All Questions
2,494 questions
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
5
votes
1
answer
316
views
connected component of the identity section in the special fiber of the Neron model under isogenies
Let $A$ and $B$ be two isogenous abelian varieties over a number field $K$ and let $\mathcal{A}$ and $\mathcal{B}$ denote their Neron models over $\mathcal{O}_K$. Let $v \in M_K^0$ denote a finite ...
- 841
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\...
- 2,842
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
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
2
answers
862
views
Can we bound the minimal degree of a field extension required to obtain semi-stable reduction
Let $K$ be a number field and let $X$ be a smooth projective geometrically connected curve over $K$.
There exists a finite field extension $L/K$ such that $X_L=X\otimes_K L$ has semi-stable ...
- 9,497
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(...
- 303
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
19
votes
2
answers
3k
views
Bertini theorems for base-point-free linear systems in positive characteristics
Suppose that $X$ is a smooth algebraic variety over an algebraically closed (uncountable if it helps) field of characteristic $p > 0$. Suppose that $L$ is a line bundle, probably ample or at least ...
- 20.5k
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
- 2,842
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
12
votes
4
answers
2k
views
Finite subgroups of $PGL_2(K)$ in characteristic $p$
Let $K$ be a field of characteristic $p$. What are the finite subgroups of $PGL_2(K)$ whose orders are divisible by $p$? And if $G$ and $H$ are two such subgroups that are isomorphic, can one say when ...
- 1,199
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
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 ...
- 13.1k
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 ...
- 647
2
votes
1
answer
690
views
Restricted universal enveloping algebra of Abelian p-Lie algebra
Question: Let $p$ be a prime. Let $k$ be a commutative ring such that $p=0$ in $k$.
Let $\mathfrak g$ be an abelian $p$-restricted Lie algebra over $k$. In other words, let $\mathfrak g$ be a $k$-...
- 33.8k
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 $...
- 839
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 ...
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)...
- 841
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" ...
- 13.1k
148
votes
4
answers
69k
views
What are "perfectoid spaces"?
This talk is about a theory of "perfectoid spaces", which "compares objects in characteristic p with objects in characteristic 0". What are those spaces, where can one read about them?
Edit: A bit ...
- 10.8k
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 ...
- 65.4k
15
votes
0
answers
779
views
Lifting varieties from char. $p$ to char. 0 after alterations
The question is related to this MO question:
Lifting varieties to characteristic zero.
Let $X$ be a projective smooth variety over $k$ alg. closed field of char. $p.$ Does there always exist an ...
- 4,265
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; ...
- 33.8k
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 $k$?
- 14.2k
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
7
votes
2
answers
636
views
The number of singular fibres of a semi-stable arithmetic surface over \Z
This is an arithmetic follow-up to my previous question Does there exist a non-trivial semi-stable curve of genus >1 with only 4 singular fibres
Let $k$ be an algebraically closed field and let $...
- 9,497
5
votes
1
answer
461
views
Given a branched cover with branch cycle description $(g_1,...,g_r)$, does $g_i$ generate some decomposition group?
Classically:
Let $a_1,...,a_r$ be points in $\mathbb{P}^1_{\mathbb{C}}$, and let $\alpha_1,...,\alpha_r$ be simple loops around the $a_i$, all counterclockwise, and none touching (so $\alpha_1...\...
- 5,929
21
votes
2
answers
5k
views
State of resolution in positive characteristic?
Heisuke Hironaka's coming talk makes me wonder how the state of the work on that theme is. So far, I noticed (but didn't read) these papers:
Kawanoue, Hiraku, Toward resolution of singularities over ...
- 10.8k
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 ...
- 297
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 $\...
- 85
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
6
votes
2
answers
671
views
Global Sections of the Identity Component of Neron model
Let $A$ be an abelian variety over a number field $K$ and consider the Neron model $\mathcal{A}$ of $A$ over $X=Spec{\mathcal{O}_K}$. If $\mathcal{A}^0$ is the identity component of $\mathcal{A}$, ...
- 297
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 $X$. ...
- 85
20
votes
3
answers
2k
views
Rational points on algebraic curves over $\mathbb Q^\text{ab}$
Motivation:
Let $\mathbb{Q}_{\infty,p}$ be the field obtained by adjoining to $\mathbb{Q}$ all $p$-power roots of unity for a prime $p$. The union of these fields for all primes is the maximal ...
- 3,867
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
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:
...
- 1,841
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 ...
- 831
8
votes
1
answer
697
views
Does combining Abhyankar's Lemma and embedded resolution give horizontal normal crossings
Let $\pi:Y\longrightarrow \mathbf{P}^1_{\mathbf{Z}}$ be a finite surjective flat morphism of schemes, where $Y$ is a normal integral flat projective 2-dimensional $\mathbf{Z}$-scheme, with branch ...
- 9,497
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\...
- 801
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 $Ext^1(\mathbb{G}_m,\mathbb{Z})=0$?
- 141
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
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}...
- 297
17
votes
1
answer
2k
views
Rational points à la Chabauty-Coleman
I have been trying to learn the method of Chabauty and Coleman to find rational points on curves; I have been reading an exposition by McCallum and Poonen which was pointed out to me by Emerton in ...
- 2,827
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 ...
- 1,141
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 ...
- 13.1k
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