All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
3
votes
0
answers
125
views
Absolute irreducibility of the connected components of certain fibers of a certain scheme
I am currently reading Serre's "Topics in Galois Theory". Specifically, I am looking at the proof of Theorem 3.6.2, and there is one aspect of the proof that is unclear to me. I'll describe the ...
5
votes
1
answer
1k
views
Analogy between Jacobian of curve and Ideal class group
It is excerpt from "Algebraic Geometry Codes Basic Notions"(https://www.google.ru/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CCoQFjAA&url=http%3A%2F%2Fwww.math.umass.edu%...
- 2,576
4
votes
1
answer
639
views
Conductor of abelian varieties
Let $A$ be a non-zero abelian variety defined over a number field $F$. Let $v$ be a finite place of $F$, and let $f_v(A)$ be the usual conductor exponent of $A$ at $v$ (defined e.g. on p.500 of the ...
- 107
0
votes
0
answers
161
views
Is there a reference book for the duality between the genus of function fields and the discriminant of number fields?
Bjorn Poonen mentions in his "Lectures on rational points on curves" the analogy between the genus of a function field and the discriminant of number fields. I'm looking for a reference book for this ...
0
votes
0
answers
124
views
field of definition of abelian varieties with extra endormorphism
Let $A$ be a complex abelian variety such that $\mathrm{End}(A)$ is strictly bigger than $\mathbb{Z}$.
Question: Is is true that $A$ is defined over $\overline{\mathbb{Q}}$?
This is of course what ...
- 1
5
votes
0
answers
308
views
Algorithm for solutions to quadratic forms over number fields
Are there any know (preferably implemented) algorithms to find solutions to quadratic forms over number fields (or global fields)?
I am especially interested in the quaternary case. There exist some ...
- 101
7
votes
1
answer
759
views
On Deligne's determinant of motives
This is a question about Deligne's conjecture on special values of L-functions. I have to confess that I've never understood the definition of the determinant which is supposed to give the right ...
- 71
18
votes
1
answer
552
views
Is there a known example of a curve X of genus > 1 over Q such that we know the number of points of X over the n-th cyclotomic field, for every n?
By Falting's theorem, these numbers are of course finite. Is there an example where we can explicitly compute them for every $n$?
Thank you!
- 3,910
21
votes
1
answer
4k
views
What makes the Cartier operator "tick"?
Let $C$ be a smooth curve over a finite field of characteristic $p$. Let $t$ be a local parameter at a point. If $f$ is a regular function on a neighbourhood of the point, one can write uniquely
$$f =...
- 40.2k
4
votes
2
answers
362
views
Does this modified Hasse principle hold for curves?
Let $C$ be a curve over $\mathbb Q$ with a point $P$ on $Pic^1$. For each $\mathbb Q$-rational point $Q$, $Q-P$ is a point on the Jacobian $J$. We can use the map $H^0(\mathbb Q, J) \to H^1(\mathbb Q,...
- 149k
7
votes
3
answers
594
views
Hyperelliptic modular curves in characteristic p
Ogg characterized the finitely many N such that $X_0(N)_{\mathbb{Q}}$ is hyperelliptic, and Poonen proved in "Gonality of modular curves in characteristic p" that for large enough N, $X_0(N)_{\mathbb{...
- 10.5k
14
votes
1
answer
2k
views
Deformations of p-divisible groups
Given a p-divisible group over $\mathbb{F}_p$, Grothendieck-Messing theory tells us that deforming the group to $\mathbb{Z}_p$ is the same as finding an admissible filtration of the Dieudonne-module ...
- 143
0
votes
0
answers
324
views
Solution of a special class of Diophantine Equations
Suppose A is Diophantine Equations which have a variable x,such that every other variable has function of x the variable as it's solution.For example,a Diophantine equation in three variables has ...
- 3,104
3
votes
0
answers
213
views
Natural construction of Hodge (Phi,Gamma)-modules
I am looking for a functor from varieties $X/\mathbf{Z}_p$ to $(\varphi,\Gamma)$-modules over the Robba ring over $\mathbf{Q}_p$ (overconvergent ones) that is contructed by differential methods (...
2
votes
1
answer
228
views
Field of definition of canonical morphism between (congruence) modular curves
Let $\Gamma\subseteq \Gamma'\subset SL_2(\mathbb Z)$ be congruence subgroups, and
$X(\Gamma)$, $X(\Gamma')$ be the associated smooth projective modular curves over $\mathbb C$. The inclusion $\Gamma\...
1
vote
2
answers
574
views
References for period matrix of abelian variety
Hi, everyone.
I am looking for some references for period matrix of abelian variety over arbitrary field, if you know, could you please tell me?
For period matrix of abelian varieties, I means that ...
- 21
28
votes
3
answers
2k
views
Is there an algebraic curve over Q which is not modular?
Every elliptic curve $E/\mathbf Q$ is modular, in the sense that there exists a nonconstant morphism $X_0(N) \to E$ for some $N$.
It is tempting to extend this definition in a naïve way to an ...
- 3,910
0
votes
0
answers
275
views
r-torsion points on elliptic curve on finite field
Let $E(\mathbb{F}_q)$ - elliptic curve, $r$ is prime, $|E(\mathbb{F}_q)[r]| > 1$.
Let $r | q-1$. Is it true that $|E(\mathbb{F}_q)[r]| = r^2$?
- 2,576
10
votes
1
answer
536
views
examples of "exotic" moduli problems for elliptic curves?
Let $\textbf{Ell}$ be the category of elliptic curves over various base schemes, and where a morphism between $E\rightarrow S$ and $E'\rightarrow S'$ is a cartesian diagram with those two maps as ...
- 10.7k
4
votes
2
answers
409
views
j-invariant duplication, triplication and quintuplication formulae... how?
I am interested in finding the derivation of the duplication, triplication and quintuplication formulae for Klein’s j-invariant, which are equations (13) – (24) of the corresponding page (Klein’s j-...
- 43
5
votes
1
answer
1k
views
equivalence between katz and classical modular forms
$\newcommand{\CC}{\mathbb{C}}$
$\newcommand{\ZZ}{\mathbb{Z}}$
$\newcommand{\PP}{\mathbb{P}}$
$\newcommand{\QQ}{\mathbb{Q}}$
$\newcommand{\hH}{\mathcal{H}}$
$\newcommand{\eE}{\mathcal{E}}$
$\newcommand{...
- 10.7k
4
votes
0
answers
290
views
Is Normalization of log smooth scheme smooth?
Let $f:Y\rightarrow X$ be a finite flat morphism between smooth schemes over $Spec k$, where $k$ is a perfect field. Let $D$ be an irreducible and smooth divisor of $X$, $U=X\setminus D$ the ...
- 93
2
votes
1
answer
339
views
Finite Flat Group Schemes for Modular Forms of Higher Weight
Let $f$ be a newform (normalized, cuspidal) of weight $k \ge 2$. Then for a prime $\ell$ there is an $\ell$-adic Galois representation associated to $f$. If $k=2$, this comes from an abelian variety, ...
- 15.4k
1
vote
1
answer
174
views
reference request for the finiteness of cuspidal subgroup of $X_0(N)$?
I've seen stated offhand in many sources that the cuspidal subgroup of the Jacobian of $X_0(N)$ is finite.
Do they mean that the subgroup of the jacobian generated by $\mathbb{Q}$-rational cusps is ...
- 10.7k
1
vote
1
answer
954
views
Are there modular elliptic curves over a field extension of $\mathbb{Z}[i]$?
Let $E$ be an elliptic curve, $\mathbb{Q}$ the field of rational numbers, $\mathbb{Z}[i]$ the ring of Gaussian integers and let $K$ be a number field that is some field extension of $\mathbb{Q}$.
Is ...
- 19
45
votes
2
answers
3k
views
What are the possible motivic Galois groups over $\mathbb Q$?
Let $E$ be a motive over $\mathbb Q$. (I should precise, that by a motive I mean
here a pure motive over $\mathbb Q$, with coefficients in $\mathbb Q$, that I see here as a conjectural object which ...
- 26k
3
votes
0
answers
146
views
P-adic Weierstrass Lemma for several variables
The p-adic Weiestrass lemma asserts that a power series $f(z)$ with coefficients in the ring of integers of a local field can be factored as $π^n·u(z)·p(z)$ where u(z) is a unit in the ring of power ...
- 31
28
votes
2
answers
5k
views
Status of (global) Langlands conjecture for $\mathrm{GL}_2$ over $\mathbb{Q}$
$\DeclareMathOperator\GL{GL}$Apologies if this question has already been dealt with on MO. I am wondering about the status of the global Langlands conjectures for $\GL_2$ over the rational numbers. ...
- 283
6
votes
2
answers
313
views
Reals with integer powers bounded away from integers?
Do there exist real numbers whose integer powers are bounded away from integers? More precisely, for an arbitrary constant 0 < $\epsilon$ < 1/2, does there exist real x such that for all ...
- 475
10
votes
0
answers
743
views
What does the tensor product of two central simple algebras correspond to geometrically?
Let $k$ be a field, assumed to have characteristic $0$ for simplicity (though this probably isn't necessary).
Let $A$ be a central simple algebra over $k$ of dimension $n^2$. Then the collection of ...
- 21.3k
30
votes
3
answers
3k
views
Elliptic curve over a scheme is a group scheme?
In Katz's article p-adic properties of modular schemes and modular forms in the Antwerp proceedings, the following definition of an elliptic curve over a base scheme $S$ is given:
By an elliptic ...
- 3,910
19
votes
2
answers
2k
views
What is the relationship between these two notions of "period"?
The motivation for this question is to understand a recent theorem of Francis Brown which implies that all periods of mixed Tate motives over $\mathbb{Z}$ lie in $\mathcal{Z}[\frac{1}{2\pi i}]$, where ...
- 9,061
8
votes
2
answers
397
views
A criterion for freeness over a local ring
Let $A=K[[X_1,\dots,X_n]]$ where $K$ is a field. Let $M$ be a finitely generated torsion-free $A$-module, such that
for all $k$, the $A[1/X_k]$-module $M[1/X_k]$ is free of rank $d$;
for every $i \...
- 6,924
19
votes
2
answers
5k
views
New Geometric Methods in Number Theory and Automorphic Forms
The MSRI is organising a programme with the above title from Aug 11, 2014 to Dec 12, 2014. Here is a short description from their website :
The branches of number theory most
directly related to ...
- 18.7k
15
votes
2
answers
2k
views
sum of three cubes and parametric solutions
The first paragraph in the following link asserts that the equation $x^3+y^3+z^3=2$ has finite many parametric solutions over $\mathbb{Q}$, i.e., there are finite many polynomial triples $(x(t),y(t),z(...
- 3,337
25
votes
1
answer
1k
views
When does a modular form satisfy a differential equation with rational coefficients?
Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least locally), and ...
- 4,593
7
votes
1
answer
803
views
Overconvergent/infinitesimal site, base change and six operations
This question is about 6 operations formalism for 'crystalline' cohomology theories - more specifically the infinitesimal cohomology of smooth $\mathbb{C}$-varieties, and the overconvergent cohomology ...
- 1,838
2
votes
0
answers
204
views
Two different definitions of $\sigma$-L-spaces in Kottwitz I and II
In his papers "Isocrystals with additional structure" I and II, Kottwitz defines the notion of $\sigma$-$L$-spaces. In the first one the situation is the following
$k$ an algebraically closed field ...
- 271
10
votes
1
answer
2k
views
Modular interpretation of nebentypus
Recall that for a subgroup $\Gamma \subset SL_2(\mathbb{Z})$ a modular form $f$ of weight $k$ is a holomorphic function from the upper-half plane into the complex numbers such that for any
$\begin{...
- 12.1k
3
votes
2
answers
375
views
critical values of motives
Hi friends,
I have some questions concerning the critical values of motives, in the sense of Deligne. I will only look at motives of the form $h^i(X)$ where $X$ is a smooth projective algebraic ...
- 31
23
votes
1
answer
4k
views
How many proofs of the Weil conjectures are there?
I hope this this is not seen as too much as jumping on the band-wagon, but here goes.
Deligne's proof of the last of the Weil conjectures is well-known and just part of a huge body of work that has ...
- 35.5k
1
vote
0
answers
293
views
P-adic Sigma Functions
Hello,
I know there is a contruction of the p-adic sigma function due to Tate and Mazur for curves with ordinary reduction. I think this has been generalized to more cases, do you know of a good (...
- 19
0
votes
1
answer
748
views
Pairing on elliptic curve
Let $E(\mathbb{F_q})$ - elliptic curve.
$G_1 = E(\mathbb{F_q})[r]$. $|G_1| = r$.
$k$ is minimal such $r | q^k - 1$.
$\pi_q$ - $q$-power Frobenius endomorphism.
$G_2 = E(\mathbb{F_{q^k}})[r] \cap ...
- 9
3
votes
1
answer
393
views
On explicit examples of the Parshin Construction
In the Proof of Mordell Conjecture by Gerd Faltings, it is famous that Parshin constructed a curve $C_P$ for each $\mathbb{Q}$-rational point $P$ on the given curve $C$ over $\mathbb{Q}$ such that the ...
1
vote
1
answer
209
views
Algebraicity of the canonical projection $X(\Gamma)\to X(1)$ and of $X(\Gamma)$
Let $\Gamma\subset {SL}_2(\mathbb Z)$ be a congruence subgroup, and let $X(\Gamma)$ be the associated compact modular curve. The inclusion $\Gamma \subset SL_2(\mathbb Z)$ induces a canonical ...
19
votes
1
answer
2k
views
What do formal group laws of height $\geq 3$ look like?
By the classification of formal groups in characteristic $p$, we know that isomorphism classes of connected smooth $1$-dimensional formal groups, equivalently group scheme structures on $\operatorname{...
- 149k
2
votes
2
answers
453
views
Any other definition for algebraic number than the root of algebraic equation? [closed]
Any other definition for algebraic number than the root of algebraic equation?
- 3,104
8
votes
2
answers
1k
views
How to see the geometry and arithmetic of tannakian fundamental groups?
The etale fundamental group is an inverse limit of automorphism groups of finite etale coverings. We can see the geometry of etale fundamental group very well from etale coverings just like ...
- 1,921
2
votes
0
answers
193
views
Regarding Bruinier's book "Borcherd Products on $O(2,l)$ ..." 3.4.1 example 2.
Hi all
I am currently trying to understand Borcherd's lifts for a part of my thesis work. I am dreadfully confused about some comments made in the example given in Bruinier's book mentioned in the ...
- 63
5
votes
1
answer
558
views
Topology in Arithmetic
Conjectured by Mordell and later proved by Faltings, a non-singular algebraic curve of genus $g$ over $\mathbb{Q}$ has finitely many rational points if $g > 1$. Since the genus of the Fermat curve $...
- 595