All Questions
1,203 questions
33
votes
5
answers
8k
views
Why no abelian varieties over Z?
Motivation
I learned about this question from a wonderful article Rational points on curves by Henri Darmon. He gives a list of statements (some are theorems, some conjectures) of the form
the set $\{...
- 1,446
5
votes
0
answers
81
views
Compute the principal polarization on $J_0(N)$ in terms of modular symbols
If we consider the modular curve $X = X_0(N)$ as a curve over $\mathbb C$ then one can describe the jacobian $J(X)$ as $H^0(X,\Omega^1_X)^\vee/H_1(X,\mathbb Z)$ as one can do for any curve $X$. ...
- 2,224
2
votes
0
answers
145
views
How to compute the character of the Steinberg module for the group $\mathrm{SL}_n$ over a field of characteristic $p$?
It is known that the Steinberg representation $V$ of the group $\mathrm{SL}_n$ over a field $k$ of characteristic $p$ (maybe one needs to assume that $k$ is perfect, I am not sure) is the irreducible ...
- 63.9k
3
votes
0
answers
127
views
Isogeny of elliptic curve over positive characteristic $p$ which does not come from characteristic $0$
Let $K$ be quadratic imaginary field. Let $E$ be an elliptic curve which has CM over $R_K$
($R_K$ is ring of integers of $K$).
According to SIlverman's ''ADvanced topics in the arithmetic of elliptic ...
- 1,541
2
votes
0
answers
47
views
Characters of simple $\mathfrak{sl}_n$-modules in positive characteristic with subregular nilpotent central character
Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$ (i.e. $\chi$ is a nilpotent matrix whose Jordan normal form has two blocks ...
- 63.9k
4
votes
0
answers
64
views
An analog of a BGG resolution in subregular case in positive characteristic
Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$. For every regular weight $\lambda$ of $\mathfrak{sl}_n$, we have the ...
3
votes
0
answers
120
views
Resolving the "wild" singularities of $\mathbb A^n/C_n$
Let the cyclic group on $n$ elements, $C_n$, act on $\mathbb A^n$ by permuting the co-ordinates (over a field $k$). If $n \neq 0 \in k$, we can resolve the singularities of $X = \mathbb A^n/C_n$ by ...
- 7,746
2
votes
0
answers
119
views
Any kind of duality between differentials and Tate modules?
Let $X$ be a curve over some algebraically closed field $k$ and let $J$ be its Jacobian. I have read that one should think of the Tate module $T_lJ$ as being the first homology group of $X$ with ...
- 12.9k
7
votes
1
answer
296
views
Reference request. Finiteness of the Selmer group
Let $K$ be a global field (ie either a number field or the function field of a curve over a finite field). Let $A,B$ be abelian varieties over $K$ and let $\phi:A\to B$ be an isogeny. Associated with $...
- 141
1
vote
1
answer
149
views
When is $R$ a direct summand of Frobenius pushforwards?
Let $(R,\mathfrak m)$ be a reduced Noetherian local ring of prime characteristic $p$. For integer $e>0$, let $F^e_* R$ denote the $R$-module which is $R$ as an abelian group, but the $R$-module ...
- 20.5k
2
votes
0
answers
177
views
How do characters of representations in cohomology depend on the (positive-characteristic) field?
The following sentence appears in Jantzen - Representations of algebraic groups, 2nd edition, p. x, where $G$ is a reductive group over an algebraically closed field $k$, $B$ is a Borel subgroup, $T$ ...
- 12.9k
3
votes
1
answer
237
views
Are there any quadratic functions on an abelian variety not from the height machine?
Let $X$ be an abelian variety defined over a number field $K$. We know that the Neron--Tate height machine associates to a class in the Picard group of $X$ a unique quadratic function which is zero at ...
- 149k
2
votes
1
answer
243
views
Intersection of translate of divisors on abelian variety
Setup. Let $K$ be an algebraically closed field of characteristic zero, and let $A/K$ be a simple abelian variety of dimension $n$. Let $\{ x_1,x_2,\dots,x_{m^{2n}}\}$ denote the $m$-torsion points of ...
- 254
5
votes
0
answers
480
views
What does Colmez's conjecture tell us?
There is the well known conjecture of Colmez, which describes the logarithmic derivative of the $L$-function of a character via the periods of CM-abelian varieties. Equivalently it describes the ...
- 4,428
4
votes
1
answer
227
views
Compute de Rham-Witt sheaves
I am really new to this, but I am having a hard time understanding all the de Rham-Witt construction.
It seems to be really difficult to compute anything with those beasts: like I cannot find any ...
3
votes
0
answers
314
views
Ampleness of the normal bundle to the Albanese image
Let $X$ be a projective surface of general type over $\mathbb{C}$, and assume that $\Omega_X$ is globally generated. Then the Albanese map $a \colon X \to \operatorname{Alb}(X)$ is a local embedding ...
- 66.3k
15
votes
7
answers
2k
views
Examples of rational families of abelian varieties.
I'd like to know examples of non-trivial families of abelian varieties over rational bases (e.g. open subschemes of the projective line P^1).
One can generate many examples as Jacobians of rational ...
- 42.9k
9
votes
0
answers
1k
views
Ample vector bundles, $H^1=0$ and global generation in characteristic $p$
This is a follow up from Ample vector bundles on curves question, in characteristic $p>0$. First, some background. It is known that in characteristic $0$, a vector bundle $E$ on say a projective ...
- 12.9k
11
votes
0
answers
420
views
Good reduction of finite etale covers of abelian varieties
Let $R$ be a dvr (whose residue characteristic is zero if it helps) with fraction field $K$.
Let $A$ be an abelian variety over $K$ with good reduction over $R$. Let $X\to A$ be a finite etale ...
- 81
6
votes
1
answer
771
views
A regular, geometrically reduced but non-smooth curve
Can anyone give an example of a projective, regular, geometrically reduced but non-smooth curve ?
Of course, the base field should be imperfect.
In Exercise 4.3.22 of Qing Liu's book Algebraic ...
- 149k
4
votes
0
answers
197
views
Inequalities between numerical invariants of nonsingular projective Varieties in positive Characteristic
It is well-known that Miyaoka and Yau-type inequalities do not hold in positive characteristic. In "a note on Bogomolov-Gieseker’s inequality in positive characteristic", however, we can ...
- 2,821
2
votes
0
answers
253
views
Künneth formula for algebraic de Rham cohomology
Let $X$ and $Y$ be finite type schemes over a field $k$, and let $H^i(X/k)$ denote the $i$-th algebraic de Rham cohomology group of $X$ over $k$. I'm interested in the extent to which a Künneth ...
- 63.9k
3
votes
0
answers
154
views
Principally polarised supersingular abelian varieties
If $A/k$ is a supersingular abelian variety, then it is $k$-isogenous to $A'/k$ a superspecial abelian variety, see "A Note on supersingular abelian varieties" by Chia-Fu Yu. And if $A'/k$ ...
- 645
3
votes
1
answer
296
views
$p$-power torsion of semiabelian variety
Let $K$ be a finite extension field of $\mathbb{Q}_p$. Let us consider a semiabelian variety $G$ defined over $K$, i.e there exists an extension of an abelian variety $B$ and a torus $T$ defined over $...
- 7,377
35
votes
3
answers
5k
views
In which ways can the isogeny theorem fail for local fields?
Fix a field $K$ with absolute Galois group $G$. By an isogeny theorem over $K$, I mean the statement that the map $\operatorname{Hom}(A,B)\otimes\mathbb{Z}_l \to \operatorname{Hom}_G(T_l A, T_l B)$ is ...
- 2,821
9
votes
1
answer
768
views
Is a complex algebraic set with a Zariski dense subset of algebraic points already defined over the algebraic numbers?
This is a cross-post! For the original post on SE (9 upvotes, no answer) see:
https://math.stackexchange.com/questions/4475853/is-a-complex-algebraic-set-with-a-zariski-dense-subset-of-algebraic-...
- 149k
12
votes
4
answers
2k
views
Quotients of Abelian varieties by finite groups
Let $A$ be an Abelian variety (over an algebraically closed field). The group $\mathbb{Z}/2\mathbb{Z}$ acts on $A$ and the quotient is called the Kummer variety of $A$. These are well studied and, ...
- 12.9k
10
votes
0
answers
294
views
Relation between Faltings height and height on moduli space
Let $E$ be an elliptic curve over a number field $K$. The difference between the semistable Faltings height $h_F(E)$ of $E$ and the height $h(j_E)$ of the $j$-invariant of $E$ can be bounded in terms ...
- 321
3
votes
1
answer
642
views
Decomposition theorem for polarized abelian varieties in positive characteristic
In characteristic zero we have the following decomposition theorem for polarized abelian varieties: it gives an isomorphism between a PPAV and a product of PPAV's of lower dimension and is valid (as ...
CommunityBot
- 1
1
vote
0
answers
116
views
Abelian subvarieties corresponding to vector subspaces
Let $S$ be a connected smooth projective surface.
Let $C$ a smooth curve on $S$
In page 9 of the paper "https://arxiv.org/abs/1704.04187v1" a read the following:
Let
\begin{equation*}
r: ...
- 519
3
votes
1
answer
283
views
Is the theta characteristic attached to an etale double cover of a plane quintic arising from a cubic threefold even in characteristic two?
We work over an algebraically closed field $k$ of characteristic $2$. Let $X$ be a cubic threefold realized as a conic bundle via $f: X \to \mathbb{P}^2$ (after blowing up some line). Let $C \to \...
- 679
1
vote
1
answer
293
views
What's a right parameter space of abelian varieties over a non algebraically closed fields?
Let $k$ be a field of characteristic not 2 or 3. Then the set of elliptic curves over $k$ can be parametrized by the affine variety $S=D(4a^3+27b^2)\subset\mathbb{A}^2_k$ via the family $E\to S$ where ...
- 149k
4
votes
0
answers
284
views
modularity of elliptic curves over function fields in positive characteristic
Let $F$ be a global function field over a finite field, and $E$ be an elliptic curve over $F$. Much like the number field case, it is natural to study the Galois representation on the Tate module of $...
- 685
1
vote
0
answers
76
views
How to construct explicitly defining polynomials of an morphism between smooth irreducible curves?
Let $\phi\!: C_1 \to C_2$ be a separable morphism of smooth irreducible curves embedded as projectively normal models by invertible sheaves $\mathcal{L}_1$ and $\mathcal{L}_2$ respectively. Theorem 4....
- 1,833
2
votes
0
answers
218
views
Borel-Weil-Bott theorem for wonderful compactification in characteristic p
Are there any known results for a Borel-Weil-Bott theorem for the wonderful compactifications over characteristic $p$ (i.e., theorems that classify the cohomologies of all line bundles on a wonderful ...
- 41
4
votes
3
answers
795
views
Quotients of Tate modules
Let $p$ be a prime number, let $K$ denote a finite extension $\mathbb{Q}_{p}$ and let
$\overline{K}$ be an algebraic closure of $K$. Let $A$ be an ellitpic curve over
$K$ and denote by $T_{p}A$ its ...
- 1,949
1
vote
0
answers
135
views
Polarization induces alternating pairing on homology
Let $A$ be an abelian variety over $\overline{\mathbb{Q}}$. We work up to isogeny (i.e., Hom sets are tensored with $\mathbb{Q}$). I am looking for a reference (and ideally, a short explanation) for ...
- 533
10
votes
1
answer
1k
views
Are there "reasonable" criteria for existence/non-existence of Levi factors or their conjugacy in prime characteristic?
Classical theorems attributed to Levi, Mal'cev, Harish-Chandra for a finite
dimensional Lie algebra over a field of characteristic 0 state that it has a Levi decomposition (semisimple subalgebra plus ...
- 2,821
4
votes
0
answers
296
views
de Rham Witt complex vs. de Rham complex of the Witt ring
I am reading the paper "Revisiting the de Rham-Witt complex" by Bhatt-Lurie-Mathew and I am a bit confused about the difference between $W\Omega_R^*$ and $\hat{\Omega}^*_{W(R)}$.
Let $\...
- 218
4
votes
0
answers
204
views
Explicit description of wonderful compactification for PGL_3
Let $k$ be an algebraically closed field of positive characteristics. Let $X$ be the wonderful compactification of $PGL_3$ (see for example section 6 of "Frobenius Splitting Methods in Geometry ...
- 163
2
votes
0
answers
147
views
Automorphism groups of "reductive" Lie algebras in positive characteristic
I put "reductive" in quotes because, of course, in positive characteristic one should speak of Lie algebras of reductive groups, not of reductive Lie algebras.
Let $G$ be a reductive group ...
- 63.9k
5
votes
1
answer
291
views
Do there exist elliptic curves over $H_K$ having everywhere good reduction and CM by $\mathcal{O}_K$?
For $K$ a number field, denote by $\mathcal{O}_K$ its ring of integers and by $H_K$ its Hilbert class field.
For which imaginary quadratic field $K$ does there exist an elliptic curve $E$, defined ...
- 37k
0
votes
0
answers
93
views
Determine the CM type of a CM elliptic curve
I have something don't understand about the CM types of CM elliptic curves. I want to determine the CM type of certain elliptic curves. Let $K=\mathbb{Q}(\sqrt{-3})$ be the CM field and $\omega=\frac{-...
- 390
20
votes
5
answers
4k
views
Equivalent statements of the Riemann hypothesis in the Weil conjectures
In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with $q$ elements says that: the eigenvalues of ...
- 4,713
2
votes
1
answer
361
views
Lie algebroid in algebraic geometry
When I did net-surfing at home, I met some geometric backgrounds of Lie algebras and encountered the concept of Lie algebroids. In differential geometry, a Lie algebroid seems to be defined as ...
- 4,008
4
votes
1
answer
169
views
Tri-homogenous polynomials of tridegree $(3,3,3)$ to add three points on an elliptic curve
Consider an elliptic curve $E \subset \mathbb{P}^2$ with the zero point $\mathcal{O}$. There are classical articles about complete systems of addition laws on $E$ (see
Lange and Ruppert - Complete ...
- 20.8k
3
votes
0
answers
87
views
Two pairings on the group $K(L)$ associated with a non-degenerate line bundle $L$ on an abelian variety
Let $A$ be an abelian variety over a field and let $L$ be a non-degenerate line bundle on $A$.
Then $L$ gives rise to a morphism $\lambda:A\to A^*$ from $A$ to its dual.
As usual, let $K(L):=\ker(\...
- 3,076
4
votes
1
answer
287
views
Difference between stabilizer and automorphism group of subvariety of an abelian variety
Let $X$ be a smooth closed subvariety of a complex abelian variety $A$. Assume $X$ is of general type and of codimension one with $\omega_X$ ample.
Often, people speak about the stabilizer $\mathrm{...
- 38k
9
votes
1
answer
356
views
Reference request for indecomposable representations of $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$
Is there a good reference on the classification of indecomposable representations of the Lie algebra $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$ (of course, in ...
CommunityBot
- 1
2
votes
0
answers
97
views
Non-noetherian Cartier Isomorphism
A result in positive characteristic is that if $R/\mathbb{F}_p$ is a smooth ring, then we have a Cartier isomorphism
$$\Omega_{R}^\bullet\cong H^\bullet(\Omega_R^\bullet)$$
which is essentially ...
- 4,428