All Questions
28 questions
5
votes
1
answer
363
views
Unramified fppf cohomology
Let $F$ be a number field, $\mathcal{O}_F$ its ring of integers and $G$ an fppf $\mathcal{O}_F$-group scheme.
See the question Unramified Galois cohomology of number fields for unramified cohomology ...
- 265
3
votes
0
answers
118
views
A question on the averages of Kloosterman sums
Sorry to disturb. Recently, I encountered a puzzle on the sums involving two Kloosterman sums. That is,
For any $h, q_1,q_2\in \mathbb{N}$ with $(q_1,q_2)=1$ and $Q>1$, how two get a bound
$$\sum_{...
- 563
11
votes
1
answer
1k
views
Hodge conjecture as the equality of arithmetic and algebraic weights of motivic L-functions
Recently I became aware of the following statement given on page 13 of this paper. First, let us recall the following definitions:
Definition 4.1. Suppose $L(s)$ is an analytic $L$-function with ...
- 533
0
votes
0
answers
92
views
A question on the evaluations of certain three-dimensional hyper-Kloostermans
There is a basic question regrading the 3-dimensional hyper-Kloosterman sum which needs some help from the experts here:
For any integers $q,h \in \mathbb{N}$, how to estimate the sum:
$$\sideset{_{}^{...
- 563
2
votes
1
answer
154
views
On the estimate for the mixed 3-dimensional hyper-Kloosterman sum
There is a basic question regrading the mixed 3-dimensional hyper-Kloosterman sum:
For any positive integer $n$ not divisible by $p$, how to prove
$$\sideset{_{}^{}}{^{\ast}_{}}\sum _{x,y ,z\bmod p}
\...
- 563
12
votes
1
answer
942
views
Comparing singular cohomology with algebraic de Rham cohomology
Let $X$ be a smooth projective variety over a number field $K$. Then there are two cohomology groups we can attach to $X$: the algebraic de Rham cohomology group
$H^k_{\text{dR}}(X/K), $
which is a ...
- 1,949
4
votes
1
answer
264
views
Reference request: Long exact sequence in profinite Galois cohomology up through $H^2$
I'm looking for a reference of the following statement. Let $G$ be the Galois group of a Galois extension $L/K$, not necessarily finite. Let $A,B,C$ be groups with a continuous $G$-action, and let
$$1\...
- 7,527
2
votes
1
answer
117
views
Effective semi-group of a singular abelian surface
Let $A$ be a singular abelian surface over $\mathbb{C}$; that is, an abelian surface of maximal Picard rank $\rho(A)=4$. By Shioda-Mitani we know $A \cong E \times E'$ where $E,E'$ are isogenous ...
- 1,701
6
votes
2
answers
926
views
Motivating the coefficient field of $\ell$-adic cohomology
It was already known to Weil that a sufficiently reasonable cohomology theory for algebraic varieties over $\mathbb{F}_p$ would allow for a possible solution to the Weil conjectures.
It was also ...
- 317
20
votes
1
answer
902
views
Double Counting: Motivic Edition
One of the most important proof techniques in combinatorics is double counting: proving that both sides of an identity count elements of some set in two different ways. This question is an attempt at ...
- 85.6k
3
votes
0
answers
113
views
Cohomologies of $[V/GL_n]$ in characteristic $p$ for a representation $V$ of $GL_n$
Let $V$ be a representation of $G=GL_n$ (or more generally any reductive group $G$) over an algebraically closed field $\mathbb k$ of characteristic $p$. Let $[V/G]$ be the corresponding quotient ...
- 790
16
votes
0
answers
532
views
Are there smooth and proper schemes over $\mathbb Z$ whose cohomology is not of Tate type
Is there an example of smooth and proper scheme $X \to \mathrm{Spec}(\mathbb Z)$, and an integer $i$ such that $H^i(X, \mathbb Q)$ is not a Hodge structure of Tate type?
Alternatively: such that $H^...
- 171
7
votes
0
answers
483
views
independence of $\ell$ for $p$-adic cohomology of varieties over finite fields
Let $X/k$ be a smooth projective geometrically integral variety ($X = A$ an Abelian variety suffices) over $k = \mathbf{F}_q$ with absolute Galois group $\Gamma$, $\bar{X} = X \times_k \bar{k}$, $q = ...
user19475
4
votes
1
answer
620
views
finitness of syntomic/fppf cohomology with coefficients in a finite flat group scheme
Let $X/k$ be a smooth projective variety over a finite field of characteristic $p$ and $\mathscr{A}/X$ be an Abelian scheme.
Is then $H^1_\mathrm{SYN}(X,\mathscr{A}[p]) = H^1_\mathrm{fppf}(X,\mathscr{...
user19475
6
votes
2
answers
2k
views
Sketch of Weil's proof of the Riemann hypothesis for curves
I was wondering if anybody could provide a sketch of Weil's proof of the Riemann hypothesis for curves that uses the Jacobian $\text{Pic}^0(X)$ and a bit of the intersection theory on $X \times X$ and ...
user97565
2
votes
1
answer
466
views
Do $PGL_n$-torsors induce elements of the Brauer group
Let $K$ be a field and let $n\geq 2$. If $n=2$, then the set of $K$-isomorphism classes of $PGL_n$-torsors is in bijection with the $n$-torsion of the Brauer group of $K$.
Is this only for $n=2$?
Is ...
- 23
5
votes
1
answer
1k
views
"Role" of cohomology of coherent sheaves in SGA 4.5, étale cohomology
As the question title suggests, what is the role cohomology of coherent sheaves plays for SGA 4.5, étale cohomology? Why are they so important for the construction and establishing properties of étale ...
user97565
32
votes
2
answers
2k
views
Etale cohomology can not be computed by Cech
It can be proven that if in a quasicompact scheme $X$ any finite subset is contained in an affine open subset then for any sheaf $\mathcal{F}$ on $X$ its Cech cohomology $\hat{H_{et}^{\bullet}}(X,\...
- 7,377
13
votes
1
answer
869
views
Rationality of zeta function and Grothendieck-Lefschetz fixed point formula, cohomology can be computed as the de Rham cohomology
Trivial example. First, suppose $X$ is finite. Then we have a finite set $S := X(\overline{\mathbb{F}}_q)$ with an action of $\text{Fr}_q$. How can one explain why the rationality of the zeta function ...
user97565
3
votes
0
answers
408
views
Finding the number of rational points effectively
Consider $\# P$ and $\oplus P$.
There is a $\# P$-hard problem: to find number of rational solutions of a system of polynomial equations over $\mathbb{F}_2$. The corresponding $\oplus P$-hard ...
- 2,576
5
votes
0
answers
224
views
Comparison of sheaves of modular forms
Let $\pi:E\to X$ the universal generalized elliptic curve over the compactified modular curve, with zero section $e: X\to E$. Now consider the following two sheaves on $X$:
$e^*\Omega^1_{E/X}$ and $\...
- 845
13
votes
2
answers
1k
views
Galois cohomologies of an elliptic curve
I asked this question on Mathematics Stack Exchange but did not get any answer and I was suggested to post the question here.
I am studying basic theory of elliptic curves. I encountered Galois ...
user46846
4
votes
2
answers
740
views
Upper bound on Betti numbers of an intersection of hypersurfaces (or quadrics)
I have a problem I have been stuck with since several weeks now, and yet I believe it should be easy to specialists.
Let $k$ be an algebraically closed field, $m$ and $n$ two integers. Let $H_1,\dots,...
- 26k
4
votes
0
answers
188
views
Complexes of arithmetic $\mathcal{D}$-modules with Frobenius structure
This is a question about the category $F\text{-}D^b_\mathrm{coh}(\mathscr{D}^\dagger_{\mathscr{X},\mathbb{Q}})$ of complexes of arithmetic $\mathscr{D}$-modules with Frobenius structure on a smooth ...
- 1,838
3
votes
0
answers
593
views
"Extended" Weil Cohomology Theories
According to Wikipedia, a Weil cohomology theory is a functor from the category of smooth projective varieties over a field $k$, to graded algebras over a field $K$ of characteristic zero, together ...
- 1,838
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 ...
20
votes
1
answer
2k
views
De Rham cohomology of formal groups
Let $G$ be some (dimension $1$, to simplify) formal group over a characteristic $0$ field $K$. The law of $G$ is denoted by $\oplus$. If $w(X) \in K[[X]] dX$ is a differential form, let $F_w(X)$ be ...
- 6,924
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