All Questions
2,494 questions
3
votes
2
answers
612
views
tamely branched cover over P^1
k is an algebraically closed field, X is a smooth, connected, projective curve over k. f: X-->P^1 is a finite morphism. Let t be a parameter of P^1, suppose f is etale outside t=0 and t=\infty, and ...
- 1,091
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
7
votes
1
answer
1k
views
Elementary questions in arithmetic geometry
In many theories there is a rough divide between elementary problems that can be solved with "one's hands", and "deep results that require powerful tools". For example, I am told that Hodge theory is ...
16
votes
2
answers
1k
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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
13
votes
1
answer
3k
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The monodromy-weight-, Ramanujan-, Langlands-landscape
The drawing on the last page of Yoshida's notes make me puzzle, perhaps you can help? It shows a "landscape" featuring the monodromy-weight conj., the general Ramanujan-conj., the Langlands ...
- 10.8k
2
votes
2
answers
2k
views
Reference of primitive root mod p
Can any body give me a reference of the result about primitive root mod p for a class of prime number p.
The result that I am looking for is something along this line:
$2$ is a primitive root mod $p$...
- 73
6
votes
2
answers
945
views
Notation/name for "Artin-Schreier roots"?
If x is an element of a field K and n is a positive integer, we have both a symbol and a name for a root of the polynomial t^n - x = 0: we denote it by x^{1/n} and call it an nth root of x.
Of course ...
- 65.4k
19
votes
3
answers
2k
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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
21
votes
1
answer
2k
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Two conjectures by Gabber on Brauer and Picard groups
In a paper I need to make reference to two conjectures by Gabber, from
Ofer Gabber, On purity for the Brauer group, in: Arithmetic Algebraic Geometry, MFO Report No. 37/2004, doi:10.14760/OWR-2004-37
...
- 30.6k
11
votes
5
answers
2k
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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
11
votes
2
answers
1k
views
Class groups of normal domains over finite fields
Let R be a local, normal domain of dimension 2. Suppose that R contains a finite field. I am interested in knowing when the class group of R is torsion. In characteristic 0, this is known to be ...
- 30.6k
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
12
votes
2
answers
1k
views
Weil Conjectures for nonprojective algebraic varieties
If we replace projective variety with algebraic variety in the statement of the Weil conjectures what happens? To me it seems the statement still makes sense. But is it still true?
- 1,462
5
votes
2
answers
1k
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Weil Conjectures for Grassmannians
To establish the Weil conjectures for $n$-dimensional projective space over a finite field is elementary. Does there exist a simple direct proof of the conjectures for finite field Grassmannians?
- 1,462
13
votes
2
answers
662
views
Evidence for $Q^{\operatorname{solv}}$ being pseudo-algebraically-closed
This is a follow-up to the following answer:
Solvable class field theory
in which it is stated as a "folklore" conjecture that the maximal solvable extension of Q is pseudo algebraically closed (...
- 671
16
votes
1
answer
4k
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Kapranov's analogies
I just wonder about Kapranov's "Analogies between Langlands Correspondence and topological QFT". I would like to read a more detailed exposition and how one turns that analogy into concrete ...
- 10.8k
10
votes
2
answers
393
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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
7
votes
5
answers
1k
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L-functions and random matrices
I am curious about the connection between properties of L-functions and random matrices, and about (if existent) function field versions of that. Do you know a survey or an other article where one ...
- 10.8k
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
6
votes
1
answer
1k
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Uniformization in algebraic/arithmetic geometry?
Jonah's question makes me wonder: What is with uniformization in algebraic/arithmetic geometry? E.g. this article by Faltings seems to be about that, the Shimura-Taniyama statement too, Mochizuki ...
- 10.8k
2
votes
4
answers
618
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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
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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
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1
answer
705
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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
56
votes
8
answers
8k
views
Questions about analogy between Spec Z and 3-manifolds
I'm not sure if the questions make sense:
Conc. primes as knots and Spec Z as 3-manifold - fits that to the Poincare conjecture? Topologists view 3-manifolds as Kirby-equivalence classes of framed ...
11
votes
4
answers
3k
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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
5
votes
1
answer
836
views
An inverse problem: Number fields attached to elliptic curves over Q
If I understand FC's remark under the post "Very strong multiplicity one for Hecke eigenforms," in the course of Faltings's proof of the Tate conjecture, Faltings proves the following statement: let E/...
- 7,072
48
votes
5
answers
15k
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Algebraically closed fields of positive characteristic
I'm taking introductory algebraic geometry this term, so a lot of the theorems we see in class start with "Let k be an algebraically closed field." One of the things that's annoyed me is that as far ...
- 12.6k
50
votes
6
answers
6k
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Intuition for the last step in Serre's proof of the three-squares theorem
Serre's A Course in Arithmetic gives essentially the following proof of the three-squares theorem, which says that an integer $a$ is the sum of three squares if and only if it is not of the form $4^m (...
- 118k
6
votes
1
answer
536
views
Finiteness of Obstruction to a Local-Global Principle
Say that a projective variety V over Q satisfies the local-global principle up to finite obstruction (#) if there are only finitely many isomorphism classes of projective varieties over Q that are not ...
- 7,072
9
votes
3
answers
2k
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Characterisation for separable extension of a field
Can someone verify this for me.. or tell me what reference shows me this... is this true:
Let $k$ be a field. Then a field extension $K$ of $k$ is separable over $k$ iff for any field extension $L \...
- 2,275
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
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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 ...
- 11.3k
6
votes
3
answers
601
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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
13
votes
5
answers
5k
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Examples and intuition for arithmetic schemes
How should a beginner learn about arithmetic schemes (interpret this as you wish, or as a regular scheme, proper and flat over Spec(Z))? What are the most important examples of such schemes? Good ...
6
votes
1
answer
777
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Existence of proper regular models for varieties over Q and other global fields
What is known about regular proper models for smooth projective varieties over Q? Results for other global fields would also be interesting, as well as general comments and suggested references for ...
- 5,637
8
votes
2
answers
8k
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What does "supersingular" mean?
Are supersingular primes and supersingular elliptic curves related?
(this was essentially a subquestion in my earlier question, but still looks sufficiently different to me to deserve a separate post)...
- 15.1k
7
votes
1
answer
718
views
Ways to characterize supersingular primes?
I've read the definition, and it basically says p is a supersingular prime iff
the fundamental domain of a group generated by \Gamma(p) and a matrix ((0, 1), (-p, 0)) is rational.
And there's a ...
- 15.1k
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
79
votes
12
answers
13k
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Is there a high-concept explanation for why characteristic 2 is special?
The structure of the multiplicative groups of $\mathbb{Z}/p\mathbb{Z}$ or of $\mathbb{Z}_p$ is the same for odd primes, but not for $2.$ Quadratic reciprocity has a uniform statement for odd primes, ...
- 118k
27
votes
7
answers
6k
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Etale covers of the affine line
In characteristic p there are nontrivial etale covers of the affine line, such as those obtained by adjoining solutions to x^2 + x + f(t) = 0 for f(t) in k[t]. Using an etale cohomology computation ...
- 52.7k
43
votes
1
answer
19k
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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
16
votes
1
answer
2k
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Reference for the `standard' Tate curve argument.
I'd like a reference (e.g. something published somewhere that I can cite in a paper) for the proof of the following:
Let $E$ be an elliptic curve over $\mathbb Q$ with minimal discriminant $\Delta$...
- 10.5k
20
votes
5
answers
4k
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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
37
votes
4
answers
12k
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Finite extension of fields with no primitive element
What is an example of a finite field extension which is not generated by a single element?
Background: A finite field extension E of F is generated by a primitive element if and only if there are a ...
- 24.1k