All Questions
1,203 questions
3
votes
1
answer
282
views
Frobenius splitting for an excellent, non $F$-finite, $F$-pure hypersurface
Let $p$ be an odd prime. Let $k$ be a field of characteristic $p$ such that $[k:k^p]=\infty$ (i.e. $k$ is not $F$-finite ) .
Also assume that $-1$ is not a square in $k$ . Consider the homogeneous ...
- 642
3
votes
1
answer
262
views
$\mathcal{M}_g$ and $\mathcal{A}_g$ have natural structures as quasi-projective varieties
Reading M. Hindry and J. H. Silverman (Diophantine Geometry-An Introduction), I find the claim that $\mathcal{M}_g$ and $\mathcal{A}_g$ have natural structures as quasi-projective varieties. Mumford ...
- 560
10
votes
1
answer
385
views
Surjectivity of the Abel-Prym map
It is well known that the Abel-Jacobi map restricted to $\text{Eff}_g(C)$ surjects onto the Jacobian $\text{Jac}(C)$, since every divisor of degree $g$ is effective.
Is there an analogous statement ...
- 147
3
votes
1
answer
276
views
Polarization of an abelian variety made by the sum of two divisors
Let $X$ be an abelian variety of dimension $n$, and let $L$ be a polarization, that is, an ample line bundle on $X$, with $\chi(L)=3$.
In my specific case, I have that $L=\mathcal{O}_X(\Theta + D)$, ...
7
votes
1
answer
478
views
Homogeneous vector bundles on Abelian varieties
I recently encountered a result about vector bundles on Abelian varieties, which I found interesting. It characterizes homogeneous (translation invariant) vector bundles on Abelian varieties. More ...
- 5,901
3
votes
0
answers
133
views
Isomorphism of certain irreducible representations over finite fields
We are given a faithful representation of a cyclic group of order 5 $\rho: C_5=G \rightarrow End_{\mathbb{F}_3}(V) $ with $dim_{\mathbb{F}_3}V=8$ as vector space. It is also known that $V=U\oplus W$ ...
- 269
1
vote
0
answers
153
views
A map on Jacobians coming from a correspondence explicitly
From this question, we know that every map of the form $J(C) \to J(C)$ for a curve $C$ and it's jacobian $J(C)$ comes from a correspondence between $C\times C$ and in fact we can take this ...
- 7,746
6
votes
1
answer
711
views
On the moduli stack of abelian varieties without polarization
(I am especially interested in abelian surfaces and characteristic 0).
How bad is the moduli stack of abelian varieties (with no polarization or level structure)? Is it an Artin stack? DM (Deligne-...
- 7,746
2
votes
2
answers
645
views
Supersingular elliptic curves and their automorphisms
If $E$ is a supersingular elliptic curve over a finite field of characteristic $p$, what is known about its automorphism group (i.e the stabilizer of a point in algebraic curves terminology).
Do all ...
6
votes
0
answers
130
views
Sections of infinite order of elliptic surfaces
Let $X\to \mathbb{P}^1$ be a non-isotrivial elliptic surface over $\mathbb{C}$ with a section and with $X$ a smooth projective connected surface over $\mathbb{C}$. Let $\sigma:\mathbb{P}^1\to X$ be a ...
- 513
6
votes
0
answers
126
views
For which (g,q) does there exist a supersingular curve?
We say a curve over a finite field $\mathbb F_q$ is supersingular if it's Frobenius eigenvalues (on $H^1(X,\mathbb Z_\ell)$) are of the form $q^{1/2}\alpha$ for $\alpha$ a root of unity.
As far as I ...
- 7,746
3
votes
1
answer
199
views
Surfaces of general type with $q=1$
Let $X$ be a smooth projective connected surface of general type over $\mathbb{C}$ with $q(X) = 1$, where $q(X) = \mathrm{h}^1(X,\mathcal{O}_X)$.
Let $E$ be the Albanese variety of $X$, and let $X\to ...
- 513
0
votes
0
answers
84
views
Proving the Immersion part of an Embedding
Trying to see the proof of embedding the Jacobian of a Compact Riemann Surface $X$ using Theta functions. So, using the Theta divisor we have the corresponding line bundle say $L$, we want to prove ...
- 954
4
votes
0
answers
100
views
The profinite topology on the Mordell Weil group
In this lecture of Serre on his open image theorem, around 6 minutes, Serre mentions the following theorem of Tate:
Let $A/k$ be an abelian variety over a number field and consider the Mordell-Weil ...
- 7,746
2
votes
0
answers
122
views
A Lefschetz style formula for the $\ell^\infty$ torsion of an Abelian variety over a finite field
Let $A/\mathbb F_q$ be an abelian variety over a finite field. Define $A_\ell = A[\ell^\infty](\mathbb F_q)$, the $\ell^n$ ($n\geq 0)$ torsion points defined over the base field. I can assume $\ell \...
- 7,746
2
votes
0
answers
167
views
Chow variety of 1-cycles on abelian surface
It is an easy exercise to show that on a K3 surface, a smooth genus $g$ curve moves in a $g$-dimensional linear system. Nearly the same exercise shows that on an abelian surface, the corresponding ...
- 1,701
2
votes
0
answers
195
views
Structure of non-big divisors in an abelian variety
Let $A$ be an abelian variety over $\mathbb{C}$. If $A$ has an effective non-big divisor, then $A$ is not simple. (In a simple abelian variety, every non-zero effective divisor is ample.)
What can ...
- 513
5
votes
0
answers
405
views
Jacobian fibration of an abelian fibration
Let $f \colon S \rightarrow C$ be a minimal elliptic surface and let $g \colon J \rightarrow C$ be its jacobian fibration. In this case, we know that the fibers of $g$ are better behaved that the ones ...
- 625
1
vote
0
answers
504
views
The Picard scheme of an ordinary singular curve
Let $k$ be an algebraically closed field, $C$ a proper reduced connected scheme over $k$ of dimension 1, whose singularity is at worse ordinary, $\pi : \tilde{C} \to C$ the normalization of $C$ and $...
- 1,364
1
vote
0
answers
100
views
When a CM abelian variety has complex multiplication by $\mathcal{O}_E$?
I'm reading Milne's note, Complex Multiplication. There are many properties, such as Shimura-Taniyama Formula provided that $A$ is an abelian variety with complex multiplication by $\mathcal{O}_E$. So ...
- 59
4
votes
0
answers
169
views
Extensions of fraction field and residue field
Let $A \subset B$ be integrally closed local domains, $K(A), K(B)$ be fields of fractions, and $k(A),k(B)$ be residue fields. How to prove $[K(B):K(A)] \ge [k(B):k(A)]$?
This question should be easy ...
- 393
0
votes
0
answers
165
views
Elliptic curves and archimedean place
here is my question :
Let $K$ be any field, $ E \to spec(K)$. Let $ v \in M_K $ an archimedean place.
We know that $ \overline{K_v} \simeq \mathbb{C}$ and there exists $\tau_v \in \mathbb{H}$ such ...
3
votes
0
answers
232
views
Lifting a Frobenius endomorphism under an étale morphism
Let $X$ be a smooth affine scheme over $\mathbb{Z}/{p^2}$ that is a complete intersection, say $X$ is the spectrum of $\mathbb{Z}/{p^2}[x_1,...x_n]/(f_1, ... f_r)$, where $n-r$ is the dimension of $X$....
- 111
2
votes
0
answers
476
views
Uniqueness of theta divisor
Let $A$ be an abelian variety (at least over $\mathbb{C}$). Suppose we have two theta divisors $\Theta_1$ and $\Theta_2$ on $A$, which give two principal polarizations on $A$.
In general, are those ...
3
votes
1
answer
251
views
Do Neron-Severi groups of smooth projective unirational varieties contain $\ell$-torsion?
Let $X$ be a smooth projective unirational variety over an algebraically closed field of characteristic $p>0$, and $\ell\neq p$ a prime. My question: can the Neron-Severi group of $X$ contain (non-...
- 16.9k
1
vote
0
answers
299
views
Application of Galois descent
I not understand an assumption done at the beginning of the proof of Rigidity lemma in Moonens and van der Geers book about Abelian variaties (page 12). Here is it:
Question: Why the assumption $k= \...
- 6,006
1
vote
0
answers
194
views
Nef divisors on abelian varieties are pullbacks of ample ones
It is well known that for any polarization ( that is, ample line bundle) $L$ on an abelian variety $A$, there is an isogeny $\phi\colon A \to B$ to another abelian variety with a principal ...
3
votes
0
answers
352
views
Example of a non log-canonical pair for an abelian variety with polarization of degree >2
Let $(A,L)$ be a polarized abelian variety of dimension $g$, with an indecomposable polarization of degree $\chi(L)=d$. There is a theorem of Debarre and Hacon about the singularities of pairs:
...
3
votes
1
answer
317
views
What is known about lower etale cohomology of unirational varieties?
Which information is currently known about $H^1_{et}(X,\mathbb{Z}_l)$ and $H^2_{et}(X,\mathbb{Z}_l)$, where $X$ is a smooth unirational variety over an algebraically closed field of finite ...
- 16.9k
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 ...
- 513
0
votes
1
answer
480
views
Why does MAGMA claim that the automorphism group of a curve is trivial?
I have been trying to compute the Automorphism group of a curve using MAGMA with no success. This is what I have tried: I have tried to compute the Automorphism group of the curve $y^3=x^4-x$ and no ...
- 269
-4
votes
3
answers
525
views
Relation between elliptic curve and Fermat's last thereom
I am looking for a elaborate explanation how the elliptic curve $E (a, b) := y^2=x(x-a)(x-b)$ is associated with the solution of $a^n+b^n=c^n$.
In 1969 Hellegouarch performed the elliptic curves $E (a,...
1
vote
0
answers
325
views
Compatiblity of completion and fibre products. (Formal completion and formal groups)
Let $S$ be a scheme (not necessarily locally noetherian), $X$ a smooth separated group scheme over $S$, and $\hat{X}$ be the formal completion along with the identity section.
Then does the group ...
- 1,364
8
votes
1
answer
774
views
A variant on characteristic $p$ de Rham cohomology
I was thinking about de Rham cohomology in characteristic $p$, and in particular the recent question about Poincare residues, and I came up with the following construction.
Let $k$ be a perfect field ...
- 156k
9
votes
1
answer
456
views
Isomorphic Jacobian Varieties Just Like Abelian Varieties — Torelli's Theorem
Torelli's theorem states:
Let $R$, $R'$ be compact Riemann surfaces of genus $g$, $J(R)$, $J(R')$ their Jacobian varieties, $\Theta$, $\Theta'$ their respective theta divisors. The Riemann surfaces ...
- 560
9
votes
1
answer
731
views
Tamagawa numbers
Let $K$ be a finite extension of $\mathbb{Q}_p$ with absolute Galois group $G_K$. Let $A$ be an abelian variety defined over $K$. The (geometric) Tamagawa number is defined as the order of the ...
- 323
6
votes
0
answers
164
views
What are the genus 4 curves with Jacobians that are 4-th powers?
Consider the moduli space of all genus $4$ curves $\overline{\mathscr M_4}$ of dimension $3\times 4 - 3 = 9$. Under the Torelli map, there is a map to $\overline{\mathscr A_4}$ (which has dimension $...
- 7,746
11
votes
1
answer
340
views
Lifting a splitting of an Abelian variety to characteristic 0
Let $R$ be the ring of integers in a (complete) algebraic closure of $\mathbb Q_p$ with maximal ideal $\mathfrak p$. Suppose I have an Abelian surface $\mathcal A/R$ such that over every $R/\mathfrak ...
- 7,746
3
votes
0
answers
212
views
Genus two curves on abelian surfaces
Considering a smooth genus two curve $C_2$, let $J(C_2)$ be its Jacobian surface, and take $p \in J(C_2)$ an $m$-torsion point. Let $A = J(C_2)/Z_m$, where $Z_m$ acts by $x \mapsto x+p$. The image of $...
- 427
4
votes
0
answers
222
views
Mordell-Weil group of an abelian variety on the perfect closure of a finitely generated field
This question is closely related the question Over which fields does the Mordell-Weil theorem hold?
I consider the following question:
(1) Let $K$ be a finitely generated field extension of $\...
- 26.1k
5
votes
0
answers
459
views
A functor on Abelian varieties corresponding to this operation on Weil numbers
Let $A/\mathbb F_q$ be an abelian variety over a finite field with Weil numbers $q^{1/2}\alpha_1,\dots,q^{1/2}\alpha_n$.
Consider the numbers $q^{d/2}\alpha_1,\dots,q^{d/2}\alpha_n$. These are still ...
- 7,746
1
vote
0
answers
246
views
Frobenius twist of a field
Let $k$ be a field of characteristic $p>0$ (not necessarily perfect). Consider the Frobenius endomorphism $F : k \to k$, $x \mapsto x^p$. I am curious about what happens when we take $k$ as a $k$-...
1
vote
1
answer
94
views
Intersection of a certain linear ideals of $K[[X_1,\ldots,X_{np}]]$ for ${\mathrm{ch}}(K) = p > 0$
Suppose ${\mathrm{ch}}(K) = p > 0$ and we consider the formal power series ring $K[[X_1,\ldots,X_{np}]]$ over $K$ in $np$ variables $X_1,\ldots, X_{np}$. Let $\Lambda$ be the set defined as follows$...
- 563
1
vote
0
answers
157
views
Application of Stein factorisation: rigidity lemma
Let $X,Y$ Noetherian schemes and $f:X \to Y$ proper map. The Stein factorisation factorizes $f$ as $X \xrightarrow{g} Spec \text{ } f_* \mathcal{O}_X \xrightarrow{h} Y$ with $h$ finite and $g$ has ...
- 6,006
3
votes
0
answers
305
views
Abelian varieties by Moonen and van der Geer: proof of rigidity lemma
I try to understand a reduction step in the proof of rigidity lemma as proved in Moonen's and van der Geer's Abelian varieties (Lemma 1.11 on page 12- if the link not work the draft version is online ...
- 6,006
7
votes
0
answers
156
views
Conditions for $p$-divisible group to come from an abelian variety
Over $\mathbb{Z}_p$ or $\mathbb{Q}_p$, are there any known sufficient conditions on a $p$-divisible group for it to come from an abelian variety?
- 726
5
votes
2
answers
478
views
Isogeny components of Jacobians of étale covers
Let us work over $\mathbb{C}$. Fix $X$ a smooth projective curve of genus at least $2$. For every simple abelian variety $A$, it is easy to come up with a ramified covering $Y\to X$ with a non-...
- 1,000
2
votes
0
answers
244
views
Standard application of Oort-Tate classification theorem
$\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\tors}{tors}$In Mazur's paper “Modular curves and the Eisenstein ideals”, on the bottom of page 159, it says that if $T$ is a open subscheme of $...
user141691
3
votes
0
answers
143
views
What kind of equivalences exist between categories of characteristic $0$ and characteristic $p$?
The tilting equivalence for perfectoid algebras gives an equivalence of categories $$K\text{-perf} \cong K^\flat\text{-perf}$$
where the left-hand-side are algebras in characteristic zero and the ...
- 4,164
0
votes
0
answers
60
views
abelian variety with all over non degenerate pairing
Suppose we have an abelian variety $A$ defined over the rational numbers $\mathbb{Q}$. It is known that the Weil pairing
$$
e_\ell: A[\ell]\times A[\ell] \rightarrow \mu_\ell
$$
is non degenerate, ...
- 389