All Questions
2,494 questions
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Dimension of fibres of moment maps in characteristic $p$
Suppose $G$ is a connected semisimple linear algebraic group with Lie algebra $\mathfrak{g}$ and $X$ is a homogeneous $G$-space with isotropy subgroup $H$ (associated Lie algebra $\mathfrak{h}$) that ...
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7
votes
1
answer
2k
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Polynomial representing prime numbers
Along the lines of Polynomial representing all nonnegative integers, but likely well-known question:
is there a polynomial $f \in \mathbb Q[x_1, \dots, x_n]$ such that $f(\mathbb Z\times\mathbb Z\...
CommunityBot
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5
votes
2
answers
983
views
finite or infinite many quadratic fields embedding into quaternion algebras?
Suppose $H$ is a indefinite quaternion algebra over $\mathbb{Q}$. Are there infinitely many quadratic fields that can be embedded into $H$?
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10
votes
3
answers
3k
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Some arithmetic terminology: "universal domain", "specialization", "Chow point"
As a non-connoisseur of arithmetic and arithmetic geometry, I would like to ask about some terminology, which meaning I haven't been able to find out on some books, nor on wikipedia, nor by google.
...
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2
votes
1
answer
228
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Is there a classification of embeddings of SL_2 into SP_6 as algebraic groups over Q and R respectively?
Is there a classification of embeddings of SL_2 into SP_6 as algebraic groups over Q and R respectively?
see also the link:mathoverflow.net/questions/36762,
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6
votes
0
answers
936
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Can you get Siegel's theorem "for free" from modularity and Mazur's Eisenstein Ideal paper?
There is a well-known theorem of Shafarevich that given a finite set $S$ of primes the number of isomorphism classes of elliptic curves over $\Bbb Q$ with everywhere good reduction outside $S$ is ...
CommunityBot
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2
votes
2
answers
452
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About a non-obvious (?) link between the jacobians of curves and differentials
To explain my problem, I must give a lemma:
Let $X$, $Y$, $Z$ be curves over $k$ (of characteristic 0) such that the genus of $Z$ is greater than 2, and $\pi : X \to Y$, $\phi : X \to Z$ two non-...
- 12.6k
7
votes
2
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883
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Rankin-Selberg convolutions of motivic L-series
Background:
Let $M_{f_i}, i=1,2$ be two modular motives associated to cusp forms
$f_i \in S_{w_i}(\Gamma_0(N_i))$ of weight $w_i$ and level $N_i$ respectively.
The Rankin-Selberg convolution ...
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23
votes
1
answer
2k
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Wanted: Quadratic Space in Characteristic 2 as a Counterexample to a Theorem of Arf
Hi. Peter Roquette sent me an email asking for an example of a quadratic space in characteristic 2 having certain features. I have no idea on this, but maybe someone reading this does.
He would ...
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11
votes
1
answer
903
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Families of genus 2 curves with positive rank jacobians
It's fairly easy to write down families of elliptic curves over $\mathbb{Q}(t)$ such that almost every (i.e. when the "height" of $t$ is sufficiently large) curve in the family has positive rank over $...
- 4,593
18
votes
1
answer
1k
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Eisenstein series as sections of line bundles on moduli spaces
It is well known that a modular form of weight k and level \Gamma is a global section of k-power of a Hodge line bundle over some modular curve. e.g. H^0(X,E^k).
My question is
How to characterize ...
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3
votes
0
answers
483
views
Questions about Shimura curves
1: Suppose $A_3 $ is the moduli space of abelian varieties of dimension 3 .Is the union of all one dimension shimura varieties in $A_3 $ connected?
2: Given a Shimura curve (explicit construction), ...
- 709
8
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1
answer
2k
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How do I visualize finite covers of curves over non-algebraically closed fields?
If $L$ is algebraically closed, fields of transcendence degree one over $L$ correspond to algebraic curves over $L$ up to birational equivalence, and finite extensions correspond to finite Galois ...
- 65.4k
10
votes
1
answer
3k
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Implications of the abc conjecture in Arakelov theory
It is apparent that the abc conjecture is deeply related to Arakelov theory. In one direction, it is shown in S. Lang, "Introduction to Arakelov Theory", that a certain height inequality in Arakelov ...
- 3,631
36
votes
1
answer
9k
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Fontaine-Mazur for GL_1
For any number field $K$, the Fontaine-Mazur conjecture predicts that any potentially semistable $p$-adic representation of the absolute Galois group $G_K$ of $K$ that is almost everywhere unramified ...
- 21.3k
17
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3
answers
1k
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PNT for general zeta functions, Applications of.
When I read it for the first time, I found the whole slog towards proving the Prime Number Theorem and the final success to be magnificent. So I am curious about more general results.
We talk of ...
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1
answer
4k
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How is the period of an elliptic curve defined exactly?
I sometimes read $\int_{E(\mathbf{R})} \frac{dx}{2y + a_1x + a_3}$ and sometimes $\int_{E(\mathbf{R})} |\frac{dx}{2y + a_1x + a_3}|$. Furthermore, one has to choose an orientation on $E(\mathbf{R})$.
...
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7
votes
3
answers
3k
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congruent to 1 mod p
This is a somewhat vague question: for a prime number p, we often see that various counts come out to be 1 modulo p. What are the possible reasons for this?
Here are some I've encountered:
For some ...
- 39.9k
16
votes
3
answers
2k
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On Category O in positive characteristic
Let $G$ be a semisimple algebraic group over an algebraically closed field $k$. In the case that $k$ has characteristic 0, there has been intensive study of the BGG category O of representations of ...
- 52.9k
4
votes
1
answer
1k
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Galois representation attached to elliptic curves
Unfortunately the question I am asking isnt very well-defined. But I will try to make it as precise as possible. Supposed I am given a mod-p representation of $G_Q$ into $Gl_2(F_p)$. I want to check ...
- 57.6k
7
votes
1
answer
701
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$p$-torsion in the Mordell-Weil group of Abelian varieties injecting in reduction
Let $K$ be a number field and $\mathfrak{p}$ be a place of good reduction. It is easy to see that the reduction map on prime-to-$p$ torsion $A(K)[p'] \hookrightarrow A_{\mathfrak{p}}(\kappa(\mathfrak{...
- 57.6k
9
votes
1
answer
777
views
Geometric (or intuitive) interpretation of additional derivatives in characteristic p > 0
In characteristic $p > 0$ there are "extra" differential operators, i.e., ones that are outside the algebra generated by first-order derivations.
Is there any interpretation of these operators in ...
- 12.4k
4
votes
2
answers
923
views
What is the correct formulation of the CDE triangle?
The CDE triangle with C Cartan matrix and D decomposition matrix is well-known for finite groups. I have not seen a full account of this for finite dimensional algebras which I find surprising as ...
- 44.7k
2
votes
1
answer
278
views
What is the family derived from the absolute Frobenius on the Hilbert scheme?
Let $f$ be a Hilbert polynomial, and $X := Hilb_h(P^d_{F_p})$ a Hilbert scheme defined over $F_p$. Then there is an absolute Frobenius map $F: X \to X$. I'm even interested in the case $f \equiv 1$, ...
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6
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1
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395
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Diameter of reduction graph of a curve over a complete discrete valuation ring
Let $R$ be a complete discrete valuation ring with field of fractions $K$ and algebraically closed residue field $k$, and let $X$ be a proper, smooth, geometrically connected curve over $K$. Take a ...
- 11.1k
7
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0
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491
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Alterations of regular varieties
Let $X$ be a regular quasi-projective variety over a perfect field $k$. The existence of a "good compactification" of $X$, i.e. a regular projective variety $\bar{X}$ with an embedding $X\...
- 4,450
3
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1
answer
185
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How many linear terms are in the Hilbert set of H(z,t), a polynomial in 2 variables over a field k(s) of transcendence degree one over a finite field?
I am looking for a good reference for Hilbert's irreducibility theorem, and ofproperties of Hilbert sets besides Serres Lectures on The Mordell-Weil Theorem. In particular, I am interested it to the ...
- 7,442
2
votes
1
answer
987
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Surjectivity of bilinear forms.
It is not uncommon to describe interesting classes of field extensions by declaring that an extension $L|K$ belongs to that class if some type of problem with $K$-coefficiens has a property over $L$ ...
- 4,015
3
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0
answers
409
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How looks the "land of Tamagawa numbers"?
Jonah Sinick's question here, other interesting ideas he mentioned, and Franz Lemmermeyer's remark make one think at Bloch and Kato's drawing + question. What's known or guessed about that "land" by ...
CommunityBot
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2
votes
1
answer
726
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Semi-stable model and Neron model for family of elliptic curves
I am looking for an "easy-to-understand" reference for Neron Models. Specifically if I have a semi-stable family of elliptic curves over $Spec {O}_K$ , with generic fibre $E_K$ and special fibre $E_k$ ...
- 65.4k
3
votes
1
answer
1k
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what is the first Galois cohomology group of the Galois module End(T_l(A)) for some abelian variety A over a finite field k and l some prime number different from the characteristic of the base field?
According to Serre's book 'Galois cohomology', Galois chomology group are always torsion, but it seems to me that H^1(k, End_{Z_l}(T_l(A)))=coker(Frob-1) on End_{Z_l}(T_l(A)), which has the same Z_l ...
- 4,015
4
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0
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390
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Is there a reference that treats principal homogeneous spaces for (say) group varieties using schemes?
I was wondering if anyone could recommend a reference that discusses principal homogeneous spaces for general finite type group schemes over a field $k$ entirely in the language of schemes (or even ...
- 3,380
33
votes
1
answer
1k
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Is the group of integer points on a finite-type group scheme over Z finitely presented?
Let $G$ be a group scheme of finite type over $\mathbf{Z}$. Must $G(\mathbf{Z})$ be finitely presented?
(The question is inspired by a not yet successful attempt to answer a question of Brian Conrad....
CommunityBot
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15
votes
2
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814
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Can the failure of the multiplicativity of Euler factors at bad primes be corrected?
Warning: This one of those does-anyone-know-how-to-fix-this-vague-problem questions, and not an actual mathematics question at all.
If $X$ is a scheme of finite type over a finite field, then the ...
- 9,418
7
votes
1
answer
799
views
Liftability of Enriques Surfaces (from char. p to zero)
Let $k$ be an algebraically closed field of characteristic $p > 0$, $X$ a variety over $k$.
We say $X$ lifts to characteristic zero, if there exists a local ring $R$ containing $\mathbb Z$ with ...
- 22.6k
3
votes
1
answer
844
views
finite generation of the Mordell-Weil group over finitely generated fields
Does anyone know a reference for the proof of the finite generation of the Mordell-Weil group over finitely generated fields?
- 8,945
5
votes
3
answers
2k
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Additive reduction of elliptic curves
Suppose $E/ \mathbf{Q}$ is an elliptic curve with additive reduction at a prime $p$. Is there an easy way to tell if $E$ is a quadratic twist of an elliptic curve $E'/\mathbf{Q}$ with good reduction ...
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744
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p-divisible groups of superspecial abelian varieties
Let $p$ be a prime and $F$ be an algebraic closure of the field with p elements. I will consider abelian varieties over F up to prime-to-$p$ isogeny. Principal polarizations will be $Q$-homogeneous ...
- 51
21
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1
answer
4k
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Crystalline cohomology of abelian varieties
I am trying to learn a little bit about crystalline cohomology (I am interested in applications to ordinariness). Whenever I try to read anything about it, I quickly encounter divided power ...
- 1,209
48
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5
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15k
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Algebraically closed fields of positive characteristic
I'm taking introductory algebraic geometry this term, so a lot of the theorems we see in class start with "Let k be an algebraically closed field." One of the things that's annoyed me is that as far ...
CommunityBot
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12
votes
3
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815
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Decomposition of Tate-Shafarevich groups in field extensions
Suppose $E/\mathbb{Q}$ is an elliptic curve with rank zero. According to the conjecture of Birch and Swinnerton-Dyer, the special value $L(1,E_{/\mathbb{Q}})$ should be equal (up to some harmless ...
- 1,680
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1
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877
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bibl. q.s on Dwork's "p-adic cycles", Mazur's "p-adic variations":
Matthew Emerton mentioned recently the relevance of Dwork's "p-adic cycles". As I wonder if I should read that, reviews of it are ambiguous, I'd be happy on remarks and possible further bibl. hints. ...
CommunityBot
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0
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716
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Lifting abelian varieties in (the closed fiber of) a fixed Neron model
Suppose that $R$ is a dvr with field of fractions $K$ and residue field $k$ and that $A_K$ is an abelian variety over $K$ with Neron model $A$ over $R$. Then the closed fiber $A_k$ is a smooth ...
- 1,609
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1
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1k
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What geometric properties do properties of ell-adic Galois representations imply?
This is the converse question to an earlier question. More precisely,
Let $X/K$ be a curve(or variety) over a global field $K$. We consider the Galois representation obtained by the absolute Galois ...
CommunityBot
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3
votes
2
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732
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If the morphism of root data induced by an isogeny of a reductive group is a Frobenius, is then the isogeny itself a Frobenius?
Let $G$ be a reductive (or just semisimple) algebraic group over an algebraically closed field $k$ of characteristic $p > 0$, let $T$ be a maximal Torus and let $f:G \rightarrow G$ be an isogeny. ...
- 7,108
8
votes
2
answers
2k
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Points of a variety defined by Galois descent
Let k be a perfect field. By a k-variety, I shall mean a geometrically reduced separated scheme of finite type over k. I think that is enough conditions that the following data determine an affine k-...
- 7,108
5
votes
3
answers
739
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Smoothness of hyperplane sections
Suppose $X\subset \mathbb{P}^n$ is a smooth hypersurface defined over $\mathbb{Q}$. For a "generic" prime $p$, what can be said about the set of hyperplanes $H$ in $\mathbb{P}^n(\mathbb{F}_p)$ for ...
- 16.9k
9
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2
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656
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How does the order of a pole of a zeta function indicate any geometric information?
Here, I'm primarily concerced about zeta functions of hypersurfaces over fields of finite characteristic.
Assume $F_q$ to be a finite field with q elements. Consider the zeta function of the ...
- 65.4k
6
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0
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456
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On periods of algebraic integers modulo rational primes
I run, somewhat indirectly, into the following problem and I have no hints where to look in the literature in search for answers or clues.
Let $K$ be a number field, which we may assume Galois if it ...
- 65.4k
2
votes
2
answers
2k
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Reference of primitive root mod p
Can any body give me a reference of the result about primitive root mod p for a class of prime number p.
The result that I am looking for is something along this line:
$2$ is a primitive root mod $p$...
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