All Questions
2,495 questions
3
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What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers $\sqrt{A}$ and the integer $A$
What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers and the integer
As we know,$\sqrt{2} = [1;2,2,2,2,…]$; while $\sqrt{14}= [3;1,2,...
- 3,104
2
votes
3
answers
476
views
How to define the input of computable function or Turing machine over real numbers
Computation or computability over $\mathbb{N}$ can be extended to computation or computability over $\mathbb{R}$ or even computation or computability over $\mathbb{C}$.The following is a formal ...
- 5,502
5
votes
1
answer
572
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What is known about the Brauer group of an arithmetic surface?
Let $X$ be an arithmetic surface over $\mathbb{Z}$, that is we have $\pi: X\rightarrow Spec(\mathbb{Z})$, $X$ is integral, two-dimensional and regular and $\pi$ is projective and flat.
What is known ...
- 644
0
votes
1
answer
630
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Obstruction and 1st order infinitesimal deformations of Generalized Elliptic Curves (Deligne-Rapoport)
We consider the deformation theory of a generalized elliptic curve $(C_0,+)$ over a field $k$. Let $D$ be the deformation functor.
And now we only consider the case that $C_0$ is irreducible as in D-R ...
CommunityBot
- 1
1
vote
0
answers
101
views
Points on the intersection of an affine quadric and cubic over a finite field
Are there absolute constants $N$ and $B$ such that the following is true?
Let $p>B$ be a prime. Let $q(x_0,\dotsc,x_n)$ and $c(x_0,\dotsc,x_n)$ be homogeneous of degree $2$ and $3$ with ...
- 3,142
9
votes
1
answer
855
views
Power operations and Lambda-structure-like lifts of Frobenius in $E_\infty$-geometry?
A $\Lambda$-structure on a commutative ring $R$ is a ring endomorphism wich restricts to the $p$-Frobenius homomorphism after localizing at $(p)$. One may think of this as a "flow" $\Phi \colon Spec(R)...
- 25.6k
1
vote
1
answer
173
views
Compact subgroups of linear groups over nonarchimedean fields
Let $n \in \mathbb{N}$, $K$ a (nonarchimedean) local field, $\overline{K}$ its algebraic closure. Take a compact subgroup $G \leq \text{GL}_n(\overline{K})$. Must there be a finite extension $F$ of $K$...
- 63.9k
3
votes
1
answer
324
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Faithful representations of free pro-p groups
Let $p$ be a prime number, $m,n \in \mathbb{N}$, $F = F(p,m)$ be the free pro-$p$ group on $m$ generators. For which $(m,n)$ there is a continuous faithful representation (embedding) $\rho : F \...
- 4,738
9
votes
3
answers
2k
views
How many integer points does my favorite ellipse go through?
The equation of the ellipse interpolating the six lattice points $(0,0)$, $(1,0)$, $(0,1)$, $(d-1,d)$, $(d,d)$, $(d,d-1)$ in the plane for a fixed $d$ (at least 3) is
$$
x^2+y^2 - \frac{2(d-1)}{d}xy-x-...
- 480
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votes
3
answers
683
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Circles avoiding rational points of height $\le h$
Q. Which origin-centered circles $C(r)$ (or spheres in dimension $d$)
of radius $r < 1$ avoid all rational points
of height $\le h$?
A rational point is a point all of whose coordinates are ...
- 12.6k
4
votes
1
answer
235
views
Which valuations of a field yield codimension $1$ subschemes of their 'models'
For a field $F$ (for example, a one generated by a finite number of its elements) there is a directed set of its 'models' (in this case those are 'arithmetic' schemes whose fraction field is $F$). It ...
- 550
0
votes
0
answers
694
views
"Descending cohomology, geometrically" by Mazur:
(Exist texts of that talk or related texts: http://ttv.mit.edu/collections/harris60/videos/13881-problem-session-barry-mazur ?) Article: http://www.math.harvard.edu/~mazur/papers/page37.pdf
- 10.8k
3
votes
1
answer
276
views
Mumford-Ramanujam examples in characteristic p [and in Arakelov geometry]
For a compact Riemann surface $B$ of genus $\geq 2$, it is a consequence of the Narasimhan-Seshadri theorem that there exist rank-$2$ vector bundles $E \to B$ of degree zero, all of whose symmetric ...
- 13.8k
8
votes
1
answer
380
views
Is the Tate conjecture known for etale covers of products of curves
Let $X$ be a (smooth projective geometrically connected) surface over a finitely generated field $k$. The Tate conjecture predicts that, for $l$ a prime number invertible in $k$, the Chern class map ...
- 83
1
vote
0
answers
183
views
Interpretation of the Gross-Zagier formula for Green function
I am reading the paper of Gross and Zagier on heights of Heegner points and would like to check with the experts whether the following (meta?)mathematical statement makes sense.
In the calculation of ...
- 5,186
4
votes
0
answers
180
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minimal conductors among elliptic curves with a fixed CM type
Let $K$ be a quadratic imaginary field. To simplify my life, let us assume
that $K$ has class number one.
Consider the following infinite set:
$S_1:=$ $\{$ $E\subseteq\mathbf{P}^2(\mathbf{C})$ is an ...
- 20.4k
2
votes
1
answer
466
views
On the conductor of the Groessencharacter of a CM elliptic curve
Let $K$ be a quadratic imaginary field. Let $L$ be a number field which contains $K$ and let $E/L$ be an elliptic curve defined over $L$ with complex multiplication by $K$, i.e. such that $End_{\...
- 105k
3
votes
2
answers
754
views
Elliptic curve E and Galois representation
Assume that an elliptic curve $E$ over $\Bbb Q$ has a reducible mod $p$ representation. Then
Q: Why is the semi-simplification of $E[p]$ the direct sum of ${\Bbb Z}/p{\Bbb Z}$ and $\mu_p$?
Next
Q:...
- 1,680
2
votes
2
answers
480
views
Lifting to char 0, references and questions
Suppose that I have a surface $S$, smooth proper over an algebraically closed (perfect?)field $k$ that lifts algebraically to some $S_W$ defined over a field of char 0. I am interested in properties ...
- 4,111
1
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0
answers
132
views
Points with minimal height
Let $K$ be an algebraically number field and $$\phi : \mathbb P^n (K) \to \mathbb P^m (K)$$ a polynomial map, such that $\forall \alpha \in \mathbb P^n$, where $\alpha = [\alpha_0, \dots , \alpha_n]$, ...
CommunityBot
- 1
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votes
1
answer
138
views
rank of Abelian schemes under ample hypersurface section
Let $k$ be an algebraic closure of a finite field, $\ell \neq \mathrm{Char}(k)$ be prime, $S/k$ a smooth projective geometrically connected surface and $C/k$ a smooth ample connected hypersurface ...
- 4,111
13
votes
1
answer
793
views
Arithmetic and moduli spaces of higher genus curves
Modular curves (as moduli of elliptic curves with level structure) play a key role in the study of the arithmetic of elliptic curves. The higher genus curves have very different arithmetic, but I ...
- 605
1
vote
0
answers
246
views
Uniqueness of lifting of very ample line bundle on smooth proper surfaces over DVR
Let $R$ be a complete Henselian discrete valuation ring of characteristic 0, $X_R$ be a surface smooth, proper and flat over $R$. Assume that the residue field $k$ of $R$ is algebraically closed of ...
- 503
4
votes
0
answers
188
views
Complexes of arithmetic $\mathcal{D}$-modules with Frobenius structure
This is a question about the category $F\text{-}D^b_\mathrm{coh}(\mathscr{D}^\dagger_{\mathscr{X},\mathbb{Q}})$ of complexes of arithmetic $\mathscr{D}$-modules with Frobenius structure on a smooth ...
- 1,838
0
votes
2
answers
487
views
Absolute Hodge implies Galois invariant?
Let $X$ be an Abelian variety defined over a number field $K$, suppose that it has a good reduction over a fine place $\mathfrak{p}$ of $K$. Let $G_{\mathfrak{p}}$ be the local Galois group for $K_{...
- 74
9
votes
2
answers
3k
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Isogeny classes of elliptic curves
Let $E \subset \mathbb{P}_\mathbb{C}^2$ be an elliptic curve. If $E$ has complex multiplication (by anything) then the theory of complex multiplication in particular tells us that if $\sigma \in \...
- 2,616
15
votes
1
answer
1k
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Number of curves over a finite field
Let $K$ be a finite field. Is there a formula for the number of isomorphism classes of genus $g$ smooth curves over $K$?
In other words does there exists a formula for the number of rational points ...
- 40.3k
9
votes
5
answers
2k
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The use of embedding a curve into its Jacobian
I'm looking for as many examples/applications as possible of the use of embedding a smooth projective geometrically connected curve $X$ over a number field $k$ with $X(k)\neq \emptyset$ into its ...
CommunityBot
- 1
5
votes
1
answer
755
views
Elliptic curve and Galois representation
For an elliptic curve $E$ over ${\Bbb{Q}}$, let us consider Serre's mod $l$ representation by
$\rho_{E,l} \colon {\mathrm{Gal}}({\overline{\Bbb{Q}}}/{\Bbb{Q}}) \to {\mathrm{Aut}}(\phantom{}_lE) = {\...
- 5,019
2
votes
0
answers
531
views
vanishing of étale cohomology of affine surface
Let $U$ be an affine smooth surface over an algebraic closure of a finite field. Let $\mathscr{A}/U$ be an Abelian scheme and $\ell \neq \mathrm{char}(k)$ be prime.
Are there vanishing results for ...
CommunityBot
- 1
5
votes
1
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820
views
Specialization Map of family of abelian varieties
In Lang's Survey on Diophantine Geometry, page 40, he said the following:
Let $F=k(Y)$ be a function field of variety $Y$ over the constant field $k$ and $X_F$ a non-singular projective variety over $...
- 1,285
1
vote
2
answers
199
views
kernel of isogeny becomes constant after base change
Let $S = Spec(O_K)$ be the spectrum of the rings of integers of a number field $K$. Let $A/S \setminus T$ be an Abelian scheme over an open subscheme $S \setminus T \subseteq S$. Does the kernel of $n$...
- 113
21
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0
answers
617
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Bounding failures of the integral Hodge and Tate conjectures
It is well know that the integral versions of the Hodge and Tate conjectures can fail. I once heard an off hand comment however that they should only fail by a "bounded amount". My question is what ...
- 21.4k
5
votes
2
answers
1k
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Local factors of Hasse-Weil zeta function - what do they have in common?
Let $X$ be a regular scheme, flat and of finite type over $Spec(\mathbb{Z})$ (add "projective" if you want). Then the Hasse-Weil zeta function of $X$ is defined as a product over all prime numbers of ...
- 1,623
13
votes
2
answers
944
views
Belyi's theorem for function fields
Belyi's theorem states that every smooth projective algebraic curve $C$ defined over $\bar{\mathbb{Q}}$
admits a map $C\to\mathbb{P}^1$ ramified only over $0,1,\infty$.
Is there an analogue of this ...
- 23k
6
votes
3
answers
811
views
Torsion group of the following elliptic curve
Let $p_1=2, p_2 = 3,\ldots,$ be the prime numbers, and define $n_i = \prod_{j=1}^i p_j$. Moreover, let $E_i $ be the elliptic curve defined by $y^2 = x^3 + n_i$.
Can one compute the torsion group $...
- 22.6k
2
votes
1
answer
122
views
Does the isotropic group of local galois representation have finite index?
Let $\rho : G\to \mathrm{GL}_n(\mathbb Z_p)$ be a crystalline representation of $G=\mathrm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)$. For any non zero element $a\in \mathbb{Z}_p^n$ , it spans a rank one ...
- 37k
9
votes
1
answer
719
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Example of a variety over a number field with non-semisimple Galois representation on $\ell$-adic cohomology
This question is inspired by the question: Example of non-projective variety with non-semisimple Frobenius action on etale cohomology?
Let $K$ be a number field (or finitely generated field of ...
CommunityBot
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5
votes
1
answer
760
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log smooth curve vs pointed node curve
F.Kato has a statement said that a log smooth curve $f:(X, M,\alpha)\to (k,N,\beta)$ where $k$ is an algebraic closed field and $N$ is some fine log structure on $k$, is equvalent to pointed node ...
- 45.7k
3
votes
0
answers
239
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log structure on Witt ring and Frobenius map
I am a little confused about the log structure of Witt ring and its Frobenius map.
Let $k=\bar{\mathbb{F}}_p$, $W:=W(k)$ the Witt ring.
We know that to deal with the semistable reduction, i.e. scheme ...
- 699
4
votes
1
answer
164
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Deforming curves to other curves over the field of rational numbers
Let $X$ and $Y$ be smooth projective geometrically connected curves over $k$ of genus $g$ at least two.
If $k$ is an algebraically closed field of characteristic zero, there exists a connected ...
- 4,111
7
votes
2
answers
614
views
Is a Deligne-Mumford curve defined over Qbar if and only if its coarse moduli space is
Let $\mathcal X$ be a smooth proper finite type Deligne-Mumford stack over $\mathbb C$ that is generically a scheme. Let $X$ be its coarse moduli space.
If $\mathcal X$ can be defined over $\overline{...
- 9,497
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0
answers
199
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Question about the "middle" intermediate Jacobian
Suppose $Y$ is a smooth projective variety of dimension $2p-1$ over $\mathbb{C}$. I have a few questions about the $p^{~\text{ th}}$ intermediate Jacobian $J^p(Y)$ of $Y$.
Does it come from (i.e. is ...
- 299
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vote
1
answer
244
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Arithmetic property of a surface of general type
In my previous post I asked about the hyperbolicity of the affine surface $S'=\{zw \neq o\}$ in the projective surface $z^2 = P(x) Q(y)$ in $\mathbb{P}^3$, where $P$ and $Q$ are two general ...
CommunityBot
- 1
3
votes
1
answer
522
views
representation of algebraic fundamental group of projective line minus three point
everyone, I want to ask is there any result in the literature
similar to the following:
Let $ X=\mathbb{P}^1\backslash \{0,1,\infty\}$, then $X$ is defined over $\mathbb{Z}$. Let $X_{\mathbb{Q}}$ ...
- 149k
4
votes
2
answers
402
views
Defining isogenies over smaller fields
I'm having some issues with abelian varieties and fields of definition. This already became clear in my previous question on Jacobians. Here's another question. If somebody can explain some nice facts ...
- 1,196
15
votes
3
answers
2k
views
What is the intuition behind the definition of cuspidal representations?
Let $\mathbb{G}$ be a reductive group defined over a number field $K$, let $Z$ be its center, and let $\mathbb{A}:=\mathbb{A}_K$ be the ring of adeles of $K$. Reasonably, we care about the $\mathbb{G}(...
- 7,037
12
votes
2
answers
424
views
Existence of local sections
I would like to know when the property of "having a section" for a morphism of varieties in characteristic $0$ can be detected by spreading out to characteristic $p$.
Take a number field $K$, and let ...
- 4,111
1
vote
0
answers
103
views
degree of isogenies between Jacobians and Abelian Varieties
Let $K$ be a local field of characteristic zero and positive residual characteristic. Let $A$ be a simple abelian variety and assume we have an isogeny $f:Jac_C\rightarrow A$ with $C$ a smooth curve ...
- 11
18
votes
3
answers
3k
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Lifting varieties to characteristic zero.
If you want to compute crystalline cohomology of a smooth proper variety $X$ over a perfect field $k$ of characteristic $p$, the first thing you might want to try is to lift $X$ to the Witt ring $W_k$ ...
- 8,998