All Questions
624 questions
5
votes
3
answers
739
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Smoothness of hyperplane sections
Suppose $X\subset \mathbb{P}^n$ is a smooth hypersurface defined over $\mathbb{Q}$. For a "generic" prime $p$, what can be said about the set of hyperplanes $H$ in $\mathbb{P}^n(\mathbb{F}_p)$ for ...
- 16.9k
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 ...
- 65.4k
6
votes
0
answers
456
views
On periods of algebraic integers modulo rational primes
I run, somewhat indirectly, into the following problem and I have no hints where to look in the literature in search for answers or clues.
Let $K$ be a number field, which we may assume Galois if it ...
- 65.4k
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 $p$...
- 65.4k
6
votes
2
answers
507
views
Concerning the dimension of a complex variety modulo a prime
Let V be a complex affine variety given as the vanishing set of a set of polynomials with integral coefficients. I have 3 questions.
1)
Under what assumption will the dimension of V over C remain ...
- 65.4k
7
votes
4
answers
736
views
Simply connected quasi-projective varieties in positive characteristic
I am looking for examples of non-projective (quasi-projective) varieties $X$ defined over a field of positive characteristic, which have trivial étale fundamental group.
It is well known that the ...
CommunityBot
- 1
9
votes
1
answer
763
views
Restriction theorems over finite fields
A short while ago, Dvir proved the Kakeya conjecture over finite fields. Does this have any implications for restriction theorems over finite fields? I am aware only of implications going in the ...
- 65.4k
25
votes
3
answers
2k
views
product of all F_p, p prime
Let $R$ be the ring $$R = \prod_{p\ \text{prime}} \mathbb{F}_p$$ where $\mathbb{F}_p$ is the field having $p$ elements.
Is it true that $R$ has a quotient by a maximal ideal which is a field of ...
- 65.4k
10
votes
3
answers
2k
views
Is $Sym^n (V^*) \cong Sym^n (V)^\ast$ naturally in positive characteristic?
Background/motivation
It is a classical fact that we have a natural isomorphism $Sym^n (V^*) \cong Sym^n (V) ^\ast$ for vector spaces $V$ over a field $k$ of characteristic 0. One way to see this is ...
- 65.4k
8
votes
2
answers
8k
views
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)...
CommunityBot
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13
votes
0
answers
943
views
Beilinson-Bernstein localization in positive characteristic
This is a follow-up to this question; in particular, I'm wondering if anyone can expand upon the interesting answers given by Kevin McGerty and David Ben-Zvi there. (In particular, in this question I'...
CommunityBot
- 1
15
votes
1
answer
4k
views
Frobenius Descent
Let $S$ be a scheme of positive characteristic $p$ and $X$ a smooth $S$-scheme. Let $F:X\rightarrow X^{(p)}$ denote the relative Frobenius. A result by Cartier (often called Cartier descent or ...
- 57.6k
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?
- 65.4k
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:...
- 65.4k
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 ...
- 65.4k
16
votes
1
answer
1k
views
Coarse moduli spaces over Z and F_p
I would like to know to what extent it is possible to compare fibers over $\mathbb{F}_p$ of coarse moduli spaces over $\mathbb{Z}$, and coarse moduli spaces over $\mathbb{F}_p$. I ask a more precise ...
- 65.4k
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 ...
- 65.4k
9
votes
1
answer
566
views
algorithm for calculating the Chow groups of a variety over a finite field
Is there an algorithm for calculating the Chow groups of a variety over a finite field?
It is know that $H^{2i,i}_\mathrm{mot}(X,\mathbf{Z}) = CH^i(X)$. In how many cases does this help us?
CommunityBot
- 1
4
votes
1
answer
412
views
F_q-structures on schemes
Let $k|\mathbb{F}_q$ be a field extension. An $\mathbb{F}_q$-structure on a $k$-algebra $A$ is an $\mathbb{F}_q$-subalgebra $A _0$ of $A$ such that $A _0 \otimes _{\mathbb{F}_q} k \cong A$ via the ...
- 65.4k
11
votes
1
answer
2k
views
Are automorphism groups of hypersurfaces reduced ?
In the following article : "H. Matsumura, P. Monsky, On the automorphisms of hypersurfaces, J. Math. Kyoto Univ. 3 (1964) 347-361", it is shown that in finite characteristic, automorphism groups of ...
- 23.8k
10
votes
2
answers
1k
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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 ...
CommunityBot
- 1
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 ...
CommunityBot
- 1
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, ...
CommunityBot
- 1
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? ...
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