All Questions
481 questions with no upvoted or accepted answers
4
votes
0
answers
144
views
How often is the rank of J_0(p)^- zero
As mentioned in this answer there is a conjecture by
Kimball Martin that, formulated slightly informally, has the following special case.
Conjecture:
On average $J_0(p)$ has 2 simple components when ...
- 2,224
4
votes
0
answers
262
views
de Rham Bloch-Ogus theory in positive characteristic
In their famous paper Gersten's conjecture and the homology of schemes, one of the results that Bloch and Ogus prove is that the second page of the coniveau spectral sequence for $X$ smooth over a ...
- 2,054
4
votes
0
answers
130
views
Castelnuovo–Mumford regularity and wedge powers in positive characterisitc
A vector bundle on $\mathbb{P}^n$ is said to be $r$-regular if
$$H^i(\mathbb{P}^n,F(r-i))=0$$ for all $i>0$.
It is always true that if $F$ is $r$-regular and $G$ is $s$-regular (both vector bundles)...
- 2,356
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
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
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
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
4
votes
0
answers
133
views
Non-cyclic Galois groups over the field of formal Laurent series in positive characteristic
This should be an easy question, but I am unfortunately not able to give an answer, so I am sorry if this is not the appropriate level for the site.
Let $C$ be an algebraically closed field of ...
- 41
4
votes
0
answers
287
views
The $p$-divisible groups which arise from an abelian variety over characteristic $p$?
Can you classify all conditions for $p$-divisible groups arising from an abelian variety? For example, $2\dim = \mbox{height}$.
- 397
4
votes
0
answers
169
views
Fibered surfaces degenerating to Frobenius
Let $R$ be a DVR with algebraically closed, positive characteristic residue field $k$. Let $X\rightarrow Spec(R)$ and $C\rightarrow Spec(R)$ be smooth projective morphisms of relative dimension 2 and ...
- 599
4
votes
0
answers
195
views
lemma II.2.4 in Harris-Taylor (about drinfeld-katz-mazur level structure on 1-dimensional $p$-divisible groups)
Lemma II.2.4 on page 82 in Harris and Taylor's "The Geometry and Cohomology of Some Simple Shimura Varieties" (or lemma 3.2 here), says that given a Drinfeld(-Katz-Mazur) level structure $\alpha:(p^{-...
- 371
4
votes
0
answers
122
views
dual of quotient abelian variety
Let $A$ be an complex abelian variety and $S\subset A$ a finite subgroup.
How to calculate the dual abelian variety of $A/S$?
- 1,891
4
votes
0
answers
115
views
Abelian variety over Q with many roots of unity
Given an abelian variety $A$ over the rational integers $\mathbb{Q}$, and a prime $p$, we know that $\mathbb{Q}(\zeta_p)$ is contained in $\mathbb{Q}(A[p])$, the $p$-division field of $A$, and where $\...
- 389
4
votes
0
answers
275
views
Symmetric power contained in tensor power?
Let $V$ be an $R$-module. Traditionally the symmetric algebra $S(V)$ is defined as a quotient of the tensor algebra $T(V)$, by the ideal generated by all $a\otimes b-b\otimes a$.
Can $S^n(V)$ also be ...
- 2,519
4
votes
0
answers
293
views
Derived categories of coherent sheaves and degenerations of abelian varieties
By the work of Burban-Drozd (https://projecteuclid.org/euclid.dmj/1076621984), we know what happens to the derived category of coherent sheaves when an elliptic curve degenerates into a nodal curve or ...
- 3,187
4
votes
0
answers
468
views
Quaternion algebras in characteristic 2
Let $k$ be a field and let $Q$ be a quaternion algebra over $k$.
It is well known that, if $\mathrm{char}\,k\neq 2$, one can define $Q$ as the $k$-algebra of dimension $4$ generated by elements $x,y$ ...
- 375
4
votes
0
answers
115
views
Relations between an projective variety and galois cohomology
Let $f_1, \cdots, f_k$ be homogeneous polynomials over $\mathbb{Q}[x_0, \cdots, x_n]$. They define an projective variety $X$ over $\mathbb{P}^n(\mathbb{C})$, namely their set of zeros $$X = Z(f_1, \...
- 423
4
votes
0
answers
184
views
Weil Pairing and Galois descent
One way the Weil pairing for an Abelian Variety $A/k$ is phrased is the following (for simplicity, let me only deal with the multiplication by $m$ map ($[m]: A\to A$) instead of arbitrary isogenies):
...
- 7,746
4
votes
0
answers
210
views
Universal vectorial bi-extension as a scheme
In 'The universal vectorial Bi-extension and p-adic heights' Coleman works with the pullback of the Poincaré biextension of an abelian variety A to its universal vectorial extension and claims this is ...
- 1,482
4
votes
0
answers
117
views
How to describe the subspace of invariants under the Rosati involution?
Consider the Jacobian $J_C$ of hyperelliptic curve
$$C\!: y^2 = x^5 + a$$
over a finite field $\mathbb{F}_p$, where $a \in \mathbb{F}_p^*$, $p \equiv 2 \ (\mathrm{mod} \ 5)$, $p > 2$. Let $\pi \...
- 1,833
4
votes
0
answers
240
views
On the class of Shimura data of Hodge type that cover a given Shimura datum of abelian type
A Shimura datum $(G,X)$ is of Hodge type if there exists an injective morphism of Shimura data $(G,X) \hookrightarrow (\mathrm{GSp}_{2g}, \mathfrak{H}^{\pm})$, for some integer $g$.
Let $(G',X')$ be ...
- 133
4
votes
0
answers
267
views
Dimension of the moduli space of abelian varieties with a prescribed endomorphism algebra
Let $D$ be a division algebra over a number field $K$, and consider abelian varieties $A$ over the complex numbers, of dimension $g$ with an action of (an order of) $D$. Is it known when this set is ...
- 2,834
4
votes
0
answers
173
views
Are there methods to compute the induced action of Frobenius map on the Neron-Severi group of a supersingular abelian surface over a finite field?
Let $A$ be a supersingular abelian surface over a finite field $\mathbb{F}_q$. In that case the Neron-Severi group $NS(A\otimes\overline{\mathbb{F}_q})$ is the lattice of rank $6$. Are there methods ...
- 1,833
4
votes
0
answers
176
views
Is a Kummer surface unirational over a sufficiently large finite field of characteristic 2?
Let $A$ be a supersingular abelian surface over a sufficiently large finite field $\mathbb{F}_q$ of characteristic $2$ and let $K_A = A/(-1)$ be the Kummer surface. Shioda ("Kummer surfaces in ...
- 1,833
4
votes
0
answers
245
views
Deformations of the moduli space of ppav's
Consider the complex algebraic moduli space $X:=\mathcal A_g^n$ of ppav's of dimension $g$ with some high enough level $n$ structure (so that it represents the corresponding functor).
Can one compute ...
- 193
4
votes
0
answers
373
views
Moduli of coherent sheaves on abelian varieties
Let $\mathcal{A}_g$ be the moduli space (stack?) of $g$-dimensional principally polarized abelian varieties.
We have the universal family of abelian varieties $\chi_g\rightarrow \mathcal{A}_g$, where ...
- 643
4
votes
0
answers
285
views
Application of Frobenius splitting in characteristic $0$
In general, Frobenius splitting only defines on field of characteristic $p$ (algebraically closed) field.
I am reading Brion and Kumar's book and I can see that there are geometric results can be ...
- 849
4
votes
0
answers
209
views
Partial differential Equation over characteristic p
I want some references on partial differential equations over characteristic $p$. If we have a first order partial differential equation, how can we check whether there exists polynomials or rational ...
4
votes
0
answers
185
views
Are these subspaces of $\mathbb{Z}/3[[x]]$ stable under the shallow Hecke algebra?
This is a characteristic $3$ analog of part of my earlier question, "Are these two subspaces of $\mathbb{Z}/2[[x]]$ the same?"
Notation
Fix a prime $N$ other than $3$. Let $F,G \in \mathbb{Z}/3[[x]]$...
- 5,422
4
votes
0
answers
164
views
Is there an analogue of distributions in characteristic p?
Some motivation: When working over $\mathbb{C}$, distributions (in the sense of generalized functions) act as natural generators for $D$-modules (in the sense that any regular holonomic $D$-module is ...
- 1,488
4
votes
0
answers
189
views
Does this space of mod 2 modular forms admit a (Z/8)* degree decomposition?
Fix an odd N>0. Let M consist of all odd elements of Z/2[[x]] that are the mod 2 reductions of elements of Z[[x]] arising as the Fourier expansions of modular forms for (Gamma_0)(N); it's easy to see ...
- 5,422
4
votes
0
answers
814
views
Adjunction Formula for Weil Divisors on a Normal Variety X
Let $X$ be a normal variety over an algebraically closed field $k$ of characteristic $p>0$ and $S$ be a prime Weil divisor on $X$ which is normal too. Now if $K_X+S$ is NOT $\mathbb{Q}$-Cartier, ...
- 165
4
votes
0
answers
136
views
A subring of the Serre Swinnerton -Dyer ring of level N modular power series
Suppose ell is prime and (N,ell)=1. Consider those power series over Z that are expansions at infinity of modular forms for gamma_0 (N) of weight a multiple of ell-1. I'll say that an element of (Z/...
- 5,422
4
votes
0
answers
289
views
Does the Albanese map satisfy Torelli's theorem
Let $M_h$ be the moduli space of canonically polarized varieties with Hilbert polynomial $h$. Let $M_h \to A_g$ be the Albanese map, with $g$ an integer which depends on $h$ and $A_g$ the moduli space ...
- 381
4
votes
0
answers
249
views
Ordinary vs Non-ordinary for GL(2)-type Abelian Surfaces over Q
Let $A_f$ be an abelian surface over $\mathbf{Q}$ of $\mathbf{GL}_2$-type arising from a weight $2$ cuspidal eigenform $f\in S_2(\Gamma_0(N))$. What is known (or expected to be true) for the size of ...
- 2,406
4
votes
0
answers
218
views
p-divisible group over an algebraically closed field of characteristic p arises from abelian variety
It may be trivially true or trivially false, just a quick ask, if $k=\overline{k}$ and char k = p>0, X is a p-divisible group over $k$, suppose the Newton polygon of $X$ is symmetric, then there ...
- 51
4
votes
0
answers
320
views
Dieudonné modules over rings of charateristic zero
Dear Colleagues,
would appreciate if you could recommend references, if such a theory exits, for the following question.
Let $A$ be an Abelian scheme over $\text{Spec}(R)$, where $R$ is a subring of ...
- 185
4
votes
0
answers
282
views
Does semi-stable reduction behave well with Weil restriction of scalars
Let $A$ be an abelian variety over a number field $K$ with semi-stable reduction over $O_K$.
Does the Weil restriction $\textrm{Res}_{K/\mathbf{Q}}A$ of $A$ to $\mathbf{Q}$ have semi-stable reduction ...
- 1,213
4
votes
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
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 ...
- 4,902
4
votes
0
answers
345
views
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
4
votes
0
answers
1k
views
level structures and moduli of abelian varieties
Hello,
In the definition of level structure of level $n$ for an elliptic curve $A$, there are two versions:
an isomorphism of group schemes $(\mathbf Z/n\mathbf Z)^2 \to A[n]$.
an isomorhpism of ...
- 647
3
votes
0
answers
39
views
p-torsion in the Tate-Shafarevich group of supersingular elliptic curves
Let $E$ be a supersingular elliptic curve over $\mathbb{F}_p(t)$. Is something known on the $p$-torsion of the Tate–Shafarevich group in this case? In particular, I would like to know if (or if known ...
- 103
3
votes
0
answers
111
views
Weil pairing for an abelian variety uniformized as torus modulo lattice over a non-archimedean field
Suppose I have an abelian variety $A$ with split toric reduction over a non-archimedean field $K$, so that $A$ can be realized as $T / \Lambda$ where $\Lambda$ is a lattice in the torus $T$. Letting $...
- 1,298
3
votes
0
answers
192
views
How can I prove this stronger version of Fedder's Criterion?
I was reading Fedder's original paper which proved what is now known as "Fedder's criterion". I noticed that the abstract stated something which is a priori stronger than what is proved in ...
- 317
3
votes
0
answers
135
views
Abel's theorem for cubic threefold
The classical Abel's theorem for curves states that the fiber of Abel-Jacobi map
$$Sym^kC\to J(C),\ (x_1,\ldots,x_k)\mapsto \sum_{i=1}^k\int_{x_0}^{x_i}$$
as a linear functional on $H^0(C,\Omega_C)$ ...
- 1,803
3
votes
0
answers
91
views
Mattuck's Theorem for abelian varieties for a non-locally compact field
Let $A$ be an abelian variety of dimension $d$ defined over a complete ultrametric field $K$ of dimension $0$. Let us put on $A(K)$ the topology induced by the one of $K$ (for example, following ...
- 95
3
votes
0
answers
171
views
Grothendieck-Messing in characteristic 0?
Let $A$ is an abelian scheme over a base scheme $S$. Let $S \rightarrow S'$ be a thickening defined by an ideal of square zero (for example).
If $p$ is locally nilpotent on $S$, then Serre-Tate and ...
- 261
3
votes
0
answers
296
views
Explicit family of polynomials describing embedded torus in complex projective space
This question is cross-posted (with modifications) from MSE. The original question is probably unfit for MathOverflow (although a professor I asked said that this is very nontrivial), but I'm hoping ...
- 1,763
3
votes
0
answers
227
views
Tate isogeny theorem over varieties?
Let $X$ be a nice scheme, $\pi:E\to X$ an elliptic curve, and $\ell$ a prime invertible on $K$. Then we can consider the "Tate module" $(R^1\pi_*\mathbb{Z}_{\ell})^\vee=\hbox{''}\varprojlim\...
- 371