terms of Hodge ...
- 1,305
4
votes
3
answers
1k
views
Normal crossings on a surface and ordinary double points
Let $R$ be a discrete valuation ring with residue field $F$, and let $X/R$ be a regular projective relative curve with closed fiber $X_0$. Then $X_0/F$ is a projective curve. We know that $X$ may be ...
- 43
7
votes
1
answer
1k
views
Status of Ihara's lemma for Shimura curves over totally real fields?
What is the status of Ihara's lemma for Shimura curves over totally real fields?
In particular, why is it not implicit in the exact sequence of Rajaei, "On the levels of mod $l$ Hilbert modular forms" ...
- 1,141
17
votes
2
answers
2k
views
Why does Tate's conjecture imply semisimplicity of crystalline Frobenius?
I'm trying to find a reference for the following fact:
If Tate's conjecture is true for all smooth projective varieties over $\mathbb{F}_p$, then the Frobenius endomorphism on the crystalline ...
- 37k
89
votes
9
answers
13k
views
Why should I believe the Mordell Conjecture?
It was Faltings who first proved in 1983 the Mordell conjecture, that a curve of genus 2 or more over a number field has only finitely many rational points.
I am interested to know why Mordell and ...
- 2,827
10
votes
1
answer
1k
views
Bad behaviour of perverse sheaves over 'general' bases?
Could one define $\mathbb{Q}_l$-perverse etale sheaves over more or less general (excellent, separated) base scheme by combining the results of Gabber and Ekedahl? Would their functoriality properties ...
- 16.9k
4
votes
0
answers
678
views
Soft proof of multiplicity one for character groups of Shimura curves?
Is it not possible to prove mutiplicity one type statements for character groups of quaternionic Shimura curves by simply using Raynaud's description for character groups at primes dividing the ...
- 1,141
8
votes
1
answer
2k
views
Geometric abelian class field theory
There is a very nice geometric proof of Deligne for the Artin Reciprocity in the geometric setting, namely for a smooth, projective, geometrically irreducible curve $C$ over a finite field $\mathbb{F}...
- 522
16
votes
2
answers
2k
views
Period rings for Galois representations
I have some questions concerning period rings for Galois representations.
First, consider the case of $p$-adic representations of the absolute Galois group $G_K$, where $K$ denote a $p$-adic field. ...
- 657
1
vote
1
answer
431
views
Is it easy to define weights for $Q_l$-sheaves over finite type $Z[1/l]$-schemes?
In her paper "Mixed perverse sheaves for schemes over number fields" A. Huber defines certain weights for certain categories of $\mathbb{Q}_l$-sheaves over a finite type $\mathbb{Q}$-scheme $...
- 16.9k
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 ...
17
votes
1
answer
2k
views
Construction of abelian varieties from Hilbert modular forms?
Some experts tell me that the construction of abelian varieties from
Hilbert modular forms is an (apparently difficult) open problem. However,
in view of the construction of $l$-adic Galois ...
- 1,141
2
votes
1
answer
528
views
Is there an easy proof of the fact that the intermediate image functor respects weights?
It was proven in BBD (see Corollary 5.3.2) that for an open immersion $j$ the functor $j_{!*}$ preserves weights of mixed sheaves. The proof relies on several previous results; it is especially ...
- 16.9k
12
votes
2
answers
811
views
Given a family of curves, when does there exist a fibered surface over Spec Z parametrizing them?
Let $X_p$ be a projective curve over the finite field $\mathbf{F}_p$ (i.e. a projective $\mathbf{F}_p$-scheme pure of dimension 1) for every prime number $p$. Let $X_\mathbf{Q}$ be a projective curve ...
- 9,497
9
votes
3
answers
929
views
Upper bounds for ranks of modular jacobians
The following question came to me earlier as a "side question"; something I'd like to know, but which is not totally necessary for what I'm thinking about or doing:
Consider the genus 32 curve $X_0(...
- 2,827
0
votes
1
answer
295
views
local statement
I have a property which is local and stable for faithfully flat base change over a base scheme $S$. So I need to prove it for $O_{S,s}$ with $s\in S$.
Why if I can prove it for a local artinian ring ...
- 141
0
votes
2
answers
2k
views
non discrete valuation ring [closed]
Hi,
I am looking for examples of non-discrete valuation rings. Could you help me?
Thanks
- 141
19
votes
3
answers
1k
views
Drawing planar graphs with integer edge lengths
It is well known that every planar graph has an embedding such that every edge is drawn as a straight line segment (Fáry's Theorem). Kemnitz and Harborth made the following stronger conjecture
...
- 32.1k
3
votes
0
answers
279
views
Tate-Shafarevich group of non-principally polarized abelian variety
Let $A/k$ be an abelian variety over a number field $k$ with a polarization of minimal degree $d>1$. (Assume all Tate-Shafarevich groups to be finite.)
What can one say about the order of $\mathrm{...
- 841
2
votes
1
answer
513
views
normalization of a stack
Hi,
how is it defined the normalization of an algebraic stack $A$ inside another algebraic stack $B$. If you do not want to write the answer could you give to me some reference?
Thank you
- 31
1
vote
1
answer
213
views
flatness of a kernel
Hi,
let $A$ an abelian scheme over a curve $C$ and $n$ an integer greather than 3 coprime with the characteristic of the ground field. Do you know why the kernel of the multiplication by $n$ is flat ...
- 31
3
votes
2
answers
547
views
less than normal
Hi,
if we could write a classification about the known regularity which is the known class of schemes that are immediately less good than normal schemes? And which properties have they?
thank you
- 73
1
vote
2
answers
393
views
Could the Kunneth decomposition of a motif depend on the choice of $l$?
Suppose that over some (algebraically closed) field $K$ of characteristic $p>0$ we have: numerical equivalence of cycles coincides with homological one with respect to ${\mathbb{Q}}_{l}$ and ${\...
- 16.9k
13
votes
1
answer
5k
views
partition functions and Galois representations?
The recent answer's link to Ono's work makes me ask and wonder if his new results on partition functions tell something about Galois representations? (Hoping that question is not a case of this)
- 10.8k
11
votes
0
answers
576
views
What's known about the mod 2 reduction of the level l Jacobi modular equation?
Motivation:
Let $\ell$ be an odd prime. Let $A$ in ${\mathbb Z}/2[[x]]$ be $x+x^9+x^{25}+x^{49}+...$, and $B=A(x^\ell)$. One can use the level $\ell$ Jacobi modular equation to get a polynomial ...
- 5,422
3
votes
1
answer
288
views
Do $Q_l$-etale Euler characteristics of Chow motives coincide for all $l$?
I am interested in (Chow) motives over (algebraically closed) characteristic $p>0$ fields. For $H$ being $\mathbb{Q}_l$-adic cohomology, one can consider $Ch_l(M)=\sum (-1)^i\dim_{\mathbb{Q}_l} H^i(...
- 16.9k
20
votes
1
answer
2k
views
De Rham cohomology of formal groups
Let $G$ be some (dimension $1$, to simplify) formal group over a characteristic $0$ field $K$. The law of $G$ is denoted by $\oplus$. If $w(X) \in K[[X]] dX$ is a differential form, let $F_w(X)$ be ...
- 6,924
10
votes
2
answers
2k
views
Roadmap to Computer Algebra Systems Usage for Algebraic Geometry
I've decided it's time to start learning how to use a computer to do calculations... I've used Singular to some small extent so far, but I want to start relying on computer algebra systems more.
...
10
votes
2
answers
2k
views
Picard group of scheme over DVR
Let $A$ be a DVR and let $X/A$ be a smooth, proper scheme with geometrically integral fibers. Is there an easy way to see that the Picard group of $X$ is isomorphic to the Picard group of the generic ...
- 5,163