All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
12
votes
1
answer
2k
views
what exactly is the moduli functor for classifying elliptic curves with (full) level N structure?
So, when people say, "the moduli problem of classifying elliptic curves over $\mathbb{C}$ with level $N$ structure", there are usually two associated functors I've seen:
$P_N : \textbf{Ell}\...
- 10.7k
1
vote
1
answer
372
views
Automorphic and modular forms for subgroups of modular group and fuchsian groups
Is there a well-understood correspondence between subgroups G of $SL_2(\mathbb{Z})$ (not necessarily of finite index) and graded algebras of modular forms invariant under G?
Given an algebra of ...
- 127
7
votes
3
answers
2k
views
moduli interpretations for modular curves
Some big picture questions -
What are some applications of the moduli interpretation for congruence curves? Specifically, the interpretations for congruence curves parametrizing elliptic curves with ...
- 10.7k
14
votes
1
answer
5k
views
Why are Galois Representations so important in Number theory ?
Dear everyone,
Motivation :
From the past few days, I have been reading about the Galois Representations . I was really amazed to see that every seminal idea in the theory of elliptic curves have ...
4
votes
2
answers
544
views
Parity dependent population inversion in Mordell elliptic curves
I've been looking at curves of the form $y^2=x^3+k$ (where k is 6th power free and not divisible by 3^3) and I've noticed that there seems to be distinct grouping in residues classes modulo 504.
One ...
- 791
9
votes
1
answer
1k
views
Modern Proof of the Theorem of the Base
I am looking for a modern proof of the so-called "Theorem of the Base"--that the Neron-Severi rank of a smooth projective variety is finite. One can prove this for varieties over $\mathbb{C}$ easily ...
- 23k
2
votes
2
answers
392
views
A unified description of zeta functions of a curve over $\mathbb{F}_q$ and Riemann $\zeta$ function
Is there a unified description (or a set of axioms) of the zeta function of an algebraic curve over a finite field $\mathbb{F}_q$ and the Riemann zeta function?
user16974
5
votes
0
answers
212
views
Rational solutions of $x (y - z) y (z - x) z (x - y) = t^2$
I am interested in finding every rational solution of $x (y - z) y (z - x) z (x - y) = t^2$ (expressed in homogenous form, to show its symmetry).
Among other approaches I am pursuing, it is clear ...
- 1,545
11
votes
1
answer
1k
views
Equivalence between statements of Hodge conjecture
Dear everyone,
I was unable to obtain the equivalence between the two statements of the Hodge conjecture. I searched for some previous questions that others asked here, to check whether someone has ...
9
votes
1
answer
2k
views
Overview of Arakelov intersection theory and the Arakelov Chow ring
I'm looking for a reference that gives an overview of the most important properties of Arakelov intersection theory (on arithmetic varieties of arbitrary dimension) and that describes basic properties ...
- 47.4k
2
votes
0
answers
178
views
Axioms for zeta function of a scheme
Is there any set of axioms that characterize completely the zeta function of a scheme over a finite field of characteristic $p$?
user16974
1
vote
0
answers
522
views
Component group of Neron model of a parametrized abelian variety
Let $A$ be an abelian variety of dimension $2$ over a $p$-adic field $K$ with (additive) valuation $v$. Assuming $A$ has multiplicative reduction, the theory of $p$-adic theta functions gives us an ...
- 15.4k
0
votes
0
answers
171
views
Weaker conditions for potential good reduction of Abelian varieties
We are concerned with slight weakening of a result of the Serre-Tate paper "Good reduction of abelian varieties" google. I think the title of the question conveys what we are in here for. So, I'll ...
- 1
1
vote
1
answer
206
views
How to define zeta function for curves over a number field
How to define zeta function for a curve over $\mathbb{Z}$ or $\mathbb{Q}$?
user16974
10
votes
2
answers
2k
views
Heuristics for the Hodge Conjecture
W. V. D. Hodge is famous for his Hodge conjecture, one of the Millennium prize problems. Hodge might have had some rough heuristics or ideas that led him to the formulation of the conjecture.
I am ...
5
votes
1
answer
753
views
Some help in digesting a paragraph in the introduction of Deligne/Rapoport's "Les Schemas de Modules de Courbes Elliptique"
http://www.springerlink.com/content/04x54gr171v556m4/fulltext.pdf
On page 149 (DeRa-7), in the middle of the page, I can translate the middle paragraph that starts "3. La surface de Riemann ..." as ...
- 10.7k
10
votes
0
answers
758
views
How does one understand geometric CFT in terms of modularity?
I have recently asked a question in a similar vein:
What makes Geometric CFT easier than CFT?
but I'm afraid I wasn't quite ripe to ask it yet. I have since consulted with the following sources:
http:/...
- 5,929
9
votes
1
answer
618
views
abelianization of adelic points of an algebraic group
Let $G$ be a connected reductive group defined over a number field $K$ and $G^{der}$ its derived subgroup.
Let $\mathbb{A}_K$ denote the adeles of $K$.
Then for $G=GL_n$ we have $[GL_n(\mathbb{A}_K),...
- 248
3
votes
2
answers
1k
views
Good reduction of abelian varieties [S-T] -- Why is this ring henselian?
First of all, I find it hard to formulate a good title for this question. Sorry that it is so vague.
Let's move on te the question itself. Lately I have been studying the article "Good reduction ...
- 5,504
5
votes
2
answers
443
views
Rank of $x (x^2 - 1) = c (c^2 - 1) y^2 $ over $\mathbb{Q}$ for given rational values of $c$
Can anything be said in general about the rank etc over $\mathbb{Q}$ of the family of Weierstrass equations (in slightly non-standard form) $x (x^2 - 1) = c (c^2 - 1) y^2$ for various given rational ...
- 1,545
13
votes
2
answers
572
views
Existence of points on varieties which avoid a given number field.
Let C be a geometrically integral curve over a number field K and let K' be a number field containing K. Does there exist a number field L containing K such that
$L \cap K' = K$, and
$C(L) \neq \...
- 10.5k
2
votes
1
answer
434
views
Weil reciprocity on abelian varieties and biextensions?
I was once told, by someone who would likely be right about such things, that the version of Weil reciprocity for abelian varieties (as in Lang, Abelian Varieties) should come out of consideration of ...
- 12.6k
14
votes
3
answers
3k
views
Quadratic reciprocity and Weil reciprocity theorem
I was told that Weil reciprocity theorem (one has two meromorphic function $f,g$ on a complex curve $C$, so $\prod\limits_{x\in C} g(x)^{ord_xf}=\prod\limits_{x\in C}f(x)^{ord_xg} \ $ where $ord_xf$ ...
- 5,063
5
votes
1
answer
544
views
What happens to factors of the resultant upon specialization?
Let $f, g$ be two polynomials in $S[t]$ where the coefficient
ring is $S = \mathbb{C}[a_1..a_n]$.
The resultant of $R(f,g)$ gives some measure as to whether or
not $f$ and $g$ share a common factor.
...
- 103
9
votes
2
answers
2k
views
Is an elementary symmetric polynomial an irreducible element in the polynomial ring?
Let $S=\mathbb{C}[x_1,x_2,\dots,x_n]$ be a polynomial ring. Let $e_a$ denotes the elementary symmetric polynomials of degree $a$ in $S$.
For $n=2$:
$e_1=x_1+x_2$;
$e_2=x_1x_2$.
For $n=3$:
$e_1=x_1+...
- 446
1
vote
0
answers
342
views
Passing from Regular sequence to Prime ideal, for power sum symmetric polynomial
Let $S=\mathbb{C}[x_1,x_2,x_3,x_4]$ be a polynomial ring. Let $p_i=x_1^i+\cdots+x_4^i$ be the power sum symmetric polynomial in $\mathbb{C}[x_1,x_2,x_3,x_4]$.
Let $I=(p_1,p_2)$ be an Ideal of $\mathbb{...
- 446
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
20
votes
4
answers
3k
views
Does $(x^2 - 1)(y^2 - 1) = c z^4$ have a rational point, with z non-zero, for any given rational c?
I need this result for something else. It seems fairly hard, but I may be missing something obvious.
Just one non-trivial solution for any given $c$ would be fine (for my application).
- 1,545
4
votes
3
answers
794
views
Quotients of Tate modules
Let $p$ be a prime number, let $K$ denote a finite extension $\mathbb{Q}_{p}$ and let
$\overline{K}$ be an algebraic closure of $K$. Let $A$ be an ellitpic curve over
$K$ and denote by $T_{p}A$ its ...
- 41
6
votes
3
answers
2k
views
What is the "Lefschetz Principle" (examples) ?
Hi there,
can anyone explain to me what the "Lefschetz Principle" is by some clear "classical"
examples (not relying explicitly on model theory, say).
Thanks !
- 4,555
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 ...
34
votes
2
answers
3k
views
Shimura-Taniyama-Weil VS Grothendieck's dessins
When listening to the beautiful lectures by Gilles Schaeffer at
the SLC68, the following (perhaps crazy) question occurred to me:
did anyone attempt (succeed?) to combinatorially prove modularity of ...
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
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
6
votes
1
answer
472
views
Existence of elements in an Eichler order whose norm is minus one
Let $B$ be an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $D$, and $\mathcal{O}_N$ be an Eichler order of level $N$. Is there an element $x\in \mathcal{O}_N$ such that its reduced ...
- 570
33
votes
1
answer
1k
views
Coefficients of Weil Cohomology Theories
A Weil Cohomology theory is a functor from the category of smooth projective varieties (over some fixed field $k$) to graded $K$-algebras (for some fixed field $K$) satisfying various axioms. For ...
- 331
16
votes
2
answers
4k
views
Elliptic Curves over Rings?
So an elliptic curve $E$ over a field $K$ is a smooth projective nonsingular curve of genus $1$ together with a point $O \in E$.
I was reading Silverman's "Arithmetic of Elliptic Curves" and it ...
- 1,458
3
votes
0
answers
741
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 ...
4
votes
0
answers
275
views
An arithmetic analogue of the discriminant curve of a conic bundle threefold
I am looking for an "arithmetic" analogue of a well known result on threefolds with a conic bundle structure. The following result can be found in [Iskovskikh - On the rationality problem for conic ...
- 21.4k
2
votes
2
answers
355
views
Points on a projective variety modulo $p$
Suppose that for $n \geq 4$ we have $F(x_1, \cdots, x_n) \in \mathbb{Z}[x_1, \cdots, x_n]$ is a homogeneous polynomial. Consider a large prime $p$, and suppose that we consider points of the variety $...
- 26.9k
21
votes
1
answer
2k
views
Polynomials for addition in the Witt vectors
The addition of $p$-typical Witt vectors ($p$ a prime number) is given by universal polynomials $S_n=S_n(X_0,\dots,X_n;Y_0,\dots,Y_n)\in\mathbb{Z}[X_0,X_1,\dots;Y_0,Y_1,\dots]$ determined by the ...
- 4,492
1
vote
1
answer
359
views
Adjoint groups of Mumford-Tate groups
Let $F$ be a sub-field of $\mathbb{C}$ and $B/F$ and $C/F$ be abelian varieties, with $C$ of CM type. Denote the Mumford-Tate groups of $B$, $C$ and $B\times_F C$ by $G_B$, $G_C$ and $G_{B\times C}$, ...
- 11
3
votes
0
answers
231
views
A small question on Carayol's paper
This question might be silly but really confuses me...
On page 31 of his paper: Sur la mauvaise réduction des courbes de Shimura , Carayol gave a modular interpretation of the actions of $G'(\mathbb ...
- 73
4
votes
0
answers
440
views
What's an example of a wild monodromy action on a high genus curve?
Given a smooth projective curve $X/\mathbb{Q}_{p}$ of genus $g \ge 2$, there is an induced monodromy action of the Galois group on the $\ell$-adic cohomology of
$$X \otimes \bar{\mathbb{Q_{p}}}.$$
...
- 3,284
5
votes
5
answers
1k
views
What is the possible usefulness of étale topology and cohomology apart from the resolution of the Weil conjecture ?
My question is as stated in the title:
What is the possible usefulness of étale topology and cohomology apart from the resolution of the Weil conjecture ?
I am particularly interested to know if it'...
- 481
0
votes
0
answers
746
views
Igusa model of modular curves
I would like to know what the "Igusa model" of the modular curve is and basic properties about it.
Can someone point me to a reference?
- 1,271
4
votes
2
answers
1k
views
Isogenous elliptic curve with integral j-invariant
Let $E$ be an elliptic curve over a local field $K$.
If $E$ has non-integral $j$-invariant, under what conditions will there exist an isogenous curve with integral $j$-invariant?
Here, saying an ...
- 41
6
votes
1
answer
1k
views
Rank of isogenous elliptic curves
I think that k-isogenous elliptic curves have the same rank as I think rank is an isogeny invariant. However, I am not sure. Does anyone know where could I find a proof? Thanks!
- 201
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