All Questions
1,203 questions
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Classification of fibres in pencils of curves of genus two
For the case of characteristic positive, there exist some clasification of families of curves of genus two over a elliptic curve?.
- 11
1
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0
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231
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Where can I find a copy of Serre's Cours au college de France 1985-1986?
Hi,
I was wondering: where might I be able to find a copy of this work online?
And are there any other resources for the proof of the open image theorem for abelian varieties with endomorphism ring ...
- 15.4k
2
votes
0
answers
179
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Notation for a canonical quotient of an abelian variety in positive characteristic
This is a light question about notation, but I received no answer in Stackexchange.
Let $k$ be an algebraically closed field of characteristic $p>0$ and let $A=A_{/k}$ be an ordinary abelian ...
- 20.5k
3
votes
1
answer
231
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upper bounds for the ranks of the minus parts of modular jacobians
Let $p$ be a prime, and $J^-(p)$ be the maximal quotient of the Jacobian of the modular curve $X_0(p)$ on which the involution acts by $-1$.
Is anything known or conjectured about upper bounds for ...
- 19.2k
1
vote
1
answer
373
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etale covers of line bundles on an abelian variety
subj: etale covers of line bundles on an abelian variety
Is there an explicit decryption of finite
etale covers of a line bundle $L$ on an abelian variety and its associated C*-bundles
$L^o = L \...
- 149k
0
votes
1
answer
162
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Terminology about Abelian varieties over finite fields
Is there a standard meaning for ordinary and supersingular Abelian varieties over finite fields? If so, where can I find it (together with basic properties about them)?
- 12.6k
6
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1
answer
1k
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Albert classification of rational endomorphism rings of simple Abelian varieties over finite fields
Recall the Albert classification of rational endomorphism rings with involution of simple Abelian varieties over arbitrary fields:
Type I: totally real, trivial involution
Type II and III: quaternion ...
CommunityBot
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0
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279
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deRham cohomoloy of CM liftings of Jacobians
Let $k$ be field of characteristic $p>0$ and $W=W(k)$ the ring of Witt vectors of $k$. We call a smooth curve over $k$, ordinary, when the Jacobian of $J(X)$ of $X$ is an ordinary abelian variety. ...
- 637
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742
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Explicit description of O^{cris}_n in Fontaine/Messing
Let $k$ be a perfect field of characteristic $p$, $W(k)$ the Witt ring and $K$ its quotient field. In their article "$p$-adic periods and $p$-adic etale cohomology" Fontaine and Messing give in II.1.4 ...
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316
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Weil pairing as an algebraic cycle?
Is there an algebraic cycle corresponding to the Weil pairing on an abelian variety (of dim>1)? Ideally I'd like to see an example as explicit as possible, e.g.
an explicitly given variety of dim>1 ...
- 468
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397
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false elliptic curves and principal polarizations
Hi,
Let $\Delta$ be a quaternion algebra over $\mathbf Q$ and let $\mathcal O_\Delta$ be a maximal order in $\Delta$.
Recall that a false elliptic curve over a field $K$ is a pair $(A/K,i)$ ...
- 2,842
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0
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172
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Dual Honda systems
Hello,
There is an equivalence of categories between p-divisible groups over the ring of Witt vectors $W(k)$ and the category of "Honda systems", that is couples $(M,L)$ formed by a Dieudonné module $...
- 51
3
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405
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lifting abelian varieties
Hello,
Let A be an abelian variety over $k=\overline{\mathbb{F}_p}$. This abelian variety is endowed with a principal polarization and an action by $\mathcal{O}_E$, the ring of integers of an ...
- 51
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1
answer
510
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hyperalgebras (positive characteristic)
The question is about commutator in integral forms. Let $A$ an associative algebra over a field of characteristic zero, $x\in A$ and $k\in \mathbb Z$, we denote $x^{(k)}=\frac{x^{k}}{k!}$.
How to ...
- 829
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1
answer
740
views
finite non-commutative local group schemes
Can I have some examples of finite non-commutative connected group schemes over a field $k$?
I would like also to see some non-trivial torsors over a $k$-scheme $X$ under such group schemes. Thanks.
- 19.6k
3
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1
answer
980
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Construction of Kummer map for abelian variety
Let $A$ be an abelian variety over the rational numbers $\mathbf{Q}$. Let $V=T_p A \otimes \mathbf{Q}_p$ be the $\mathbf{Q}_p$-Tate module of $A$. Let $G$ be the absolute Galois group of $\mathbf{Q}$. ...
CommunityBot
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321
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CM abelian variety from an algebraic Hecke character?
Hi,
Given an algebraic Hecke character $\chi$ of a number field $k$ there should be a
"rank 1 CM-motive" $M$ with
$\overline Q$-coefficients such that $L(s,M) = L(s,\chi)$. This follows from the ...
- 2,842
8
votes
1
answer
982
views
Is there a really big ring of differential operators in characteristic p?
$k$ is a field of characteristic $p$.
$k[t]$ has canonical first-order differential operator $\partial$
As an endomorphism of $k[t]$, $\partial^p=0$.
First way to fix it:
Use the divided power ...
CommunityBot
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0
answers
403
views
Does every Abelian variety have a finite resolution by Jacobians?
One knows that every Abelian variety is a quotient of a Jacobian. Does every Abelian variety have a finite resolution by Jacobians?
CommunityBot
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2
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1
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570
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Generalization of singular moduli
$j$-invariants of CM curves $E$ over (say) the complex numbers are known as singular moduli. As the theory of complex multiplication explains, singular moduli are algebraic integers of great ...
- 2,406
14
votes
3
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2k
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Representations in characteristic p
Let G be a finite group and let F be an algebraically closed field. If the characteristic of F is 0, then the number of irreducible F-representations of G is given by the number of conjugacy classes ...
- 6,357
2
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0
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225
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Question on division field of abelian variety
I am wondering if the following holds or not.
Let A be an abelian variety of dimension $d\geq 1$ over $\mathbb{Q}$.
Then there is a positive number c depending on d and A such that
$[\mathbb{Q}(A[n])...
- 3,320
11
votes
5
answers
1k
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Schottky locus in genus 2
Let $\phi_g : \mathcal{M}_g \rightarrow \mathcal{A}_g$ be the period mapping from the open moduli space of genus $g$ Riemann surfaces to the moduli space of $g$-dimensional principally polarized ...
- 12.4k
5
votes
2
answers
556
views
Existence of certain identities involving characteristic 2 "thetas"
Let l=2m+1 be prime. In my previous MO question, "What are the polynomial relations between these characteristic 2 thetas?", I defined a subring of Z/2[[x]] as follows:
The subring, S, is generated ...
- 5,422
6
votes
0
answers
436
views
Do there exist weak Lefschetz-type statements that were proved for varieties over complex numbers, but were not proved in finite characteristic?
As far as I understood the situation (reading section 3.5B of Lazarsfeld's "Positivity in algebraic geometry" and also some of the references), some of weak Lefschetz-type statements known rely on '...
- 156k
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1
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Minimal Model Program for surfaces over algebraically closed fields of characteristic p
Let $k$ be an algebraically closed field of characteristic $p>0$.
I have been trying to find out unsuccessfully if there is a mmp for algebraic surfaces over $k$. I know minimal surfaces are ...
- 656
8
votes
3
answers
570
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Variations on a theme of O'Bryant, Cooper and Eichhorn concerning power series over $\mathbb Z/2\mathbb Z$
Define 2 power series over the field $\mathbb Z/2\mathbb Z$ by $f=1+x+x^3+x^6+\dots$, the exponents being the triangular numbers, and $g=1+x+x^4+x^9+\dots$, the exponents being the squares. Write $f/g$...
CommunityBot
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6
votes
2
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945
views
Notation/name for "Artin-Schreier roots"?
If x is an element of a field K and n is a positive integer, we have both a symbol and a name for a root of the polynomial t^n - x = 0: we denote it by x^{1/n} and call it an nth root of x.
Of course ...
CommunityBot
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0
answers
464
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understanding Milne's article "Duality in the flat cohomology of a surface"
I have trouble understanding a point of Milne's article "Duality in the flat cohomology of a surface" http://jmilne.org/math/articles/1976a.pdf
see the "Alternatively" on p. 177, paragraph before ...
CommunityBot
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2
answers
1k
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Explicit way to construct simple complex tori/abelian varieties of dimension at least 2
The following question was motivated by one of the earliest exercises of Complex Abelian Variaties by Birkenhake and Lange during my presentation last year.
It can be shown that any complex torus $X$...
- 47.5k
3
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1
answer
805
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Finite connected groups over a perfect field of characteristic p
In 14.4 of "Introduction to Affine Group Schemes" it is proved (!) that if $A$ represents a finite connected group scheme over a perfect field $k$ of characteristic $p$ then $A$ has the form $k[X_{1}, ...
- 163
3
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0
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308
views
Invertible Hasse-Witt for non-ordinary curves
Assume that $S$ is a smooth curve of over a field $k$ of characteristic $p>0$ and $f\colon X\to S$ is a relative curve over $S$ (i.e., the fibers are curves). It is well-known that when all the ...
- 8,467
5
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1
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641
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Mumford-Tate groups of abelian varieties with potentially good reduction everywhere
Let $A$ be an abelian variety defined over a number field, and let $MT(A)$ be its Mumford-Tate group. It is a conjecture of Morita that if $MT(A)$ is anisotropic-mod-center (that is, it has no $\...
- 10.5k
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4
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2k
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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 ...
CommunityBot
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368
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modern reference for Néron's "Quasi-fonctions et Hauteurs sur les Varietes Abeliennes"
Is there a modern reference for Néron's "Quasi-fonctions et Hauteurs sur les Varietes Abeliennes" http://www.jstor.org/pss/1970644 i.e. using Grothendieck's language of schemes and in English?
CommunityBot
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1
answer
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Mumford-Tate group and Galois representations
Could someone please point me towards a proof of why the image of a Galois representation on the Tate-module of an abelian variety is limited by its Mumford-Tate group?
- 3,032
4
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0
answers
325
views
Good reduction of isogenous abelian varieties over finitely generated fields
Let $K$ be a finitely generated field over $\mathbb{Q}$.
Let $A$ and $B$ be abelian varieties over a field $K$, isogenous over some finite extension $L$ of $K$.
I want to ask if they have the same ...
- 1,500
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1
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818
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To what extent does Poincare duality hold on moduli stacks?
Poincare duality gives us, for a smooth orientable $n$-manifold, an isomorphism $H^k(M) \to H_{n-k}(M)$ given by $\gamma \mapsto \gamma \frown [M]$ where $[M]$ is the fundamental class of the manifold,...
- 3,272
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256
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The Schrodinger representation on the space of sections of a general $(1,3)$-polarized abelian surface
This question arose while I was studying some finite covers of abelian surfaces.
Let $(A, \mathscr{L})$ be a $(1,3)$-polarized abelian surface over the complex numbers and consider the vector space $...
- 66.3k
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454
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If $p=0$ and $df=0$, is $f$ a $p$th power?
This question is a follow-up to When does the relative differential $df=0$ imply that $f$ comes from the base?. There it was asked, for an $A$-algebra $B$, under what conditions does $df=0$ (in the ...
CommunityBot
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7
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5k
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Chevalley's Theorem on Constructible Sets
I'm having a hard time understanding the theorem in the title, more specifically the proof of the related fact that the image of a dominant morphism contains a dense open set of it's closure. (My ...
- 1,298
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3
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2k
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Elliptic curves on abelian surface
Let $Y$ be an abelian surface. Is it true that for every general point $P \in Y$, there exists an elliptic curve passing through $P$?
CommunityBot
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2
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386
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Zariski closures of one parameter additive maps in positive characteristic
Suppose we are in characteristic $p$, and that the field, $K$, that we are working over is imperfect. We have a map $\Theta: K \to K^{\delta}$ where each coordinate function $\Theta_i$ is an additive ...
- 30.5k
5
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1
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710
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Log resolutions on surfaces and 3-folds in characteristic p
If $X$ is a normal projective variety and $D$ a divisor in it, we say that $\pi\colon (\widetilde X,\widetilde D)\rightarrow (X,D)$ is a log resolution if $\widetilde X$ is a resolution of $X$, the ...
- 10.5k
6
votes
2
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1k
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CM abelian varieties and potential good reduction
Let $F$ be a number field and $A$ an abelian variety over $F$. It is known that if $A$ has complex multiplication, then it has potentially good reduction everywhere, namely there exists a finite ...
- 10.5k
8
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873
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Resolution of singularities in positive characteristic
I am currently trying to make some small parts of the minimal model program work for some very explicit varities in positive characteristic. I have such a variety $X_1$ and I know that there is a ...
- 783
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2
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536
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What are the polynomial relations between these characteristic 2 "thetas" ?
Suppose $\ell=2m+1$, $m>0$. Define $[i]$ in $\mathbb{Z}/2\mathbb{Z}[[x]]$ to be $$\sum_{n\equiv i\mod l} x^{n^2}.$$ Note that $[0]=1$, and that $[i]=[j]$ whenever $\ell$ divides $i+j$ or $i-j$.
...
- 5,422
4
votes
0
answers
345
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relationship between pairings on principally polarized abelian varieties
Let $X$ be a $g$-dimensional principally polarized abelian variety over $\mathbb{C}$, for example the jacobian of a curve of genus $g$. Let $X = \mathbb{C}^{g}/\Lambda$ where $\Lambda$ is a full $\...
- 51
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0
answers
560
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Function fields of characteristic p modular curves, and mod p reductions of the classical modular equation
Let l and p be distinct primes, l>2. There are "characteristic p modular curves" X_0(l) and X(l), defined over an algebraic closure, K, of Z/p, solving moduli problems for elliptic curves with some ...
- 5,422
9
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1
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683
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Can we always find a curve which doesn't have semi-stable reduction
Let $K$ be a number field and let $g$ be a positive integer. Does there exist a smooth projective geometrically connected curve $X/K$ of genus $g$ such that $X$ does not have semi-stable reduction ...
- 11.1k