All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
11
votes
1
answer
875
views
An arithmetic highest weight theory?
I apologize if these questions seem naive or loaded.
Is there an analogous theory of highest weights for irreducible finite-dimensional representations of Lie algebras of algebraic group (or perhaps ...
- 436
4
votes
2
answers
336
views
Methods of showing a map has integral or good reduction
Question
Say we have a map, C->D, of relative curves over a Dedekind scheme, S. What are some of the available methods for showing that this map has good reduction, or integral reduction, at some s&#...
4
votes
2
answers
2k
views
Brauer-Manin obstruction and Tate-Shafarevich group of an Abelian variety
I read that the Brauer-Manin obstruction $A(\mathbb{A}_K)^{\mathbf{Br}}$ of an Abelian variety $A$ over a number field $K$ equals (naturally?) its Tate-Shafarevich group $\mathrm{III}(A)$.
Is this ...
user19475
9
votes
2
answers
656
views
How does the order of a pole of a zeta function indicate any geometric information?
Here, I'm primarily concerced about zeta functions of hypersurfaces over fields of finite characteristic.
Assume $F_q$ to be a finite field with q elements. Consider the zeta function of the ...
- 141
14
votes
3
answers
1k
views
Non-simply-connected smooth proper scheme over Z?
Source
This question came up in the discussion between Kevin Buzzard and Minhyong Kim in the comments to Smooth proper scheme over Z. It was 2 weeks ago, so I took the liberty of posting it as ...
3
votes
1
answer
2k
views
Conditions that allow unique solutions for Linear Diophantine equations
(This posting became very long, so I should note that there are two alternative but nearly equivalent formulations of the same question being given. The first one asks for the optimal strategy for ...
33
votes
5
answers
8k
views
Why no abelian varieties over Z?
Motivation
I learned about this question from a wonderful article Rational points on curves by Henri Darmon. He gives a list of statements (some are theorems, some conjectures) of the form
the set $\{...
- 15.1k
5
votes
1
answer
513
views
Field of Definition of a Meromorphic Function
Question
Let X be a smooth, projective curve over the algebraic closure of ℚ. Let f:X->ℙ1 be a meromorphic function. Assume that the zeros and the poles are defined over some number field,...
- 1,522
18
votes
1
answer
1k
views
A Galois Theory Computation
Excuse me for the specificity of this question, but this is a silly computation that's been giving me trouble for some time.
I want to explicitly realize the order 21 Frobenius group over ℂ(x), ...
- 1,522
14
votes
2
answers
728
views
A local-to-global principle for being a rational surface
Let $k$ be a number field and $F$ a $1$-variable function field over $k$ (a finitely generated extension of $k$, of transcendence degree $1$, in which $k$ is algebraically closed). If $F$ becomes the ...
- 18.7k
3
votes
1
answer
723
views
A strange logical implication in algebraic geometry
So there's an old theorem of Lang and Weil showing that the Riemann hypothesis for curves over finite fields implies a kind of quasi-riemann hypothesis for surfaces over finite fields.
I am wondering:...
- 13.1k
18
votes
1
answer
2k
views
Eichler-Shimura isomorphism and mixed Hodge theory
Let $Y(N),N>2$ be the quotient of the upper half-plane by $\Gamma(N)$ (which is formed by the elements of $SL(2,\mathbf{Z})$ congruent to $I$ mod $N$). Let $V_k$ be the $k$-th symmetric power of ...
- 23.5k
63
votes
1
answer
7k
views
Smooth proper scheme over Z
Does every smooth proper morphism $X \to \operatorname{Spec} \mathbf{Z}$ with $X$ nonempty have a section?
EDIT [Bjorn gave additional information in a comment below, which I am recopying here. -- ...
- 23.8k
32
votes
9
answers
5k
views
Do there exist modern expositions of Klein's Icosahedron?
Reading Serre's letter to Gray
, I wonder if now modern expositions of the themes in Klein's book
exist. Do you know any?
- 10.8k
11
votes
2
answers
738
views
Building elliptic curves into a family
Suppose $E/ \mathbb{Q}$ is an elliptic curve whose Mordell-Weil group $E(\mathbb{Q})$ has rank r. When can we realize E as a fiber of an elliptic surface $S\to C$ fibered over some curve, with ...
- 13.1k
16
votes
2
answers
1k
views
Is the tangent space functor from PD formal groups to Lie algebras an equivalence?
The previous version of this question was rather badly broken, and I hope this version makes some sense.
There have been a few questions on MathOverflow about how much representation-theoretic ...
- 45.7k
16
votes
5
answers
2k
views
Elliptic Curves over F_1?
Is there an notion of elliptic curve over the field with one element? As I learned from a previous question, there are several different versions of what the field with one element and what schemes ...
- 27.5k
19
votes
3
answers
2k
views
Elkies' supersingularity theorem in higher dimension
The following is a theorem of Elkies:
Let $X$ be an elliptic curve over $\mathbb{Q}$. Then there are infinitely many primes $p$ such that the action of Frobenius on $H^1(\mathcal{O}, X)$ is zero.
...
- 156k
11
votes
5
answers
2k
views
Geometry Vs Arithmetic of schemes
Let's suppose we have a Scheme $X$ over the the field $k$, where such a field can be though to be either $\mathbb{C}$ or a finite field $\mathbb{F}_q$. Then having this in mind,
Where do we find some ...
- 2,132
37
votes
7
answers
8k
views
Model theoretic applications to algebra and number theory(Iwasawa Theory)
One of my favorite results in algebraic geometry is a classical result of AX (see http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/) I'll recall ...
- 2,396
10
votes
2
answers
1k
views
Does a universal Frobenius map exist?
For any prime p, one has the Frobenius homomorphism Fp defined on rings of characteristic p.
Is there any kind of object, say U, with a "universal Frobenius map" F such that for any prime p and any ...
- 2,399
14
votes
4
answers
7k
views
"Counter"-example for Gauss's Lemma on irreducible polynomials
Gauss's Lemma on irred. polynomial says,
Let R be a UFD and F its field of fractions. If a polynomial f(x) in R[x] is reducible in F[x], then it is reducible in R[x].
In particular, an integral ...
- 5,052
6
votes
1
answer
875
views
What should Spec Z[\sqrt{D}] x_{F_1} Spec \bar{F_1} be?
What should be $\text{Spec } \mathbb{Z}[\sqrt{D}] \times_{\mathbb{F}_1} \text{Spec } \overline{\mathbb{F}}_{1}$?
Sure, there's more than one definition.
I'm looking for any answer that uses at least ...
- 4,593
29
votes
2
answers
4k
views
What is the algebraic closure of the field with one element?
If doing geometry over $\mathbb F_p$ means also using its algebraic closure, it must be interesting to talk about the algebraic closure of $\mathbb F_1$ - the field with one element.
I saw that the ...
- 4,593
10
votes
2
answers
393
views
Counting points on varieties of low codimension
The graduate students here at MIT have been thinking about questions like the following: Over $\mathbb{F}\_q$, how many symmetric matrices are there with nonzero determinant and $0$'s on the diagonal? ...
- 156k
25
votes
3
answers
2k
views
Least number of non-zero coefficients to describe a degree n polynomial
I'd be grateful for a good reference on this, it feels like a classic subject yet I couldn't find much about it.
Polynomials in one variable of the form $x^n+a_{n-1}x^{n-1}+\dots +a_1 x+a_0$ can be ...
- 2,901
4
votes
1
answer
530
views
Counting solutions to x^{p+1}=y^4 in a finite field
I need to compute the number of solutions to the equation $x^{p+1} = y^4$ in the field with $p^2$ elements (for p sufficiently large). The form of the equation suggests to me that the solution would ...
- 2,537
86
votes
4
answers
15k
views
Etale cohomology -- Why study it?
I know (at least I think I know) that some of the main motivating problems in the development of etale cohomology were the Weil conjectures. I'd like to know what other problems one can solve using ...
- 2,799
16
votes
4
answers
1k
views
K3 surfaces with good reduction away from finitely many places
Let S be a finite set of primes in Q. What, if anything, do we know about K3 surfaces over Q with good reduction away from S? (To be more precise, I suppose I mean schemes over Spec Z[1/S] whose ...
- 19.2k
2
votes
4
answers
617
views
A question on function fields (extending my previous question)
Consider the extension $\mathbb Q(a,b)$ of the field of rationals, where $a$, $b$ are algebraically independent transcendentals. To $\mathbb Q(a,b)$ adjoin the roots of the polynomials $x^5+a^5=1$ and ...
- 161
15
votes
5
answers
3k
views
Can we count isogeny classes of abelian varieties?
Let's fix a finite field F and consider abelian varieties of dimension g over F. Can we say how many isogeny classes there are? Is it even clear that there's more than one isogeny class? For g=1, ...
- 1,988
11
votes
1
answer
705
views
a question on function fields
Consider the transcendental extension Q(t) of the field of rationals.
To Q(t) adjoin the root of the polynomial x^5+t^5=1. The resulting
field Q(t)[x] is a radical extension of Q(t). Is it true that ...
- 161
11
votes
4
answers
3k
views
What does ramification have to do with separability?
Does ramification have anything to do with inseparability? It feels like an extension of Q in which p ramifies should somehow correspond to an extension of F_p(t). Does totally ramified <--> purely ...
- 15.4k
55
votes
3
answers
5k
views
What are the higher homotopy groups of Spec Z ?
The homotopy groups of the étale topos of a scheme were defined by Artin and Mazur. Are these known for Spec Z? Certainly π1 is trivial because Spec Z has no unramified étale covers,...
- 8,004
12
votes
4
answers
715
views
Behaviour of Zeta-function under Finite Morphism
Let X ---> Y be a finite surjective morphism of smooth, projective, connected varieties over a finite field F_q. Can one describe the zeta function Z(X, t) in terms of the zeta-function Z(Y,t) of ...
- 671
13
votes
3
answers
1k
views
A comprehensive overview of finite fields
I've read numerous introductions to finite fields, but I feel like my intuition about them is fairly lacking. Considering that finite fields are the the most "inert" objects in algebraic geometry, I ...
- 11.2k
6
votes
3
answers
601
views
Solving "a, b, a+b have given divisors" problem
I've read an interesting article, math.NT/0409456 where you're just trying to solve a simple problem:
For a given (finite) set of primes S find all solutions to an equation ...
- 15.1k
8
votes
1
answer
1k
views
Learning about Galois representations
My goal was to learn about l-adic representations on some example — I'm a newbie in these topics.
Thus take pt = Spec F_q, ...
- 15.1k
50
votes
5
answers
10k
views
Definition and meaning of the conductor of an elliptic curve
I never really understood the definition of the conductor of an elliptic curve.
What I understand is that for an elliptic curve E over ℚ, End(E) is going to be (isomorphic to) ℤ or an ...
- 5,530
13
votes
1
answer
1k
views
What are the higher $\mathrm{Ext}^i(A,\mathbf{G}_m)$'s, where $A$ is an abelian scheme?
Let $S$ be a base scheme, let $A/S$ be an abelian scheme, and let $\mathbf{G}_m/S$ be the multiplicative group; consider $A$ and $\mathbf{G}_m$ as objects in the abelian category of sheaves of abelian ...
- 2,270
20
votes
2
answers
4k
views
"Fermat's last theorem" and anabelian geometry?
Do I remember a remark in "Sketch of a program" or "Letter to Faltings" correctly, that acc. to Grothendieck anabelian geometry should not only enable finiteness proofs, but a proof of FLT too? If yes,...
- 10.8k
43
votes
1
answer
19k
views
What is inter-universal geometry?
I wonder what Mochizuki's inter-universal geometry and his generalisation of anabelian geometry is, e.g. why the ABC-conjecture involves nested inclusions of sets as hinted in the slides, or why such ...
- 10.8k
10
votes
2
answers
944
views
Logarithmic structures on moduli of elliptic curves over Z
I've heard it stated that if you take the moduli of elliptic curves with some level structure imposed (as a moduli scheme over Spec(Z)), there is a logarithmic structure that you can impose at the ...
- 52.6k
2
votes
3
answers
1k
views
What is the base change in number theory?
I'm somewhat familiar with base change in scheme theory: sometimes a property of a morphism X \to Y survives a base change ...
- 15.1k
20
votes
5
answers
4k
views
Equivalent statements of the Riemann hypothesis in the Weil conjectures
In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with $q$ elements says that: the eigenvalues of ...
- 671
13
votes
3
answers
1k
views
How to topologize X(R) when R is a topological ring?
Given a topological ring $R$, under what conditions and in what way, can one induce a topology on the $R$-points of a scheme $X$? For example, if $X$ is $P^n$ or $A^n$, one has natural topology on ...
- 671