All Questions
1,203 questions
4
votes
1
answer
280
views
Semisimplicity of the étale cohomology mod $p$
Let $X$ be a smooth projective variety over a field $k$. Then if $\ell\neq \text{char} k$, $k$ is finite, and $X$ is an abelian variety it was shown by Weil that the $\ell$-adic cohomology of $X_{k^{...
- 4,428
2
votes
0
answers
175
views
Covering abelian varieties over finite fields with the product of curves
Question. Given an $n$-dimensional abelian variety $X$ over a finite field, is it possible to find smooth projective curves
$C_1,\ldots, C_n$ such that there exists a finite regular map
$C_1\times \...
- 5,901
8
votes
0
answers
688
views
An Azumaya algebra from a vector bundle, and a construction of Belov-Kanel and Kontsevich
Let $S/k$ be a scheme over a perfect field $k$ of characteristic $p>0$.
In Automorphisms of the Weyl Algebra, Belov-Kanel and Kontsevich write down the map
$$\alpha: H^0(\Omega^1_{S/k}/d\mathcal O) ...
- 2,368
11
votes
1
answer
618
views
Canonical lift of the deformation of an ordinary abelian variety
If $A/k$ is a principally polarised ordinary abelian variety ($k$ a perfect field of characteristic $p$, we may assume it is finite for simplicity), we have a canonical lift $\hat{A}/W(k)$.
Now if I ...
- 645
2
votes
0
answers
61
views
The intersection form on a Jacobian
$\DeclareMathOperator{\End}{End}$
Let $J=\mathop{Jac}(C)$ be a Jacobian, $\Theta$ the theta divisor. Since it is principal, it induces a bijection between the Néron-Severri group $NS(J)$ and the ...
- 645
8
votes
1
answer
172
views
Non-degenerate points on a Jacobian surface
Let $C$ be a (hyperelliptic) genus $2$ curve over a number field $K$ with a $K$-rational Weierstrass point $\infty$. We embed $C$ in its Jacobian $J$ via $\infty$.
Question: Is there a quadratic ...
- 93
5
votes
0
answers
160
views
The image of a curve under the multiplication endomorphism of its Jacobian
Let $X$ be a complex smooth projective curve of genus $g\geq 2$. Embed $X$ in its Jacobian
${\rm{J}}(X)\cong{\rm{Div_0}}(X)/{\rm{Div_p}}(X)$ where ${\rm{Div_0}}(X)$ is the group of degree zero ...
- 3,599
8
votes
0
answers
196
views
Simple abelian varieties of $\mathrm{GL}_2$ type with positive rank and large dimension
$\DeclareMathOperator\GL{GL}$I would like know if there are known constructions of simple abelian varieties of $\GL_2$ type of arbitrarily large dimension and positive Mordell-Weil rank, whose rank is ...
- 2,224
4
votes
1
answer
428
views
p-torsion in the Picard group of a regular projective curve
Let $K$ be a field of characteristic $p>0$ and $C$ a regular projective geometrically integral curve over $K$.
If $C$ is smooth, then the connected component ${\rm Pic}^0_C$ of the Picard scheme of ...
- 2,051
8
votes
1
answer
943
views
Automorphisms over finite field that do not lift to an automorphism in characteristic zero
My main question is the following: is there an automorphism of the affine space $\mathbb{A}^n$ (automorphism of an algebraic variety) defined over a finite field which does not lift to an automorphism ...
- 7,722
4
votes
0
answers
213
views
Computing homology class of curve in product of elliptic curves
I have a smooth, projective curve $X/\mathbb{C}$ of genus $g$, embedded in a product of elliptic curves $A = \prod_{i=1}^g E_i$. Since $H_*(A; \mathbb{Z})$ with the Pontryagin product is isomorphic to ...
- 1,856
3
votes
0
answers
197
views
How to write down an explicit equation of given degree yielding a smooth hypersurface in a projective space?
Let F be a field of positive characteristic $p$ and let $d,n$ be two positive integers.
Can we explicitly write down an equation defining a smooth hypersurface $X_d⊂\mathbb P^n_F$ of degree d ?
This ...
- 417
1
vote
1
answer
430
views
Kernel of dual isogeny of elliptic curve
Let $E$ be an elliptic curve defined over an algebraic number field $K$ and $X$ be a cyclic subgroup of $E(K)$ of order $p^n$, then we have the isogeny $E\rightarrow E/X$ and the dual isogeny $ E/X\...
user169802
3
votes
0
answers
368
views
Homomorphisms of abelian varieties and Tate modules
Let $A$ and $B$ be abelian varieties over a field $k$ and $\ell$ be a prime different from the characteristic of $k$, we have an injection $Hom(A,B) \otimes_\mathbb{Z} \mathbb{Z}_\ell \to Hom (T_\ell ...
- 647
11
votes
4
answers
1k
views
Explicit large finite fields in characteristic $2$
Every finite field of characteristic $2$ ist given by $\mathbb{F}_2[x]/P(x)$ for some irreducible polynomial $P\in \mathbb{F}_2[x]$.
For small degree, a simple algorithm gives a way to find $P$. Is ...
- 7,722
5
votes
0
answers
148
views
algebraic de Rham cohomology of toric varieties (reference request)
I haven't been able to find anything workable yet, but I'm looking for a reference on the de Rham cohomology of toric varieties, where as many as possible of the following conditions are handled:
...
- 1,142
3
votes
3
answers
423
views
Are there characteristic-dependent Betti numbers in characteristic not equal to two?
Let $\Delta$ be a finite simplicial complex on $n$ vertices. Let $S = \mathbb{k}[\mathbf{x}]$ be a polynomial ring over a field $\mathbb{k}$ in $n$ variables and $I$ be the Stanley-Reisner ideal of $\...
- 984
1
vote
0
answers
51
views
Do we have $\cap_{P\in S}\left(\mathcal{O}_P+(1-\tau)(K)\right)=\left(\cap_{P\in S}\mathcal{O}_P\right)+(1-\tau)(K)$?
Let $\mathbb{F}$ be a finite field and let $q$ be its number of elements. Let $(C,\mathcal{O}_C)$ be a geometrically irreducible smooth projective curve over $\mathbb{F}$ and let $K=\mathbb{F}(C)$ be ...
- 1,479
1
vote
0
answers
192
views
Integer solutions of Diophantine equation $y^2= 1+4n^{\underline k} $
I am looking for the integer solutions for the diophantine equation $y^2 =4n(n-1)(n-2)\cdots (n-k+1)+1$ for a given $k$ where $n>k+1>5$.
In other words,
$$y^2=1+4n^{\underline k},\tag{I}$$
where ...
2
votes
0
answers
540
views
Polarizations in algebraic and symplectic geometry
In context of Abelian varieties there are a couple of equivalent ways to
introduce the polarization of a algebraic variety. One way is to
choose a line bundle $\mathcal{L}$ which satisfies certain ...
- 6,006
1
vote
0
answers
96
views
Polarization of Prym varieites
I'm trying to understand polarization and rational Hodge structure of spectral curves and Prym varieties.
Excuse me that this is similar to my previous question.
I want to prove the following,
Let $X$...
- 297
3
votes
0
answers
218
views
Is there a proper smooth variety in characteristic $p$ whose Hodge-to-de Rham spectral sequence does not degenerate at $E_1$?
By Deligne-Illusie, such a variety has no lifting to $W_2(k)$. In their paper they state that they do not know if such an example exists. Has this question been answered since then?
- 4,164
3
votes
0
answers
110
views
Using principal polarisation to "cancel" Jacobian summands in isomorphism
I'm working through the sketch proof of irrationality of cubic threefolds in Huybrechts' The geometry of cubic hypersurfaces.
Let $J(X)$ denote the intermediate Jacobian of a cubic threefold $X \...
- 305
9
votes
1
answer
732
views
A question about $p$-adic monodromy of abelian varieties
Let $S_0$ be a smooth (projective?) and (geometrically) connected scheme over a finite field of characteristic $p$ and let $S$ be its base change to an algebraic closure of the finite field. Let $\pi:...
- 537
12
votes
2
answers
660
views
What is the correct notion of representation for abelian varieties?
Zeroth question - am I right that in the "ordinary" sense an abelian variety does not possess any representations at all?
More precisely, a representation of an algebraic group $G$ (over an ...
- 18.4k
13
votes
2
answers
591
views
A geometric definition of the addition law on abelian surfaces
Most people will have see a geometric "explanation" of the addition law on elliptic curves given by embedding it as a cubic in the projective plane and cutting it with lines.
Is there a ...
- 7,746
14
votes
1
answer
1k
views
If it quacks like an abelian variety over a finite field
Consider smooth projective varieties over a finite field. If a curve "looks like" an elliptic curve (i.e. has genus $1$) then it can be made into an elliptic curve.
Is there something ...
- 117
13
votes
0
answers
749
views
Rings whose Frobenius is flat
Let $R$ be a ring of characteristic $p>0$. The (absolute) Frobenius is the map of rings $F_R:R\rightarrow R$ defined by $x\mapsto x^p$.
I am interested in rings for which $F_R$ is flat (hence ...
anon1432
4
votes
4
answers
853
views
Which schemes are divisors of an abelian variety?
Let $X$ be a smooth, projective ireducible scheme over an algebraically closed field $k$. I'm trying to understand when there exists an abelian variety $A$ such that $X$ is isomorphic to a prime ...
- 4,428
2
votes
0
answers
162
views
Faltings' height theorem for isogenies over finite fields
For an Abelian scheme over a ring of integers in a number field, Faltings has a theorem that describes how the Faltings' height changes through an isogeny. There are multiple references for this ...
- 7,746
6
votes
1
answer
348
views
Extensions of (semi-)abelian schemes
Let $S$ be a regular Noetherian scheme, and let $U\subset S$ be the complement of a divisor. If $A\to B$ is an isogeny of abelian schemes over $U$, and $A$ extends to a semi-abelian scheme over $S$, ...
- 63
7
votes
1
answer
498
views
Weyl algebra as an Azumaya algebra over its centre
Assume that $k$ is an algebraically closed field of positive characteristic $p$. On page 3 (page 6 of the PDF file) of Bezrukavnikov, Mirković, and Rumynin - Localisation of Modules for a semisimple ...
- 111
7
votes
0
answers
245
views
Albanese morphism induces an isomorphism on global $1$-forms
Let $X$ be a smooth projective variety over a field $k$ of characteristic zero equipped with a point $e\in X(k)$. There is Albanese morphism $a:X\to \mathrm{Alb}\,X$ which is initial among pointed ...
- 7,377
2
votes
1
answer
295
views
The size of endomorphism rings and the relation to ordinariness of Abelian surfaces
For Elliptic curves over a finite field, there is a very useful characterization of ordinary elliptic as those with commutative, quadratic endomorphism rings and of supersingular curves as those with ...
- 7,746
9
votes
1
answer
546
views
Showing subgroups with equal Lie algebras are equal
Let $k$ be a field. It might as well be algebraically closed, but I do not want to assume that it has characteristic $0$. I will write "group" for "affine group scheme over $k$", ...
- 12.9k
9
votes
0
answers
439
views
Uncountably many non-isomorphic Tate modules
Do there exist uncountably many abelian surfaces with good reduction over $\mathbb{Q}_p$ with pairwise non-isomorphic rational $p$-adic Tate modules?
If we took $l$-adic Tate modules there would be ...
user158636
10
votes
0
answers
438
views
Boundary of Siegel modular variety
The moduli space of curves has a compactification whose boundary can be understood as the product of moduli spaces of curves of lower genus. Therefore (perhaps naively) one might hope that there ...
- 4,428
16
votes
1
answer
984
views
Reconstruct a variety from its crystalline topos
Let $k$ be a perfect field of positive characteristic. Let $X$ be a smooth projective geometrically connected $k$-scheme with a $k$-point.
Can we reconstruct $X$ from its small crystalline topos $((X/...
user158636
4
votes
2
answers
918
views
Katz's proof of Cartier's (descent) theorem
I am trying to understand the proof of Cartier’s theorem on pages 370-371 (pages 17-18 of the PDF file) of Katz’s “Nilpotent connections and the monodromy theorem: applications of a result of ...
- 121
2
votes
0
answers
176
views
Outer Galois representations and Tate modules of Jacobian varieties
Let $X$ be a proper smooth curve over a field $k$. Then we have an exact sequence of profinite groups
\begin{equation*}
1 \to \pi_1(X_{\overline k}) \to \pi_1(X) \to G_k \to 1,
\end{equation*}
...
- 988
4
votes
1
answer
269
views
Fourier transform on finite groups in characteristic $p>0$
Is there a Fourier theory for finite groups in characteristic $p>0$? Assume that $p$ divides the order $|G|$ of finite groups (or just work with $p$-groups), i.e., in a modular representation-...
- 41
47
votes
1
answer
1k
views
Summing infinitely many infinitesimally small variables makes sense in algebra
There is an identity $e^x=\lim_{n\to \infty} (1+x/n)^n$, and I always thought it is a purely analytic statement. But then I discovered its curious interpretation in pure algebra:
Consider the ring of ...
- 3,772
2
votes
0
answers
100
views
Composing equal characteristic and mixed characteristic deformations
Suppose $k$ is an algebraically closed field of characteristic $p>0$ and $A$ is a $k$-point on a moduli space of certain objects over $k$. Suppose, moreover, that $A$ deforms to a $k[[t]]$-point $B$...
- 245
4
votes
2
answers
460
views
Are Chow groups invariant under universal homeomorphisms?
Let $f:X\to Y$ be a universal homeomorphism of schemes, $R$ a coefficient ring. Which assumptions on $f$ and $R$ are suffient to ensure that the pullback map $f^*$ of $R$-linear Chow groups is ...
- 16.9k
6
votes
1
answer
268
views
Hochschild cohomology of an Azumaya algebra
Let $k$ be a field. Given a commutative $k$-algebra $Z$ and an associative algebra $A$ that is Azumaya over $Z$, do we have an isomorphism of Hochschild cohomologies: $HH^*(A) \cong HH^*(Z)$?
This is ...
- 410
6
votes
0
answers
270
views
Tropical abelian variety as a limit
A tropical abelian variety is given by a quotient of a real vector space $V \cong \mathbb{R}^g$ with a fixed integral structure $\Gamma_2$, by a lattice $\Gamma_1$, equipped with some aditional ...
- 61
1
vote
1
answer
441
views
About the type of a polarization of an abelian variety
The following is a question I posted about a week ago on Maths stackexchange there, but it didn't bring any discussion nor comment. For this reason I am posting it here also.
Let $X$ be an abelian ...
- 769
9
votes
2
answers
2k
views
Some basic questions on quotient of group schemes
Let $S$ be a fixed base scheme and $G, H$ be group schemes over $S$. Since I am mainly interested in commutative group schemes over fields, we may assume that $G,H$ are commutative and $S$ is a field ...
- 603
12
votes
2
answers
605
views
Conceptual explanations of the class numbers for the first few $\mathbb{Q}(\sqrt{p})$ with odd conductor
It's known that the class number of $\mathbb{Q}(\sqrt{p})$ is $1$ for all primes $p<229$.
Question: What would it be like for conceptual explanations of $h(\mathbb{Q}(\sqrt{p}))=1$ for the first ...
- 9,501
6
votes
1
answer
373
views
For an abelian scheme, $R^pf_* \Omega^q$ is locally free and its formation is compatible with any base change
Let $k$ be a field, $\bar{R} \to R$ a local homomorphism of artinian local rings with the residue fields $k$, $I$ its kernel, $A/R$ an abelian scheme, and $\mathscr{T}$ its tangent sheaf.
Let $A_0 = A ...
- 1,364