All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
17
votes
1
answer
2k
views
Hecke operators acting as correspondences?
This question is inspired by Relation between Hecke Operator and Hecke Algebra
I remember having heard of yet another way of looking at Hecke operators acting on the spaces of modular forms for ...
- 23.5k
7
votes
0
answers
538
views
Some sort of descent theory
Let $X$ be a smooth, projective curve defined over a finite field $k$. We denote by $\overline{X}$ it's extension to the algebraic closure $\overline{k}$. All the questions are about coherent sheaves.
...
- 887
18
votes
4
answers
4k
views
Why are topological ideas so important in arithmetic?
For example, Wikipedia states that etale cohomology was "introduced by Grothendieck in order to prove the Weil conjectures". Why are cohomologies and other topological ideas so helpful in ...
- 4,351
2
votes
1
answer
474
views
Automorphism of algebraic group preserving a hyperspecial maximal compact
Suppose that $K/\mathbb{Q}_l$ is a finite extension, with ring of integers $\mathcal{O}_K$. Suppose $\mathcal{G}/K$ is a (linear) algebraic group (connected+reductive), and $\Gamma\subset \mathcal{G}(...
- 1,233
3
votes
1
answer
184
views
How many linear terms are in the Hilbert set of H(z,t), a polynomial in 2 variables over a field k(s) of transcendence degree one over a finite field?
I am looking for a good reference for Hilbert's irreducibility theorem, and ofproperties of Hilbert sets besides Serres Lectures on The Mordell-Weil Theorem. In particular, I am interested it to the ...
37
votes
3
answers
5k
views
Is there a nice proof of the fact that there are (p-1)/24 supersingular elliptic curves in characteristic p?
If $k$ is a characteristic $p$ field containing a subfield with $p^2$ elements (e.g., an algebraic closure of $\mathbb{F}_p$), then the number of isomorphism classes of supersingular elliptic curves ...
- 45.7k
3
votes
2
answers
2k
views
Question on determining the minimal polynomial for an algebraic quotient
I need to determine the minimal polynomial for a quotient in (1).
(1) B = C / A
C is known as a root of a 36th degree polynomial and A is known as a root of a 24th degree polynomial.
However I ...
- 130
3
votes
0
answers
546
views
Determining polynomial coefficients correctly
I am working with a dot product of 2 unit vectors in R3 that are algebraic.
I am trying to recover their original format in a number field, but I am not sure of how to go about doing this. I do use ...
- 130
14
votes
1
answer
1k
views
Geometry for Anderson's motives?
Anderson's $t$-motives satisfy most of what is expected of a reasonable category of mixed motives, except of course that everything is in positive characteristic. For instance, it is a linear category ...
- 4,015
9
votes
2
answers
1k
views
modularity of algebraic varieties
Hello,
Are there any examples of varieties which are not Shimura varieties or abelian varieties
and whose L-functions have been shown to be a product of automorphic L-functions?
Thanks.
N
- 2,842
14
votes
3
answers
2k
views
"Nice" definition of discriminant as alluded to in an answer of Qing Liu
In his answer
here
Qing Liu mentioned "the 'discriminant' of X which measures the defect of a functorial isomorphism which involves powers of the relative dualizing sheaf of X/R."
Could ...
- 646
3
votes
2
answers
3k
views
Opinions about the book "Lectures on Algebraic Geometry 1: Sheaves, Cohomology of Sheaves, and Applications to Riemann Surfaces" [closed]
I'm not sure whether it is appropriate to ask this question here. I apologize if it isn't.
I am deciding whether to buy the book "Lectures on Algebraic Geometry 1: Sheaves, Cohomology of Sheaves, and ...
- 39
5
votes
1
answer
196
views
Understanding different Q-models of a curve over C
These are parametrized by $H^1(Gal(\mathbb{Q}), Aut X)$, where X is some $\mathbb{Q}$-model of the curve.
It was established in Confusion about how the first cohomology classifies torsors
that fiber ...
- 5,929
4
votes
1
answer
935
views
SGA1 Chapter XIII (tamely ramified sheaves)
I'm trying to read chapter XIII of SGA1, and I'd appreciate some help about a few issues I'm having.
Definition 2.1.1. is of tamely ramified sheaves. The definition is as such: if $U$ is an open ...
- 1,522
18
votes
1
answer
594
views
Does every hyperbolic curve over a finite field have an etale cover with a real Frobenius eigenvalue?
More precisely: let X/F_q be a smooth projective algebraic curve of genus at least 2. Does there always exist a curve Y/F_{q^d} with a finite etale projection Y -> X, such that one of the Frobenius ...
- 19.2k
7
votes
1
answer
603
views
Picard groups of reductive group schemes
There exists information on the Picard (and Brauer) group of a reductive algebraic group over a number field k. For example, Sansuc shows (in his big Crelle paper of 1980) that if G is connected and ...
4
votes
0
answers
243
views
Chow groups of arithmetic surfaces
Given an arithmetic surface $S$, I would like to know the following properties of its first and second Chow groups $CH^1(S), CH^2(S)$:
Are they finitely generated? If so, what is the rank?
What is ...
- 4,593
35
votes
2
answers
3k
views
Finiteness property of automorphism scheme
Some time ago I mentioned a certain open question in an MO answer, and Pete Clark suggesting posting the question on its own. OK, so here it is:
First, the setup. Let $X$ be a projective scheme over ...
48
votes
6
answers
5k
views
Smooth linear algebraic groups over the dual numbers
It is a standard and important fact that any smooth affine group scheme $G$ over a field $k$ is a closed $k$-subgroup of ${\rm{GL}}_n$ for some $n > 0$. (Smoothness can be relaxed to finite type, ...
12
votes
3
answers
2k
views
Is there a schemetical construction for modular curves over the rationals?
One can get modular curves by the following procedures: first take the uper half plane and the rationan numbers on the x-axis, then we consider the quotient by a congruence subgroup. Now we get a ...
- 121
7
votes
3
answers
3k
views
The etale fundamental group of a field
Background and motivation:
I am teaching the "covering space" section in an introductory algebraic topology course. I thought that, in the last five minutes of my last lecture, I might briefly sketch ...
- 7,328
12
votes
4
answers
2k
views
Context for intersection theory
This is a pretty basic question. Hartshorne defines "intersection multiplicity" for any two divisors on a surface. Fulton has an impressive framework of generalizing this in his book (my understanding ...
- 5,929
17
votes
1
answer
1k
views
Does there exist a number field, unramified over a predetermined finite set of primes of Q, such that the inverse regular Galois problem is correct for that number field?
The question is: for any finite group, $G$, and any finite set of primes (of $\mathbb{Z}$), $P$, is there a number field $K$, such that there is a regular $G$-Galois extension of $\mathbb{P}^1_K$, and ...
- 1,522
3
votes
1
answer
844
views
finite generation of the Mordell-Weil group over finitely generated fields
Does anyone know a reference for the proof of the finite generation of the Mordell-Weil group over finitely generated fields?
user19475
76
votes
2
answers
6k
views
Is it known that the ring of periods is not a field?
I have just learned here that we know numbers that are not periods; is it known meanwhile that the ring of periods is not a field? I know that it is conjectured that $1/\pi$ is not a period, but the ...
- 32.6k
33
votes
3
answers
4k
views
Is there an explicit example of a complex number which is not a period?
Kontsevich and Zagier proposed a definition of a period (see for example http://en.wikipedia.org/wiki/Ring_of_periods ). The set of periods is countable, so not all of $\mathbb{C}$. I heard a rumour ...
- 1,010
3
votes
3
answers
958
views
solutions to equation mod a prime
I know that characterizing the solutions to an equation in a finite field is generally difficult, but I was wondering if anyone had anything to say about the equation
(ab)^2 + a^2 + b^2 = 0 mod p
I ...
- 39
21
votes
3
answers
2k
views
Surprising Analogue of Q
I was describing Manish Kumar's work a few weeks ago to a fellow graduate student, and she stumped me with a big-picture question I couldn't answer.
Manish Kumar proved that the commutator subgroup ...
- 1,522
3
votes
2
answers
378
views
Upper bound on greatest prime of bad reduction for a plane curve
Background
We are given a curve with integer coefficients. I want to make a suggestion in another question (Computationally bounding a curve's genus from below?) into a deterministic algorithm ...
- 4,593
3
votes
1
answer
335
views
Decomposition of primes, where the residue field extensions are allowed to be inseparable
I've been dealing with the following situation:
Let $R\subseteq S$ be an extension of Dedekind rings, where $Quot(R)=:L \subseteq E:=Quot(S)$ is a $G$-Galois extension. Let $\mathfrak{p}$ be a prime ...
- 1,386
10
votes
3
answers
650
views
Computationally bounding a curve's genus from below?
Background
In the course of answering another question (Infinite collection of elements of a number field with very similar annihilating polynomials) I found myself with a curve, that if it had a ...
- 4,593
8
votes
3
answers
2k
views
How much complex geometry does the zeta-function of a variety know
From Weil conjecture we know the relation between the zeta-function and the cohomology of the variety, however it appears that there are certainly more information containing in the zeta-function, and ...
- 1,525
9
votes
3
answers
1k
views
A question on liftings of supersingular elliptic curves over the prime fields
Let $p$ be a fixed prime number $>3$. The motivation for asking the question below is the coincidence of the following two numbers. Namely, the number $h_p^{(1)}$ of supersingular $j$-invariants ...
- 2,406
8
votes
1
answer
1k
views
Rational numbers as an extension of the field with one element?
Greetings.
I would love to have a field $\mathbb F$ which is a subfield of the field of rational numbers $\mathbb Q$, and such that the Galois group $Gal (\mathbb Q / \mathbb F)$ has preferably ...
- 3,897
9
votes
3
answers
2k
views
Supersingular elliptic curves and their "functorial" structure over F_p^2
In a letter to Tate from 1987, Serre describes a beautiful Theorem relating mod p modular forms to quaternions ("Two letters on quaternions and modular forms (mod p)", Israel J. Math. 95 (1996), 281--...
- 2,406
9
votes
4
answers
3k
views
reduction of CM elliptic curves
Can someone indicate how to prove the following equivalences for a CM elliptic curve $E$:
(i) $p$ is inert in End($E$)
(ii) $E_p$ is supersingular
(iii) The trace of the Frobenius at $p$ is $0$ [...
user19475
13
votes
4
answers
3k
views
When is the Galois representation on the étale cohomology unramified/Hodge-Tate/de Rham/crystalline/semistable?
Let $X/K$ be a variety over a global field $K$. When (and why) is the Galois representation $H^i_{et}(X \times_K \bar{K}, \mathbf{Q}_\ell)$ unramified at a place $v$ of $K$?
I guess this is true if $...
user19475
4
votes
2
answers
667
views
Deformations of Tame Coverings
To say that I am a novice at deformation theory is to grossly overestimate my abilities in this area. I've come across the following theorem in a paper, and I'd like to know how far one is able to ...
- 1,386
32
votes
4
answers
4k
views
Modular curves of genus zero and normal forms for elliptic curves
This is maybe the first question I actually need to know the answer to!
Let $N$ be a positive integer such that $\mathbb{H}/\Gamma(N)$ has genus zero. Then the function field of $\mathbb{H}/\Gamma(N)...
- 118k
7
votes
1
answer
264
views
For a given finite group G, is there a cover of P^1 over Q s.t. over C it's G-Galois?
For any finite group, G, we can find a cover of ℙ1ℂ which is G-Galois. The regular inverse Galois problem is equivalent to there existing such a cover that descends with action to ℚ. ...
- 1,522
35
votes
4
answers
8k
views
What would a "moral" proof of the Weil Conjectures require?
At the very end of this 2006 interview (rm), Kontsevich says
"...many great theorems are originally proven but I think the proofs are not, kind of, "morally right." There should be better proofs......
- 1,764
8
votes
1
answer
1k
views
Geometric Intuition for Big Monodromy
In various contexts, I have come across results referred to as "big monodromy." A standard arithmetic example is the open image theorem for the image of Galois action on non-CM elliptic curves. A ...
- 671
4
votes
0
answers
2k
views
How to learn about Shimura varieties?
Possible Duplicate:
What is a good roadmap for learning Shimura curves?
What's the best way (in your opinion) to learn the theory of Shimura varieties?
73
votes
2
answers
8k
views
The inverse Galois problem and the Monster
I have a slight interest in both the inverse Galois problem and in the Monster group. I learned some time ago that all of the sporadic simple groups, with the exception of the Mathieu group $M_{23}$, ...
- 4,994
24
votes
5
answers
6k
views
Wild Ramification
The question is, loosely put, what is known about wild ramification?
Is there a semi-well-established theory of wild ramification that can be furthered in various specific situations? Or maybe there ...
4
votes
2
answers
448
views
Can an abelian variety be represented as the cohomology of some other object?
Question
Given an abelian variety $V$ and an integer $n$, is there a natural abelian category with a natural object $X$ and natural coefficients $F$ so that $V\simeq H^n (X,F)$?
Motivation
Studying ...
- 4,593
20
votes
4
answers
2k
views
Are there Néron models over higher dimensional base schemes?
Are there Néron models for Abelian varieties over higher dimensional ($> 1$) base schemes $S$, let's say $S$ smooth, separated and of finite type over a field?
If not, under what additional ...
user19475
12
votes
2
answers
2k
views
What, precisely, is the relationship between "fields of moduli" and "moduli spaces"?
Notation
The term "field of moduli" comes in up in different scenarios, but let's consider the following: Let X->ℙ1 be a G-Galois cover, where everything is over the algebraic closure of ...
- 1,522
8
votes
2
answers
2k
views
(nontrivial) isotrivial family of elliptic curves
I think it should be a standard procedure to construct such things, can anyone give a reference or give a hint? Can this be done over any base scheme?
- 1,503
17
votes
2
answers
3k
views
Why is one interested in the mod p reduction of modular curves and Shimura varieties?
Why is one interested in the mod p reduction of modular curves and Shimura varieties?
From an article I learned that this can be used to prove the Eichler-Shimura relation which in turn proves the ...
user19475