All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
1
vote
1
answer
106
views
Actions of torsionfree discrete subgroups on hermitian symmetric domains
Let $D$ be a bounded hermitian symmetric domain with automorphism group $G(\mathbb R)$. In the example I have in mind, $D$ is Siegel upper half-space of degree $g$ and $G(\mathbb R) = \mathrm{Sp}(2g,\...
- 13
1
vote
1
answer
264
views
Discontinuous subgroups of $PGL_2(\mathbb{Q}_p)$
I'm trying to read about Mumford curves. I've barely begun and I've already encountered a stumbling block. I'm sure this is probably a basic question that an expert could resolve quickly. I would very ...
- 7,527
6
votes
1
answer
185
views
If $X$ is a genus $g\geq 2$ curve over a number field $K$, then is there a bound on $X_L(L)$ for $L/K$ that depends only on $X$ and $[L:K]$?
If $X$ is a genus $g\geq 2$ curve over a number field $K$, then $X(K)$ is finite by Falting's Theorem. My question is how does $X_L(L)$ behave for finite field extensions $L/K$? In particular, is ...
- 2,071
6
votes
1
answer
1k
views
Relationship between motivic Galois groups and Langlands program [duplicate]
I would like to know if there is any relationship between the motivic Galois groups and the Langlands program.
Many thanks.
- 1,720
3
votes
0
answers
132
views
Arithmetic version of "Attaching maps" for moduli of curves
I am looking for a reference for attaching maps of moduli of curves with marked points. Especially I would like to know whether they descend over $\mathbb{Z}$. On one hand this seems very hard to ...
- 845
0
votes
1
answer
131
views
Radical of modules [closed]
Let $R$ be a local ring with the unique maximal ideal ${\frak m}_R$ and $M$ be a $R$-module. Define
$I(M) \colon= \cap ~({\mathrm{all~ proper~ maximal ~submodules~ of}}~M)$,
where proper means ...
- 781
10
votes
1
answer
923
views
What are the $j$-invariants of the genus 1 modular curves?
I believe there are only finitely many congruence subgroups $\Gamma\le SL_2(\mathbb{Z})$ such that the compactification of $\mathcal{H}/\Gamma$ is genus 1.
Is there somewhere I can find a list of ...
- 7,527
0
votes
2
answers
2k
views
Tensor products of two domains
Let $R$ be an integral noetherian regular local ring. Let $S$ be a noetherian integral domain such that $S/R$ is finite.
That is, $R \subset S$ and the surjection $R^{\oplus n} \twoheadrightarrow S$ ...
- 781
3
votes
0
answers
412
views
action of complex conjugation on Tate modules of elliptic curves
I'm wondering about the notion of "oddness" in the theory of Galois representations. Usually, a Galois representation
$$\rho : G_\mathbb{Q}\rightarrow GL_2(\mathbb{Q}_\ell)$$
is said to be odd if the ...
- 7,527
3
votes
1
answer
310
views
Vojta's conjecture on the bounded degree algebraic points over projective line?
I want to know the accurate version of the Vojta's conjecture on the bounded degree algebraic points over projective line, which is known as an extension of the Roth's theorem for bounded degree ...
- 81
8
votes
2
answers
429
views
Arakelov divisor on $\operatorname{Spec } O_F$: places or embeddings?
Let $F$ be a number field such that $[F:\mathbb{Q}]=n$ and with ring of integers $O_F$. Let's put $B=\operatorname{Spec } O_F$, then an Arakelov divisor is an element of:
$$Div(X)\times \bigoplus_\...
- 1,237
5
votes
1
answer
1k
views
"Role" of cohomology of coherent sheaves in SGA 4.5, étale cohomology
As the question title suggests, what is the role cohomology of coherent sheaves plays for SGA 4.5, étale cohomology? Why are they so important for the construction and establishing properties of étale ...
user97565
15
votes
1
answer
474
views
Dirichlet's unit theorem for reductive schemes
Let $O_{K,S}$ be the ring of $S$-integers in a number field $K$. Dirichlet's unit theorem implies that the group of units in $O_{K,S}$ is a finitely generated group. In other words, the group $\mathbb ...
- 151
1
vote
2
answers
471
views
Prime ideal of $A[X_1,...,X_d]$
Let $A$ be a UFD domain, i.e. integral and for any height one prime
${\frak p}$ of $A$, we have ${\frak p} = (u_{\frak p})$ for some $u_{\frak p} \in A$.
Once and for all, we fix the algebraic ...
- 781
11
votes
3
answers
552
views
When $k = \mathbb{F}_q$ finite field, $X$ always has $k$-rational point, and so $A \simeq X$?
Let $k$ be an arbitrary field. Let $(A, e)$ be an abelian variety over $k$, and let $X$ be a torsor for $A$, i.e. $X$ is a proper smooth $k$-variety, and there is an $A$-action acting $:A \times X \to ...
user97565
12
votes
0
answers
283
views
A local global principle over function fields
For each prime $p$, fix a place of $\bar{\mathbb{Q}}$ extending $p$ so we can reduce elements of $\bar{\mathbb{Q}}$ which are integral over $\mathbb Z$ and get elements of $\bar{\mathbb{F}}_p$. Now ...
- 30.5k
9
votes
0
answers
910
views
Grothendieck's motivation of crystalline cohomology
Here Illusie mentions Grothendieck's observation that using Gauss-Manin connection one can give a non-canonical isomorphism between de Rham cohomology of smooth schemes over $W(k)$ with isomorphic ...
- 7,377
2
votes
1
answer
237
views
Centralizer of Shimura datum defining a Shimura curve in $A_2$
Let $B$ be an indefinite quaternion algebra over the rationals, let $G$ be the reductive algebraic group defined by $G(A) = (B\otimes A)^*$ for ${\bf Q}$-algebras $A$; hence $G({\bf R}) = GL_2({\bf R})...
- 547
22
votes
1
answer
669
views
For which $n$ is it true that all surjections $SL_2(\mathbb{Z})\rightarrow SL_2(\mathbb{Z}/n\mathbb{Z})$ have kernel $\Gamma(n)$?
For which integers $n$ does every surjection $SL_2(\mathbb{Z})\twoheadrightarrow SL_2(\mathbb{Z}/n\mathbb{Z})$ have kernel $\Gamma(n)$?
(this is the usual kernel, ie, the subgroup of matrices ...
- 7,527
12
votes
1
answer
642
views
are the congruence subgroups $\Gamma(n)$ characteristic inside $\mathrm{SL}_2(\mathbb{Z})$?
For which $n$ is the "principal congruence subgroup" $\Gamma(n)\le \mathrm{SL}_2(\mathbb{Z})$, the subgroup consisting of matrices congruent to the identity modulo $n$, characteristic? I.e., for ...
- 7,527
-3
votes
1
answer
233
views
Is $\sharp E((\mathbb{F}_{p^{2}})/E(\mathbb{F}_{p}))=1$ for almost all primes $p$? [closed]
Let $E$ be an elliptic curve over $\mathbb{Q}.$
Is $\sharp E((\mathbb{F}_{p^{2}})/E(\mathbb{F}_{p}))=1$ for almost all primes $p$?
- 1,805
-2
votes
1
answer
271
views
Any galois covering of $P^{1}$ over rationals are of the form $\mathbb{P}^1_L\to\mathbb{P}^1_\mathbb{Q}$
I recently came across the following statement,
The Galois coverings of $\mathbb{P}^1_\mathbb{Q}$ are all of the form
$$\mathbb{P}^1_L\to\mathbb{P}^1_\mathbb{Q}$$ where $L$ is a number field.
How ...
- 591
5
votes
1
answer
1k
views
Reference to "bounds of Weil and Deligne"
In the this paper by Friedlander and Iwaniec, it is said that they are "able to avoid much of the high-powered technology frequently used in modern analytic number theory such as the bounds of Weil ...
- 1,890
1
vote
0
answers
199
views
Class number of the cyclotomic tower
Let ${\Bbb Q}(\zeta_{\infty})$ be the field obtained by adjoining all roots of unity. We define
Cl(${\Bbb Q}(\zeta_{\infty})$)$\colon= \underset{m > 1}{\varinjlim}~{\mathrm{Cl}}({\Bbb Z}[\zeta_m])...
- 781
3
votes
0
answers
285
views
What is known about the prime-to-$p$ etale fundamental group of $\mathbb{P}^1_{\mathbb{F}_p}$ minus $\mathbb{F}_p$-rational points?
Is it known to be (the prime-to-$p$ part of the profinite completion of) a finitely presentable group?
Is such a presentation known? Is there a guess for what it is? What is known about it?
- 2,071
6
votes
1
answer
408
views
Is the ring of meromorphic modular forms on a fine modular curve generated in degree 1?
Consider a torsion-free congruence subgroup $\Gamma\le SL_2(\mathbb{Z})$ (for example, $\Gamma(N)$ for $N\ge 3$, or $\Gamma_1(N)$ for $N\ge 4$).
By a meromorphic modular form for $\Gamma$ of weight $...
- 7,527
32
votes
2
answers
2k
views
Etale cohomology can not be computed by Cech
It can be proven that if in a quasicompact scheme $X$ any finite subset is contained in an affine open subset then for any sheaf $\mathcal{F}$ on $X$ its Cech cohomology $\hat{H_{et}^{\bullet}}(X,\...
- 7,377
8
votes
0
answers
603
views
A Generalization of the Tate-Shafarevich/Tate/Fontaine-Mazur Conjectures
Let $A$ be an abelian variety over a number field $k$. The Tate-Shafarevich conjecture says that the Tate-Shafarevich group of $A$ is finite.
A weakening of this conjecture states that the $\ell$-...
- 15.4k
12
votes
1
answer
929
views
Dwork's proof of rationality of zeta function, crux of his generalization of a result of Borel along the way
In this article by Katz and Tate here, there's a nice account of Dwork's argument for showing the rationality of the zeta function part of the Weil conjectures. Here is an excerpt.
To recapitulate, ...
user97565
6
votes
1
answer
696
views
Selmer Group versus Selmer Variety
For an abelian variety $A$, the $p$-adic Selmer group is defined to be the subset of $H^1(G_k,H_1^{et}(A;\mathbb{Q}_{p}))$ whose restriction to $G_{k_v}$ is in the image of $A(k_v)$ for all places $v$ ...
- 15.4k
2
votes
0
answers
76
views
Behaviour of densities of places of finitely generated fields under specialisation
This question is a follow-up on question 2, posed in:
On the distribution of roots modulo primes of an integral polynomial
In appendix B of [1] by Pink, and in [2,3] by Serre, there are definitions ...
- 83
8
votes
0
answers
286
views
Functorial classes in Brauer group
For a smooth variety $X$ over a perfect field of characteristics $p$ the sheaf of differential operators is an Azumaya algebra(etale locally is isomorphic to endomorphisms of its center, which is ...
- 7,377
5
votes
0
answers
328
views
Definition of logarithm for universal vector extension
Let $S$ be a topological $\mathbb{Z}_p$-algebra and $R\to S$ a surjection (where $R$ has $p$ nilpotent) with topologically nilpotent kernel which has a PD structure.
We know that if $G/R$ is a $p$-...
- 843
13
votes
1
answer
869
views
Rationality of zeta function and Grothendieck-Lefschetz fixed point formula, cohomology can be computed as the de Rham cohomology
Trivial example. First, suppose $X$ is finite. Then we have a finite set $S := X(\overline{\mathbb{F}}_q)$ with an action of $\text{Fr}_q$. How can one explain why the rationality of the zeta function ...
user97565
10
votes
0
answers
490
views
Non-rationality of minimal del Pezzo surfaces
Let $k$ be a field. Recall that a del Pezzo surface $S$ over $k$ is a smooth projective surface with ample anticanonical divisor $-K_S$. We define its degree to be $d = K_S^2$.
We say that $S$ is ...
- 21.3k
7
votes
1
answer
557
views
Does Chabauty-Coleman method give an algorithm for finding rational points?
Let $X$ be a curve of genus $g\geq 2$ over a number field $K$. If $\mathrm{rk} \,\mathrm{Jac}\, X$ is less than $g$ there is a $p$-adic method of bounding $\# X(K)$ due to Chabauty and Coleman (see ...
- 7,377
41
votes
2
answers
3k
views
Perfectoid universal covers
It is often said, with varying degrees of rigor or enthusiasm, that every rigid space (say over $\mathbb{C}_p$) has a pro-etale cover which is 'topologically trivial' in some sense. For example, this ...
- 843
14
votes
1
answer
479
views
Weaker version of Dwork's rationality of zeta function, what is needed to beef up into a complete proof?
This is a followup to my question here.
Here is a note of Michael Larsen where he gives a very simple proof of a slightly weaker result than Dwork's rationality of the zeta function.
http://mlarsen....
user97565
26
votes
3
answers
3k
views
Crux of Dwork's proof of rationality of the zeta function?
As the question title suggests, what is the crux of Dwork's proof of the rationality of the zeta function? What is the intuition behind the proof, what are the key steps that the proof boils down to?
user97565
19
votes
1
answer
1k
views
Ehresmann's theorem over the $p$-adics
I am looking for a version of Ehresmann's theorem for analytic manifolds over the $p$-adic numbers $\mathbb{Q}_p$ or, more generally, local fields. I follow the conventions from Serre's book "Lie ...
- 21.3k
5
votes
0
answers
206
views
Real field of definition of an abelian variety of CM-type?
Question 0. Can a field of definitions (without automorphisms) of an (almost arbitrary) abelian variety of CM-type, originally defined over ${\mathbb{C}}$,
be chosen to be a totally real number ...
- 14.2k
6
votes
1
answer
225
views
Higher Chow groups: a basic case
Let $Y$ be a smooth irreducible variety over a field $k$. I know that $\mathcal{O}_Y^\times$, the group of invertible functions on $Y$, embeds in the higher Chow group $\mathrm{CH}^1(Y,1)$ [or the ...
- 61
14
votes
0
answers
664
views
$\zeta(2n)$ and amoebas
Mikael Passare showed how to compute $\zeta(2)$ (How to compute $\sum 1/n^2$ by solving triangles) using the amoeba of $1 + z + w = 0$. Has this ever been generalized to higher zeta-values? How ...
- 22.8k
6
votes
1
answer
937
views
Relationship between Tate-Shafarevich group and the BSD conjecture
The finiteness of the Tate-Shafarevich group is known to be equivalent to BSD for elliptic curves over function fields over $\mathbb{F}_{q},$ this result is due to Kato and Trihan if I am not mistaken....
- 1,805
2
votes
0
answers
127
views
Truncated hypergeometric functions modulo prime power
The question arises when I was trying to understand some congruence equations of Zhi-wei Sun. C. Herbert Clemens shows an amazing connection between the number of points on Legendre's Family $$E_\...
- 3,337
25
votes
8
answers
3k
views
Relatively concise English expositions of the proofs of the various Weil conjectures
Where can I find relatively concise (i.e. not excessively wordy and waxing poetic about history and intuitions and such, doesn't spend an eternity carefully developing various parts of the theory of ...
user97565
3
votes
0
answers
394
views
Calculation of Cartier-Manin matrix
Let $\mathbb{F}_q$ be a finite field of characteristic $p$ and let $C$ be a plane projective nonsingular curve over $\mathbb{F}_q$ ,
with function field $K = \mathbb{F}_q(C)$. Let $K^p$ denote the ...
- 2,576
3
votes
1
answer
288
views
Elliptic curve with CM by $(1+\sqrt{-11}) /2$
Can someone explain to me on how to obtain the endomorphism for elliptic curve with CM by $(1+\sqrt{-11}) /2$?
Given the elliptic curve over $F_{p}$ as $y^2=x^3-13824/539 x + 27648/539 \dots$ how do ...
- 41
10
votes
1
answer
575
views
Does every Shimura variety contain a generic point defined over a number field?
This question is related to my previous question, to which I got a partial answer.
Consider the cyclotomic field $L={{\mathbb{Q}}}(\zeta_8)={{\mathbb{Q}}}(\sqrt{2},i)$, where $\zeta_8$ is a primitive ...
- 14.2k
11
votes
2
answers
653
views
Abelian variety with prescribed endomorphism ring
Consider the cyclotomic field $L={{\mathbb{Q}}}(\zeta_8)={{\mathbb{Q}}}(\sqrt{2},i)$, where $\zeta_8$ is a primitive 8-th root of unity. Let $\Lambda={{\mathbb{Z}}}[\zeta_8]$ denote the ring of ...
- 14.2k