All Questions
481 questions with no upvoted or accepted answers
2
votes
0
answers
128
views
Local duality for abelian varieties
Let $A$ be an abelian variety over a p-adic field $K$. Let $I$ be the inertia group of $K$. There is a Yoneda pairing $$H^n(\hat{\mathbb{Z}},A^I) \times Ext^{2-n}_{\hat{\mathbb{Z}}}(A^I,\mathbb{Z}) \...
- 213
2
votes
0
answers
191
views
degenerate abelian surfaces
I am wondering if the family of degenerate abelian surfaces constructed by K. Hulek, C. Kahn and S.H. Weintraub in "Moduli spaces of Abelian Surfaces: Compactification, Degenarations, and Theta ...
- 357
2
votes
0
answers
442
views
Marten's proof of torelli theorem
I am trying to read the proof of torelli theorem by Henrik H.Martens "A new proof of torelli's theorem" Annals of mathematics vol78 no. 1 .The proof seems to me like using mysterious combination of 3 ...
- 2,106
2
votes
0
answers
160
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Must the coordinates of a polynomial iteration have about the same size?
Original post. The following statements seem plausible (not to say intuitively obvious), but I do not see how to prove them.
Let us say that a polynomial mapping of $\mathbb{C}^2$ is reducible if ...
- 13.8k
2
votes
0
answers
434
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algebraicity of Néron-Tate canonical height for Abelian varieties over global function fields
(transcendence of canonical heights)
Is the Néron-Tate canonical height for an Abelian variety $A$ over a global function field $K$, $\hat{h}: A(K) \times A^\vee(K) \to \mathbf{R}$ known to always ...
user19475
2
votes
0
answers
244
views
Descent theory of line bundles on abelian varieties under isogenies (in char p>0)
I have a couple of questions regarding the descent theory of line bundles on abelian varieties under isogenies in positive characteristic.
Let $X$ be an abelian variety and $L\in Pic(X)$ a line ...
- 614
2
votes
0
answers
139
views
Subgroups-ideals correspondence for abelian varieties over $\mathbf{F}_p$
The question I have arose while reading Waterhouse's Thesis (Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. (4) 2 1969 521–560.), and motivates another question I recently asked.
...
- 2,406
2
votes
0
answers
412
views
a reference for Kummer theory, with proofs ?
What is a standard reference for Kummer theory of semi-Abelian varieties ?
I need a complete exposition with detailed proofs. Also in prime characteristic,
although I am not sure what the statement ...
- 1,299
2
votes
0
answers
255
views
Lang isogeny for group stacks
Let $G$ be a commutative algebraic group stack over $\mathbb{F}_q$ (I don't really care about the precise definition: I'm secretly thinking about the Picard stack of a projective curve). To what ...
- 3,605
2
votes
0
answers
606
views
Kawamata-Viehweg Vanishing Theorem for Excellent Surfaces
Is there any version of Kawamata-Viehweg vanishing theorem that holds for excellent surfaces? I will be very glad if there is one such, but if not then is it at least true for a surface over a non ...
- 165
2
votes
0
answers
212
views
About Alexeev and Nakamura's paper "on Mumford's construction of degenerating abelian varieites"
Is there anyone familiar with this paper? It seems to me it contains "some" typos and even some small elementary mistakes, which makes my reading very slow. Of course the key reason for my slow ...
- 997
2
votes
0
answers
489
views
what are the possible CM-fields of PEL type shimura varieties ?
In the paper "Travaux de Shimura" section 6, Deligne had defined a PEL- type shimura variety, for the following datum $(F,E,D,\psi)$, with $F$ a totally real cubic field, and $E$ a imaginary ...
- 709
2
votes
0
answers
377
views
Which curves cut the Hyperelliptic locus?
Consider the moduli space $\mathcal{A}_{g,n}$ of abelian varieties with some level $n \geq 3$ structure. For simplicity, we just denote it by $\mathcal{A}_{g}$ and drop $n$. Denote the locus of ...
- 637
2
votes
0
answers
279
views
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
2
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0
answers
179
views
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 ...
- 797
2
votes
0
answers
172
views
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
2
votes
0
answers
321
views
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
2
votes
0
answers
225
views
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
2
votes
0
answers
464
views
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 ...
user19475
2
votes
0
answers
368
views
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?
user19475
2
votes
0
answers
644
views
canonical bundle of Abelian surface fibrations
For minimal surfaces admitting an elliptic fibration over a smooth curve,
there is a famous analysis of possible singular fibers and a canonical bundle formula due to Kodaira.
There are two papers of ...
2
votes
0
answers
214
views
structure of $T_\ell A$ for $A/\mathbf{F}_q$ an abelian variety
Can someone give me references for the structure of the $G_{\mathbf{F}_q}$-module $T_\ell A$, $A/\mathbf{F}_q$ an abelian variety?
- 417
2
votes
0
answers
254
views
Projectivized Normal Cone to Satake Compactification
Let $\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties over $\mathbb{C}$.
There exists a compactification, the Satake compactification, which is minimal and has the ...
- 16k
2
votes
0
answers
1k
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deformation of abelian varieties
$k$ is a field of characteristic p, $C_k$ is the category of all artinian local rings with residue field an extension of $k$. $A$ is a dim-$g$ abelian variety over $k$, $L$ is a CM field with $[L:\...
- 1,091
2
votes
0
answers
163
views
The automorphism group of a particular ppav
Let $E$ be the elliptic curve $y^2=x^3-x$ defined over $\mathbb F_5.$ It is ordinary with $j=-2$ and $End_{\mathbb F_5}(E)=\mathbb Z[i],$ where $i:(x,y)\mapsto(-x,2y).$ So $Aut_{\mathbb F_5}(E\times E)...
- 4,265
2
votes
0
answers
321
views
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 ...
- 3,545
2
votes
0
answers
290
views
quasi-trigonal curves
I have read in the literature about quasi-trigonal curves. Such a curve C is a hyperelliptic curve X with two points p,q identified (basically a pinch). They seem to be pretty important in the theory ...
- 3,779
1
vote
0
answers
89
views
Generic reducedness of geometric generic fibre
Let $f:X\to Y$ be a surjective morphism between two projective schemes over a field of characteristic $p>0$. Also assume that $X$ is smooth,$Y$ smooth & irreducible and $f_*\mathcal{O}_X=\...
- 6,006
1
vote
0
answers
82
views
Behavior of translation functors in characteristic $p$
Let $G$ be a semisimple and simply connected algebraic group over an algebraically closed field of characteristic $p>0$, and let $\mathfrak g$ be the Lie algebra of $G$. Let $U_\chi(\mathfrak g)$ ...
- 2,974
1
vote
1
answer
280
views
Moduli space of complex and anti-complex tori?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SL{SL}$By Will Sawin's answer to Moduli Spaces of Higher Dimensional Complex Tori the moduli space of complex $d$-tori is $X ...
- 341
1
vote
0
answers
104
views
Reference about the semiabelian variety associated to a stable curve
If a stable curve $C$ of genus $g$ is such that its dual graph has Betti number $1$, then the Jacobian of $C$ is a semiabelian variety of torus rank $1$. It is described by Mumford that the data of a ...
1
vote
0
answers
82
views
Bounding the Bloch-Kato Selmer group of a twisted symmetric power of a Tate module
Let $E$ be an elliptic curve over $\mathbb{Q}$, with good reduction at $p$, and let $V = H^1_{et}(\overline{E}, \mathbb{Q}_p)$ be (the dual of) its (rationalized) Tate module.
Let $S^nV$ denote its $n$...
- 2,907
1
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0
answers
219
views
What is the relation between two abelian varieties that have the same formal group?
Consider two abelian varieties $A$ and $B$ over the $p$-adic number field $\mathbb{Q}_p$. Let $\hat{A}$ and $\hat{B}$ be the associated formal groups of $A$ and $B$, respectively. Assume that $\hat{A}=...
- 195
1
vote
0
answers
127
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Does the torsion points of abelian varieties transfer to their formal group laws (upon suitable choice of coordinates)?
Let $A$ and $B$ be two abelian varieties over any algebraically closed field. Let $A[p^n]$ and $B[p^n]$ denotes the set of $p$-power torsion points of $A$ and $B$. Assume that $A[p^n]$ and $B[p^n]$ ...
- 195
1
vote
0
answers
195
views
When does the formal group of an abelian variety possess integral coefficients?
I am looking for a sufficient condition that ensures the formal group associated with an abelian variety has integral coefficients.
In precise, let $A$ be an abelian variety over a number field $K$ ...
- 195
1
vote
0
answers
52
views
Abelian surface with CM by a prescribed quartic field
Given a quartic field $K$, is it possible to exhibit an explicit abelian surface $A$ defined over a number field with CM by the field $K$?
For example, let's take the non-Galois CM-field $K=\mathbb Q(...
- 23
1
vote
0
answers
70
views
Simplicity of abelian varieties and localization
Let $A$ be an abelian variety defined over a number field $K$. Let $v$ be a place of $K$ and denote by $K_v$ the $v$-adic completion of $K$ with respect to $||\cdot||_v$.
Assume $A$ is simple, is it ...
- 2,907
1
vote
0
answers
166
views
Reference request showing that a very general Abelian variety $ A $ of genus $ g>1 $ has cyclic class group with ample generator
In Example of a $ \mathbb{Q} $-factorial, CM normal, projective, Mori dream space $ Z $ such that $ \operatorname{Cox}(Z) $ is integral and not CM I asked for an example of a Cohen Macaulay, normal, ...
- 912
1
vote
0
answers
184
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Calculation of de Rham cohomology of abelian varieties/ jacobian varieties
It's known that for a elliptic curve like $E:y^2=x(x-1)(x-t)$ we have a basis $\frac{dx}{y}, \frac{xdx}{y}$ for $H_{dR}^1(E)$. But find such a basis is not an easy thing. I wonder for a general ...
- 785
1
vote
0
answers
125
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Abelian varieties with endomorphism structure
Let me stick to principally polarised abelian varieties $X$ over $\mathbb C$.
I have seen several definitions of what it means for $X$ to have real multiplication by a totally real field $F$:
There ...
1
vote
0
answers
152
views
Vanishing of pushforwards under Fourier-Mukai
In the book Fourier-Mukai and Nahm transforms in geometry and mathematical physics page 85 corollary 3.5 there is the following claim:
First some notation.
Let $X$ be an abelian variety of dimension $...
- 303
1
vote
0
answers
88
views
Bad primes of twists of modular curves $X_E^{-1}(p)$
I am interested in a good reference for reading about primes of bad reduction for the modular curve $X_E^{-}(N)$, which is a twist of $X(N)$ parametrizing all elliptic curves $E'$ whose $N$-torsion ...
- 637
1
vote
0
answers
141
views
Smooth symmetric divisors in abelian varieties without points of order $2$
Let $X=V/\Lambda$ be a complex abelian variety of dimension $g$, endowed with a polarization $M$ of type $(d_1, \ldots, d_g)$. A divisor $D \in |M|$ is called symmetric if $(-1)_X^*D=D$, namely if it ...
- 66.3k
1
vote
0
answers
91
views
Is it possible to lift a pair of points on an elliptic $\mathbb{F}_{\!q}$-curve to a pair of short points on an elliptic $\mathbb{F}_{\!q}(t)$-curve?
Let $E$ be an (ordinary) elliptic curve over a finite field $\mathbb{F}_{\!q}$ of a (quite large) characteristic. For simplicity, suppose that $E(\mathbb{F}_{\!q})$ is a prime group. In addition, let $...
- 1,833
1
vote
0
answers
150
views
Relative compactification without resolutions of singularities
Let $Y$ be a smooth proper variety over a field $k$, let $X$ be a smooth variety over $k$, $U\hookrightarrow X$ the complement of a strict normal crossing divisor and $\phi\colon U\to Y$ a map. By ...
- 309
1
vote
0
answers
269
views
Moduli space of abelian surfaces
Let $K$ be a finite field with algebraic closure $\overline{K}$. The $j$-invariant gives a bijection between the the affine line $\mathbb{A}_K^1$ and $\overline{K}$-isomorphism classes of elliptic ...
- 411
1
vote
0
answers
141
views
Height on $\mathbb G^n_m$ and Néron–Tate heights
Let $A$ be an abelian variety over $\overline{\mathbb{Q}}$ and let $L$ be a line bundle on $A$. Then a Néron–Tate height $\hat h_L$ can be defined by taking a model $\mathcal A$ of $A$ (over the ring ...
- 321
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
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
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