All Questions
2,494 questions
8
votes
1
answer
812
views
The Galois representation of a p-divisible group is crystalline
Can someone explain (or give a reference) why the Galois representation attached to a p-divisible group over the ring of integers of a p-adic ring is Crystalline?
- 1,645
4
votes
1
answer
221
views
Do permutation modules of solvable groups have self-dual socle in characteristic 2?
I was searching through the small groups database in GAP to find counterexamples to a certain conjecture (which is not important here). I was checking non-nilpotent solvable groups and noticed that ...
- 44.4k
2
votes
1
answer
568
views
Can we construct rational functions with prescribed ramification on an algebraic curve over \Qbar
Let $C$ be a smooth projective connected curve of genus $g$ over $\bar{\mathbf{Q}}$. Fix a finite non-empty (Edit) set of closed points $S$ in $C$ and let $U$ be the complement of $S$ in $C$.
Q1. (...
- 9,497
1
vote
2
answers
438
views
global complete intersection and independence of $l$
Hello,
I remember reading that if $X/\mathbf F_q$ is a projective smooth global complete intersection, then the characteristic polynomial of the $\mathbf F_q$-linear Frobenius of $X$
on $H^i_{et}(X\...
- 16.9k
3
votes
0
answers
193
views
rational points of component group of the special fiber of the Neron model
Let $A$ be an abelian variety over a number field $K$ and let $\mathcal{A}$ denote its Neron model over $\mathcal{O}_K$. Let $v \in M_K^0$ denote a finite prime of $K$, $k_v$ its residue field, $\...
- 841
4
votes
0
answers
623
views
A 'standard patching argument' in Mazur's Eisenstein Ideal paper
On pp 46 of his Eisenstein Ideal paper, Mazur states Theorem I.4 and in the discussion that follows, he mentions 'a standard patching argument' that completes the proof. I was wondering whether this ...
- 297
8
votes
2
answers
775
views
Hecke algebra generated by a single element
Let $\mathbb{T}_{\mathbb{Z}}$ be a $\mathbb{Z}$-module
generated by Hecke operators $T_n$ acting on the space of cups forms $S_{k}(\Gamma,\mathbb{C})$ for the congruent subgroup satisfying $\Gamma_1(...
- 2,406
5
votes
0
answers
234
views
Modular reduction of exceptional complex reflection groups
I am interested in reducing reflection representations of complex reflection groups modulo a prime $p$. For the infinite family $G(m,r,n)$, it is straightforward to get "good reduction" provided that $...
- 10.7k
7
votes
1
answer
2k
views
Can proper-smooth base change be used to show that varieties cannot be lifted to characteristic zero?
Recall the following corollary to the proper and smooth base change theorems:
Let $\pi: X \to S$ be a proper, smooth morphism. Then the direct images $R^i \pi_* \mathcal{F}$ are locally constant ...
- 25.6k
8
votes
1
answer
1k
views
Is the Galois x Hecke action on cohomology of Shimura varieties semi-simple?
Given a reductive group $G/\mathbf Q$ (+ additional data), and a compact open subgroup $K\subset G(\mathbf A^\infty)$, there is a standard construction that produces a Shimura variety $S$ and if we ...
- 647
5
votes
1
answer
1k
views
Eichler-Shimura for Shimura curves
Hi,
What is the statement of the Eichler-Shimura relation for Shimura curves? And where
can one find a proof?
Thanks
- 406
11
votes
0
answers
855
views
Points of bounded height in a number field
Let $K$ be a number field of absolute degree $d$, let $B$ be a positive real number, and write $S(K, B) = \{x \in K : H(x) \leq B\}$. Here $H$ is the absolute multiplicative height of an algebraic ...
- 1,199
1
vote
1
answer
556
views
Poitou-Tate dualities for Galois representations into power series rings?
Suppose $K$ is a finite extension of $\mathbf{Q}_p$, $A=K[[T_1,\dots,T_n]]$, $V$ a finite-rank free $A$-module, and $\rho:G_{\mathbf{Q}} \to \mathrm{GL}(V)$ a continuous Galois representation. Are ...
- 3,867
5
votes
0
answers
672
views
choice of local system in Deligne's construction of $l$-adic Galois representations
Hello,
Deligne famously constructed $l$-adic representations of $G_\mathbf Q = Gal(\overline{\mathbf Q}/\mathbf Q)$ starting form cusp modular forms of weight $k$ by looking inside the cohomology ...
- 2,842
2
votes
1
answer
977
views
Rapoport-Zink proof of purity of monodromy
Hi,
Does anyone know if the article
"Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindene Zyklen in ungleicher Charakteristik", INvent. Math, 68 (1980)
by ...
CommunityBot
- 1
7
votes
1
answer
514
views
O-linear Weil-pairing on abelian varieties with real multiplication
Let $A/k$ be an abelian variety with real multiplication by some ring of integers $\mathcal O \subset F$. Let $n$ be an integer prime to the characteristic of $k$.
We have the standart $e_n$ pairing $...
- 5,050
13
votes
2
answers
1k
views
Is the Characteristic of a Field Detectable from the Topology of a Topological Vector Space?
Motivation
A topological vector space is a vector space over a (topological) field, K, that carries a topology such that addition and scalar multiplication are continuous maps, e.g., all normed vector ...
CommunityBot
- 1
9
votes
2
answers
5k
views
Picard number and torsion of Neron-Severi group of abelian varieties over a number field
Let $A$ be an abelian variety over a number field $k$ and let $NS_A$ denote its Neron-Severi scheme. Then the group of $k$-rational points of $NS_A$ is a finitely generated abelian group, i.e. $NS_A(k)...
- 20.8k
7
votes
1
answer
549
views
Which $\mathbf{Q}_p$-varieties come from $\mathbf{Q}$-varieties?
This is a very naive question. Fix a prime $p$ and consider the forgetful map from varieties over $\mathbf{Q}$ to varieties over $\mathbf{Q}_p$. Is there a conjectural "purely $p$-adic" ...
- 30.5k
13
votes
1
answer
651
views
Help wanted with Chebotarev condition in characteristic 2
Having promised a longtime collaborator that I would clear my plate to finish up some joint work of ours, I am swallowing my pride and tossing up the following technical point of function field ...
- 50.7k
4
votes
2
answers
604
views
Adem-Wu relations from Bullett-Macdonald identities
Question. Let $p$ be a prime. Let $q$ be a power of $p$. Let $P^0$, $P^1$, $P^2$, ... be elements of some associative $\mathbb F_q$-algebra $A$. (Here, $P^i$ does not mean $P$ to the $i$-th power; ...
- 20.7k
3
votes
1
answer
367
views
Rational points over completions of a number field
Let $X$ be a smooth geometrically irreducible $k$-variety over a number field $k$.
I do not assume that $X$ has a $k$-point.
Is it true that $X$ has $k_v$-points for almost all places $v$ of $k$?
- 65.4k
20
votes
3
answers
2k
views
Geometric construction of depth zero local Langlands correspondence
Dear community,
In light of the recent work of DeBacker/Reeder on the depth zero local Langlands correspondence, I was wondering if there is an attempt to "geometrize" the depth zero local Langlands ...
- 1,000
6
votes
2
answers
1k
views
On a Theorem of Fontaine
Let $S=Spec(R)$ where $R$ is a Henselian local ring with fraction field $K$. Let $G$ and $G'$ be finite, flat group schemes of odd order over $S$ with isomorphic generic fibers (over $Spec(K)$). Does ...
- 3,032
4
votes
2
answers
694
views
Ample line bundle and Frobenius morphism on smooth toric variety
Let $k$ be an algebraically closed field of $\mathrm{char}(k)=p>0$, $X$ a smooth toric projective variety of $\dim X=n$, $F_X:X\rightarrow X$ the absolute Frobenius morphism of $X$. Then for any $\...
- 16.2k
6
votes
1
answer
459
views
How locally ubiquitous are totally real fields?
Let $p$ be a fixed prime number.
Question 1: Given a finite extension $K$ of $\mathbb{Q}_p$ is there a totally real extension $F$ of $\mathbb{Q}$ and a place $v$ of $F$ over $p$ such that $F_v = K$?
...
- 1,645
2
votes
1
answer
332
views
Ample bundle under Frobenius morphism
Let $k$ be an algebraically closed field of char($k$)=p>0, $X$ a smooth projective variety over $k$, $F:X\rightarrow X^{(1)}$ the relative Frobenius morphism. Let $E$ be an ample vector bundle on $X$. ...
- 35.2k
9
votes
3
answers
3k
views
Elliptic Curves over Global Function Fields
I am currently thinking about a problem, and I feel that by knowing more about elliptic curves over extensions of $\mathbb{F}_q(T)$, for $q$ a power of $p$ say, might lead to insight. I am also ...
- 406
2
votes
1
answer
385
views
Tame covers of arithmetic schemes
I'm a bit confused concerning tamely ramified covers of arithmetic schemes. I guess they would reduce to tamely ramified extensions of number fields, but they don't seem to do so. Let me elaborate:
...
- 45.7k
0
votes
0
answers
352
views
Liftability in positive characteristic
What clsses of algebraic varieties over field of positive characteristic can be lift to $W_2(k)$?
- 85
2
votes
0
answers
530
views
Fontaine-Mazur conjecture for higher local fields
Hello,
For a $p$-adic local field K, the Fontaine-Mazur conjecture characterises the $p$-adic representations of the Galois group of $K$ which arise from geometry. Is there a conjectural ...
- 3,867
7
votes
2
answers
513
views
Tameness for the Galois closure of a map of curves
Say we are working over some $K=\overline{K}$, of characteristic $p>0$. Let $\phi: Y\rightarrow X$ be a nonconstant map of smooth projective curves. To this map we can associate a map $\psi: Z\...
- 839
0
votes
0
answers
524
views
DeRham cohomology
The Poincare lemma fails in positive characteristic, since pth powers vanish under differention. My question is : is there still some kind of resolution of the local system k by considering some ...
- 1
0
votes
2
answers
806
views
extensions of group schemes
Hi,
I have the following question: why $Ext^1(\mathbb{G}_m,\mathbb{Z})=0$?
- 16.2k
3
votes
1
answer
369
views
Maps on the identity components of Neron models
Any map $A \to B$ of abelian varieties of the same dimension over a global field $K$ induces a map $\mathcal{A} \to \mathcal{B}$ on the corresponding Neron models over $X$ (where $X=Spec{\mathcal{O}_K}...
- 24.1k
7
votes
0
answers
273
views
Do Scharaschkin's results on Brauer-Manin obstructions on curves generalize to non-projective curves?
Theorem: Let X be a smooth projective curve over a number field K, and let $\delta$ be the index of X (i.e., the minimal degree of a K-rational divisor on X). Then V. Scharaschkin proved in this ...
- 10.5k
6
votes
1
answer
825
views
Characterization of algebraic points on Shimura varieties?
Is there any (conjectural) characterization of $\overline{\bf{Q}}$-points
on Shimura varieties?
The question of course does not always make sense
for ${\bf{Q}}$-points: a theorem of Shimura shows ...
- 57.6k
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(...
CommunityBot
- 1
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
10
votes
1
answer
1k
views
Which primes can divide orders of Tate-Shafarevich groups?
Heuristic arguments due to (I believe) Delauney predict that every prime divides the order of the Tate-Shafarevich group of infinitely many elliptic curves over $\mathbf{Q}$. However, is it even ...
- 65.4k
0
votes
1
answer
1k
views
on connectness and normality
Hi,
the situation is the following: I have a projective scheme $\tilde{P}\rightarrow S=Spec(A)$ with $A$ excellent and $I$-adically complete for some ideal of $A$. A group $Y$ acting on $\tilde{P}$ ...
- 141
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 ...
CommunityBot
- 1
1
vote
1
answer
569
views
references for abelian schemes
Hi,
I have a very basic question.
I am looking for references explaining how to construct explicitily a Jacobian starting from a curve or examples of projective equations for an abelian scheme. I ...
- 231
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?
- 20.5k
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 ...
- 27k
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
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
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 ...
CommunityBot
- 1
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 \...
- 42.9k
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 $k$...
- 57.6k