All Questions
2,495 questions
3
votes
1
answer
235
views
Interpreting Frobenius pullback as an invariant differential in the case of an elliptic curve
Let $S$ be an affine scheme of characteristic $p > 0$, let $E \rightarrow S$ be an elliptic curve over $S$, and let $F$ denote the absolute Frobenius. Since $E$ is its own $\mathrm{Pic}^0$ there is ...
- 2,663
2
votes
1
answer
153
views
A question about decomposing mod 2 modular forms of level p^2
Fix an odd prime $p$. Each $f \in \mathbb{Z}/2[[x]]$ can be written as $f_{+} + f_{-} + f_0$ where each exponent k of $x$ appearing in $f_{+}$ (resp. $f_{-}$, $f_0$) has Legendre symbol $(k/p)$ equal ...
- 5,422
8
votes
3
answers
1k
views
Deciding a quadratic diophantine equation
Given $a,b\in\Bbb Q_+$, is there an easy way to decide if $$S_{a,b}=\{(x,y)\in\Bbb Z^2:ax^2 + by^2=1\}=\emptyset?$$
I am more interested in seeing if there is a quick way to test for case when ...
- 13.9k
0
votes
0
answers
724
views
Extension of a complete discrete valuation ring
My question came when I was reading the famous Tate's paper on $p$-divisible groups. At the beginning of chapter $(2.4)$ he cites this fact as obvious. If you take a complete discrete valuation ring $...
- 445
4
votes
1
answer
213
views
Subgroups of $Sp_{2g}$ giving rise to Shimura data
Consider the Shimura datum $(GSp_{2g},\mathcal{H}_g)$. Let $G$ be a reductive $\mathbb{Q}$-subgroup of $Sp_{2g}$. I want to know under what condition there exists a point $x\in\mathcal{H}_g$ such that ...
user70064
11
votes
2
answers
930
views
Equivalence of various definitions of arithmetic Chow groups
If I understand correctly, $n$-th arithmetic Chow group of arithmetic variety $X$ is defined as a quotient of the group of pairs of the form $(\sum\limits_in_iZ_i, g)$ where $Z := \sum\limits_in_iZ_i$ ...
- 441
0
votes
0
answers
95
views
Rationality of intersection of algebraic groups
Suppose that $G$ (defined over $\mathbb{Q}$) and $H$ (defined over $\mathbb{R}$) are two algebraic subgroups of a larger algebraic group defined over $\mathbb{Q}$. Assume that $G(\mathbb{R})$ and $H(\...
user70017
2
votes
1
answer
387
views
Is Ш a good parameter for the failure of Global-Local principle for abelian varieties?
(Comparing to class group cases: we have an isomorphism
$Cl(K)\rightarrow \prod \left(K^\times \backslash K_p^\times /O_p^\times \right)$ for a number field $K$.
Similarly, for an elliptic curve $E/\...
1
vote
1
answer
309
views
Dieudonné modules -reference request
I need a reference to start learning about Dieudonn\'e modules, and their application to the arithmetic of abelian varieities. I know that this is a copy of Reference for Dieudonné modules, but.....
- 645
8
votes
1
answer
1k
views
Representability of morphism of stacks
A morphism of Artin stacks $f:X\to Y$ over $\mathbb Q$ is representable by algebraic spaces if and only if its geometric fibres are algebraic spaces. I would like to know if one can use this to prove ...
- 81
6
votes
0
answers
195
views
Non-embeddable varieties
Suppose that $k$ is a perfect field of characteristic $p>0$, $\mathcal{V}$ is a complete discrete valuation ring with residue field $k$ and quotient field $K$, of characteristic $0$.
Then when ...
- 1,838
8
votes
1
answer
774
views
Abelian varieties with good reduction everywhere over function fields
There is a famous theorem due to J.-M. Fontaine,
Il n'y a pas de variété abélienne sur Z
(and independently by V.A. Abrashkin) that there are no abelian varieties over Z. I was wondering whether ...
- 83
4
votes
0
answers
493
views
Bruhat Tits buiding to visualize closed points of affine flag varieties?
In his survey "affine springer fibers and affine Deligne-Lusztig varieties", Goertz gives us a tutorial session on how to use Bruhat Tits buildings to visualize subsets of affine Grassmannians or of ...
- 41
8
votes
0
answers
381
views
Degeneration of wildly ramified local monodromy representations - near or far from Deligne?
Suppose we have a surface $S$ mapping to a curve $C$ and a finite cover $Y/S$ that is ramified at a divisor $D$. For each point $x \in C$ we get a ramified cover of the curve $S_x$, and we can study ...
- 149k
6
votes
2
answers
516
views
Obstruction and rational points on curves
Is etale-Brauer the only obstruction to the existence of rational points on projective plane curves over number fields?
- 11.3k
7
votes
2
answers
650
views
How to compute class number of a torus
Let $T$ be an algebraic torus over a number field $K$.
Following notations in Ono's The Arithmetic of Tori,
http://www.jstor.org/discover/10.2307/1970307?sid=21105671135711&uid=3739888&uid=2&...
- 213
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 $$H^n(\hat{\mathbb{Z}},A^I) \times Ext^{2-n}_{\hat{\mathbb{Z}}}(A^I,\mathbb{Z}) \...
- 213
2
votes
1
answer
463
views
integral basis for the Lie algebra of the Neron model of an abelian variety
Let $A$ be an abelian variety over a number field $K$. Let $\mathcal{A}$ be the Neron model of $A$ over $O_K$. Let $\Omega_{\mathcal{A}/O_K}$ be the sheaf of invariant differential forms on $\mathcal{...
- 213
6
votes
0
answers
438
views
Brauer-Manin obstruction to surfaces of Kodaira dimension 1
Roughly speaking, the Kodaira dimension is an invariant of a variety that corresponds to curvature. One can show that curves of genus $\geq 2$ have Kodaira dimension 1 using Riemann-Roch. In Corollary ...
- 998
26
votes
4
answers
1k
views
Variety acquiring rational point over any quadratic extension
Does there exist a variety $X$ over $\mathbb{Q}$ (or a number field) such that it has no rational points over $\mathbb{Q}$ but acquires points over any quadratic extension $\mathbb{Q}(\sqrt{d})$?
If ...
- 463
4
votes
1
answer
502
views
Motivic integration in positive characteristic: how much is known?
It seems that in papers on motivic integration people usually assume the base field to have characteristic $0$ (and algebraically closed?). My question is: how much can one prove over a positive ...
- 16.9k
4
votes
0
answers
438
views
When an elliptic curve is a quotient of $\mathbb{G}_a$?
I want to know when an elliptic curve $E \rightarrow S$ is a quotient of $\mathbb{G}_a$.
When $S$ is an analytic space, there is an exact sequence $$0 \rightarrow R^1 \mathbb{Z}^{\vee} \rightarrow e^{...
- 2,227
11
votes
0
answers
491
views
Can an abelian variety/Q have no non-trivial points over Q_sol?
Let $A/\mathbb{Q}$ be an abelian variety. Must there be a finite solvable
extension $K/\mathbb{Q}$ such that $A(K)$ is nontrivial?
This follows from the conjecture that the maximal (pro-)solvable ...
- 11.3k
8
votes
1
answer
814
views
Is Scholl construction of modular motives related to Deligne's construction of $\ell$-adic representations?
First of all, I need to declare my extreme ignorance on the topic of modular forms, so, please, does not assume that I know Deligne's construction in details.
In Motives for modular forms, Scholl ...
- 2,227
3
votes
1
answer
223
views
Some clarifications regarding Deligne's paper on $\ell$-adic representations arising from modular forms
I've posted this question few days ago on math.stackexchange because it seems quite superficial. However, since I've got no responses at all, I'm posting it here. If the question is not suitable, ...
- 2,227
15
votes
0
answers
458
views
Is the absolute Galois group of the rationals Hopfian?
Is every continuous epimorphism from the absolute Galois group of $\mathbb{Q}$ to itself injective?
- 11.3k
77
votes
1
answer
15k
views
What is an étale theta function?
Let me start out by urging you to take seriously that whatever I write about the papers surrounding IUTT really are questions. If you would like to use it as a guide to the mathematics in any way, ...
- 13.6k
10
votes
0
answers
706
views
Paths in $\mathrm{Spec} \, \mathbb{Z}$ and Kim's proof of Siegel's theorem for $\mathbb{P}^1 \setminus \{0,1,\infty\}$
This is motivated by a basic number theory question I asked the previous day:
Can there be arbitrarily many cubic fields unramified outside $\{p,\infty\}$? I noted there that the answer to the ...
- 13.8k
150
votes
2
answers
22k
views
What is a Frobenioid?
Since there will be a long digression in a moment, let me start by reassuring you that my intention really is to ask the question in the title.
Recently, there has been a flurry of new discussion ...
- 13.6k
3
votes
2
answers
520
views
cohomological obstructions and rational points
Let $X$ a (nice) scheme over $\mathbb{Q}$. Are there cohomolgical obstructions answering the following questions:
1) is $X(\mathbb{Q})$ an empty set ?
2) is $X(\mathbb{Q})$ a finite (non empty) set ...
- 205
4
votes
1
answer
699
views
When does a hyperelliptic Riemann surface admit a map of degree 3
Let $X$ be a hyperelliptic curve of genus $g>1$.
For which $g$ does $X$ admit a map $X\to \mathbb P^1$ of degree $3$?
I think a genus two curve $X$ admits a map of degree $3$.
Proof: Pick $P$ ...
- 41
11
votes
3
answers
1k
views
References for general Hasse-Weil zeta function
Most research on the Hasse-Weil zeta function focuses on some particular type of algebraic variety, and general surveys usually deal mostly with the better understood elliptic curve case.
I am ...
- 17.6k
7
votes
0
answers
204
views
Groupoid cardinality of DM stack and point counting on coarse moduli spaces
Let $X$ be a finite type DM stack over a finite field $k$ with a coarse moduli space $X_c$. (We only assume $X_c$ is an algebraic space and $X$ might have infinite inertia stack.)
Under which ...
- 91
2
votes
1
answer
234
views
Difference between Frobenii on Tate modules of special and generic fibre
Let $E$ be elliptic curve over $\mathbb Q$ and $p$ a prime of good reduction for $E$. Fix $\ell \neq p$.
If $E_p$ is ordinary then we have Frobenius $F_p$ on $E_p$. Assume $F_p$ lifts to ...
- 91
2
votes
1
answer
191
views
rational point of a curve [closed]
Let $X$ be a smooth projective curve over $\mathbb{Q}$. I heard (if I did not misunderstood) that the geometry of the complex points $X(\mathbb{C})$ (flat, hyperbolic case) dicts the shape (group ...
- 205
10
votes
1
answer
346
views
Is there a solvable point on any variety over the field of complex rational functions?
Let $K = \mathbb{C}(T)$ be the field of complex rational functions in one variable, and let $V$ be a variety defined over $K$.
Must $V$ have a solvable point?
The variety $V$ is assumed ...
- 11.3k
4
votes
1
answer
337
views
Are quaternion algebras from Witt's theorem endomorphism rings of vector bundles?
Let $k$ be a field with char $k \neq 2$. For $a,b \in k^{\times}$, let $(a,b)$ denote the quaternion algebra with $i^2=a$ and $j^{2}=b$, and let $C(a,b)$ denote the projective plane conic given by $...
- 331
5
votes
1
answer
1k
views
Excellent schemes
I noticed that many results in positive characteristic assumes that the object of the theorem is excellent. I have looked up the definition of excellent and have tried to get a feeling for it, but all ...
- 53
24
votes
1
answer
2k
views
When is "independence of l" known?
My question is for which varieties over local fields is "independence of l" known for
etale cohomology. Say $X/{\mathbb Q}_p$ is a complete non-singular variety and $W_l$ is the (complex) Weil-...
- 5,363
5
votes
1
answer
220
views
The generic fiber pullback for $p$-divisible groups in characteristic $p$
Let $R$ be a discrete valuation ring with the field of fractions $K$ and the residue characteristic $p$. If $K$ is of characteristic $0$, then a celebrated theorem of Tate says that the pullback ...
- 2,663
4
votes
1
answer
321
views
Torsors in the analytic topology versus torsors in the etale topology
Let $S= \mathbb A^1_{\mathbb C}$ be the affine line, and let $G$ be a smooth connected reductive group over $S$, e.g., $G = \mathbb G_m, \mathrm{SL}_n$ or $SO_n$.
Is every analytic $G$-torsor over $S$...
16
votes
1
answer
2k
views
Applications of $p$-adic Hodge theory
I am trying to learn $p$-adic Hodge theory. I found some materials explaining main theorems (or aspects) of the theory. However, I could not find references which explaining how to use the theory. ...
7
votes
1
answer
419
views
Weil height of an Abelian Variety with everywhere (potentially) good reduction
Background: Suppose that $E$ is an elliptic curve over $\mathbb{Q}$ with everywhere (potentially) good reduction. there are many ways to define the height of $E$, and I will be concerned with the ...
- 2,834
7
votes
1
answer
1k
views
"Weight-monodromy" for open varieties
Suppose that $X/\mathbb{Q}_p$ is a smooth, projective variety, and choose a prime $\ell\neq p$. Then the weight-monodromy conjecture says that the graded pieces $\mathrm{Gr}_k^M$ of the monodromy ...
- 1,838
9
votes
5
answers
4k
views
Good lecture notes/books on Jacobian of hyperelliptic curve
I want to understand what the Jacobian variety is from an algebraic (or arithmetic?) perspective.
I want to know:
What is the definition of the Jacobian?
Widely know facts about it.
Why the Jacobian ...
- 424
8
votes
2
answers
657
views
Adjoining torsion points from abelian varieties
Let $L/\mathbb{Q}$ be the field generated over $\mathbb{Q}$ by all of the (projective) coordinates of all of the torsion points of all abelian varieties defined over $\mathbb{Q}$. Is $L$ algebraically ...
- 11.3k
48
votes
4
answers
4k
views
Fermat's last theorem over larger fields
Fermat's last theorem implies that the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}$ is finite.
Is the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}^{\text{ab}}$ finite?
Here $\...
- 11.3k
3
votes
2
answers
743
views
Computing the nonsingular projective model of a plane curve
Is there an implemented algorithm available in standard software systems (Sage, Magma, Macaulay, etc.) that will compute the nonsingular projective model of a plane curve over $\mathbb Q$?
- 1,021
3
votes
1
answer
501
views
Trivial Weil-Châtelet group
Does there exist an elliptic curve over a number field $K$ such that $WC(E/K)\cong H^1(G_K, E)$ is trivial?
- 463
0
votes
1
answer
285
views
The number of solutions of a Diophantine equation [closed]
Is $\lim_{n \rightarrow \infty} |\{(x,y) \in \mathbb{Q}(\zeta_n)^2 : y^3 = x^3 + x + 1\}| < \infty ?$ where $\zeta_n$ is a primitive $n$-th root of unity.
That is, I am asking whether the number ...
- 11.3k