All Questions
2,494 questions
17
votes
1
answer
1k
views
On the Hasse-Weil L-function of $P^n$
So let us start with the "simplest" scheme over $Spec(\mathbf{Z})$ namely $X_0=Spec(\mathbf{Z})$. Then the (reciprocal) Weil zeta function of $X_0$ at a prime $p$ is given by $Z_p(T)=1-T$ (a ...
- 7,609
23
votes
1
answer
2k
views
Does smooth and proper over $\mathbb Z$ imply rational?
Does smooth and proper over $\mathbb Z$ imply rational?
I think someone told me that this is a standard conjecture. Is it a widely held? held at all? Did someone in particular make this conjecture? ...
- 8,727
21
votes
3
answers
3k
views
Stacks in modern number theory/arithmetic geometry
Stacks, of varying kinds, appear in algebraic geometry whenever we have moduli problems, most famously the stacks of (marked) curves. But these seem to be to be very geometric in motivation, so I was ...
0
votes
1
answer
223
views
Smooth quadric over p-adic integers
Let $k$ be a $p$-adic field with ring of integers $\mathcal{O}_K$ and residue field $\mathbb{F}$. Say I have a (projective) quadric $Q$ which is smooth over $\mathcal{O}_K$, such that the reduction $\...
- 5,163
4
votes
0
answers
269
views
Tate's theorem about abelian variteies in case of abelian scheme
For $k$ a finite field , $A,A'$ an abelian varieties over $k$, $G$ the Galois group of $k$, $l$ a prime number different from the characteristic of $k$ . Tate has proved that:
$Q_l\otimes Hom_k(A,A')...
- 709
1
vote
1
answer
435
views
Elliptic subfields of a function field
Let $C$ be a curve and $K(C)$ be its function field of genus 2, where $K$ = $\mathbb{C}$.
The number of essential elliptic subfields of $K(C)$ is 0 or 2 or $\infty$.
Edit: I am looking for a proof. ...
- 471
3
votes
1
answer
586
views
Conductor of an elliptic curve
Given any elliptic curve over $\mathbb{Q}$ of conductor $N$, by modularity of elliptic curves,
there exists a surjective morphism from $X_0(N)$ $\rightarrow$ $E$.There may be several such 'N' and ...
- 471
14
votes
2
answers
2k
views
"Purely local" proof of local Langlands
As from this website
http://math.uchicago.edu/~lxiao/workshop_site/
My question is: What does it mean by "purely local"?
Also, I heard about this phrase "purely local" in other problems as well, ...
- 1,503
6
votes
1
answer
1k
views
Albert classification of rational endomorphism rings of simple Abelian varieties over finite fields
Recall the Albert classification of rational endomorphism rings with involution of simple Abelian varieties over arbitrary fields:
Type I: totally real, trivial involution
Type II and III: quaternion ...
user19475
2
votes
1
answer
223
views
density of conjugate of arithmetic subgroup
$K=Q(\sqrt{d} ) , d<0 $, $\Gamma $ an arithmetic subgroup of $G=SU(2,1)(K)$ . Is $\cup_{g\in G}(g^{-1}\Gamma g)$ dense in G for the complex topology?
- 709
4
votes
2
answers
500
views
density in SU(2,1)
Let $K=Q(\sqrt{-3})$ , is $SU(2,1)(K)$ dense in $SU(2,1)(C)$ for the complex topology?
- 709
7
votes
1
answer
334
views
Rational points on smooth compactifications
Let $X$ be as smooth variety over a field $k$ of characteristic $0$.
Consider the following statements:
The variety $X$ has no $k((t))$-rational points.
No smooth compactification of $X$ has a $k$-...
- 5,163
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
17
votes
4
answers
2k
views
What are supersingular varieties?
For varieties over a field of characteristic $p$, I saw people talking about supersingular varieties.
I wanted to ask "why are supersingular varieties interesting". However, as I don't want to ask an ...
- 2,040
8
votes
1
answer
2k
views
Order of Ш (Sha)
To prove the BSD conjecture, one has to know about 'the finiteness of the Shafarevich Tate group'. But, an example of an elliptic curve of rank 2 (whose Sha group $Ш(E/\mathbb{Q})$ is finite) is not ...
- 471
4
votes
1
answer
604
views
A question about the Tannakian etale fundamental group of a curve
Let $X$ be a smooth connected quasi-projective curve over $\mathbf{Q}$. Let $U$ be the pro-unipotent etale fundamental group of $X$ over $\mathbf{Q}_p$.
$U^1 = U$ and let $U^n =[U,U^{n-1}]$.
Let $n\...
- 1,213
2
votes
1
answer
353
views
Pro-affine varieties over a local field
Let $K$ be a (perfect) local field, and let $S = \lim (\operatorname{Spec} A_i)_{i=0}^\infty$ be a pro-affine variety over $K$. This means that each $A_i$ is a finite type $K$-algebra and that the ...
- 1,213
3
votes
0
answers
742
views
Explicit description of O^{cris}_n in Fontaine/Messing
Let $k$ be a perfect field of characteristic $p$, $W(k)$ the Witt ring and $K$ its quotient field. In their article "$p$-adic periods and $p$-adic etale cohomology" Fontaine and Messing give in II.1.4 ...
2
votes
0
answers
131
views
why find a good field extension such that the curve has semi-stable model is important?
Hello everyone,
I'd like to ask some question about semi-stable reduction of curves.
The Deligne-Mumford theorem tell us "Let $A$ be an Dedekind domain, $K=K(A)$, for any smooth curve $X$ over $K$ ...
- 1,921
6
votes
1
answer
2k
views
understanding the main theorem of complex multiplication (of elliptic curves)
I want to understand the meaning of the main theorem of complex multiplication (of elliptic curves) as given in Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, II.8, or Shimura, ...
user19475
4
votes
1
answer
453
views
Why is the kernel of reduction a pro-p group?
Let $A$ be an abelian variety over a $p$-adic field $K_v$, i.e., $K_v$ is a finite field extension of $\mathbb Q_p$, for $p$ a prime number. Denote by $k_v$ the residue field of $K_v$ and let $\...
- 841
5
votes
0
answers
397
views
false elliptic curves and principal polarizations
Hi,
Let $\Delta$ be a quaternion algebra over $\mathbf Q$ and let $\mathcal O_\Delta$ be a maximal order in $\Delta$.
Recall that a false elliptic curve over a field $K$ is a pair $(A/K,i)$ ...
- 2,842
17
votes
2
answers
2k
views
Geometric interpretation of Hida isomorphism
[EDIT]: After getting a very nice answer by Kevin Buzzard I realize that my question was a little bit too vague and I try to restate it more precisely.
As the title says, I would like to understand ...
- 5,670
26
votes
1
answer
2k
views
Why does the Section Conjecture exclude curves of genus 1?
Let $X$ be an integral proper normal curve over a (perfect) field $F$, of genus $\geq 2$. One variant of Grothendieck's "section conjecture" states that the sections $G_F \rightarrow \pi_1(X)$ of the ...
- 435
15
votes
1
answer
807
views
Has the Weil conjectures been proved using other (Weil) cohomology theory?
According to Weil, the Weil conjecture should follow once one has a sufficiently powerful cohomology machine. And it is proved using one of them, namely étale cohomology.
My question is, has there ...
- 2,040
6
votes
2
answers
2k
views
Q-factorial and rational singularities on surfaces
Let $X$ be a normal surface. Is any rational singularity $\mathbf{Q}$-factorial? I've seen this somewhere for surfaces over fields, but what about the general case of an integral 2-dimensional ...
- 1,213
9
votes
5
answers
2k
views
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 ...
- 1,213
14
votes
1
answer
2k
views
A curve with bad reduction for which the jacobian has good reduction
Let $K$ be a number field. If $X$ is a curve over $K$ with good reduction at a place $v$ of $K$, then the Jacobian of $X$ also has good reduction at $v$. This follows from the functoriality of the ...
- 141
2
votes
0
answers
445
views
Are torsors over unipotent groups trivial
I might have misunderstood something I heard somewhere.
Are torsors over unipotent groups trivial?
I couldn't find this in some standard references.
- 1,213
33
votes
3
answers
3k
views
Arithmetic geometry examples
(This is inspired by Algebraic geometry examples.)
I want to collect here (counter)examples in arithmetic geometry.
Curves violating the Hasse principle: The Selmer curve $3X^3 + 4Y^3 + 5Z^3 = 0$. ...
5
votes
1
answer
4k
views
Trace of Frobenius over $F_q$
Let $q_0$ be a prime and $q$ = $q_0^n$.
Let $a(F_q/F_{q_0})$ denote any integer which is trace of Frobenius over the field $F_q$ for some elliptic curve which can be defined over $F_{q_0}$.
It is ...
- 471
3
votes
1
answer
980
views
Construction of Kummer map for abelian variety
Let $A$ be an abelian variety over the rational numbers $\mathbf{Q}$. Let $V=T_p A \otimes \mathbf{Q}_p$ be the $\mathbf{Q}_p$-Tate module of $A$. Let $G$ be the absolute Galois group of $\mathbf{Q}$. ...
- 1,213
1
vote
0
answers
457
views
Why do twists of an algebraic group over k correspond to k-torsors over G
Let $G$ be an algebraic group over a field $k$. Let $k^s$ be the separable closure of $k$.
I can't seem to figure out why isomorphism classes of twists of $G$ correspond to $k$-torsors over $G$.
It'...
- 1,213
5
votes
1
answer
819
views
Does the Mordell conjecture imply the Shafarevich conjecture
The base field is a number field.
It is known that the Shafarevich conjecture implies the Mordell conjecture (Kodaira-Parshin).
Is the converse also true?
Note that both conjectures are now ...
- 51
4
votes
0
answers
409
views
Every curve is a Hurwitz space in infinitely many ways
Diaz, Donagi and Harbater proved that every curve over $\overline{\mathbf{Q}}$ is a Hurwitz space.
A Hurwitz space is a connected component of the curve $H_n$. The curve $H_n$ is (the ...
- 163
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
8
votes
1
answer
982
views
Is there a really big ring of differential operators in characteristic p?
$k$ is a field of characteristic $p$.
$k[t]$ has canonical first-order differential operator $\partial$
As an endomorphism of $k[t]$, $\partial^p=0$.
First way to fix it:
Use the divided power ...
2
votes
0
answers
213
views
algebraic de Rham cohomology of hypersufaces
For a smooth hypersurface $X\subset\mathbb{P}^n_k$, where $k$ is an algebraic closed field of charactersitc $p>0$. How to compute its algebraic de Rham cohomology explicitly? or equivalently its ...
- 41
4
votes
2
answers
349
views
Number of connected components of the Hurwitz space $H_n^o$ and subgroups of the fundamental group
A cover (of $\mathbf{P}^1_{\overline{\mathbf{Q}}}$) is a finite morphism $X\to \mathbf{P}^1_{\overline{\mathbf{Q}}}$, where $X$ is a smooth projective connected curve over $\overline{\mathbf{Q}}$. ...
- 163
4
votes
1
answer
1k
views
Bound for the number of rational points on the modular curve
By Mazur's theorems (Reference:- Modular curves and the Eisenstein ideal, 1977),
we know that the only rational points of X_0(N) for N any prime > 163 are
the two cusps (o) and (oo) (|X_0(N)(Q)| = 2 )....
- 471
4
votes
1
answer
796
views
Deformation space of non-ordinary abelian varieties
It is a well known result of Serre and Tate that if $A$ is an ordinary abelian variety over a field $k$ of characteristic $p>0$, then the deformation space $\mathcal{M}$ of $A$ to an abelian ...
- 395
7
votes
2
answers
516
views
Zograf's bound on the index of a modular curve for Shimura curves
I've been reading Voight's paper on Shimura curves and it prompted the following question; see http://www.cems.uvm.edu/~voight/articles/shimura-clay-proceedings-071707.pdf for which notes I'm talking ...
- 9,497
0
votes
0
answers
82
views
Extending functions on curves to functions on abelian varieties
Say I have a function $f_g$ on the moduli space of curves $M_g$ for all $g\geq 1$. Is there some way of extending this to the moduli space of abelian varieties $A_g$ in a nice way?
What if I have ...
- 163
10
votes
6
answers
2k
views
Proofs in the same vein as Ax-Grothendieck
I would like to see other examples of (ideas of) proofs and results in the same vein as the proof of the Ax-Grothendieck theorem. To explain what I mean by "in the same vein", I will quote from the ...
- 23.4k
2
votes
1
answer
501
views
Etale group schemes over a local ring
Let $p$ be a prime number and $R$ be a Noetherian local ring of characteristic $p$ with residue field $k$. Let $G$ be a finite etale subgroup scheme over $R$ of order $p$. Suppose that the etale $k$-...
- 163
7
votes
1
answer
756
views
$p$-adic uniformization not from the Drinfel'd spaces?
It seems that when we talk about the $p$-adic uniformization, we typically mean those uniformized by either the Drinfel'd upper spaces (for which we think of the examples of Mumford curves and some ...
- 1,305
5
votes
4
answers
518
views
What is the obstruction for a local set of models of a curve to come from a global model?
If $X_{\mathbb{Q}}$ is a curve over $\mathbb{Q}$, we get a curve $X_{\mathbb{Q}_p}$ over $\mathbb{Q}_p$ for every prime $p$.
My question is about the reverse process. Say we are given curves $X_{\...
- 6,348
4
votes
0
answers
124
views
Detecting linear dependence on multiplicative groups
Let G = $\mathbb{G}_m^2/\mathbb{Q}$ and let $\Gamma \subseteq G(\mathbb{Q})$ be a free abelian group of rank 2. Assume that the set of primes $p$ for which $\Gamma \mod p$ is cyclic has positive ...
- 527
6
votes
0
answers
436
views
Do there exist weak Lefschetz-type statements that were proved for varieties over complex numbers, but were not proved in finite characteristic?
As far as I understood the situation (reading section 3.5B of Lazarsfeld's "Positivity in algebraic geometry" and also some of the references), some of weak Lefschetz-type statements known rely on '...
- 16.9k
6
votes
4
answers
1k
views
Nonstandard Methods ( or Model Theory ) and Arithmetic Geometry
I hear that the nonstandard methods are applied to many problems in various fields of mathematics such as functional analysis, topology, probability theory and so on.
So, I have become interested in ...
- 945