29
votes
Algebraic geometry over the complex numbers, and beyond
Algebraic geometry began over the field of reals. What Apollonius of Perga did would be certainly qualified today as algebraic geometry: he classified real plane curves of second order and studied ...
- 91.8k
18
votes
Accepted
Deligne's theorem on exponential sums
Yes, smoothness is equivalent to the gradient being nonzero for every $x \in \overline{\mathbb F}_q^n \setminus \{0\}$.
I would define smoothness of the hypersurface defined by $Q_d$ as the condition ...
- 149k
12
votes
Weil conjectures for higher dimensional cycles?
As Jason Starr notes in his comment, the series I ask about cannot be rational in general. There's more to the story though. I recently stumbled across several papers where Elizondo and Kimura study ...
- 954
11
votes
Effective weight-monodromy conjecture
You can bound the filtration length (assuming the WM conj) using the weight spectral sequence of Rapoport--Zink. This is a sp seq converging to $H^*(X_{\overline{k}})$; and if the WM conj holds, then ...
- 37k
10
votes
Accepted
Zeta function of $X = \mathbb{F}_p \mathbb{P}^1$
Deligne proved the hard Lefschetz theorem as a consequence of the Weil conjectures, not vice versa.
The RH of the Weil conjectures is a statement about the Weil zeta function. For the projective line, ...
- 149k
9
votes
Accepted
The connection between the Weil conjectures and Ramanujan's conjecture
Basically, the coefficients of an holomorphic cusp form are related to the number of points on a certain smooth projective variety over $\mathbb{F}_p$, and the Weil-Riemann hypothesis gives the ...
- 17.6k
9
votes
Could the Weil zeroes of curves be evenly distributed?
For $q$ odd, and any natural number $n$, let $\alpha$ be an element of $\mathbb F_{q^n}$ of order $q^n-1$ and let $ \sum_{i=0}^n a_i x^i$ be the minimal polynomial of $\alpha$.
Then I think the ...
- 149k
9
votes
Accepted
Could the Weil zeroes of curves be evenly distributed?
If $q$ is a prime for which $2$ is a primitive root, then I claim that the Frobenius eigenvalues of the curve $y^2-y = x^q$ over $\mathbb{F}_2$ have spacing exactly $\tfrac{2 \pi}{q-1}$. This, if ...
- 156k
8
votes
Asymptotic estimate of the number of points of variety over finite field
You want $X$ to be geometrically irreducible here.
This is a theorem of Lang and Weil, proven well before the Weil conjectures. It relies only on Weil's proof of the Riemann hypothesis for curves.
The ...
- 149k
7
votes
Direct proof of special case of Hasse's theorem for elliptic curves
Hasse bound for elliptic curves of the form $y^2=x^3+b$ can be proven without algebraic geometry. It was done by Schrutka in the article Ein Beweis für die Zerlegbarkeit der Primzahlen von der Form $...
- 12.3k
7
votes
What is the automorphic interpretation of the Weil conjectures over finite fields
This is a brief answer; possibly others have different opinions about this.
Question 1: The Langlands conjectures gives a correspondence between Galois representations and automorphic forms. So a ...
- 21.4k
6
votes
Are degrees of polynomials in Weil's zeta function equal/bounded to/by dimensions of SOME cohomologies in non-smooth or non-projective case?
The number of points always has an interpretation as a supertrace of Frobenius on the compactly supported cohomology, by the Grothendieck trace formula. The issue is that it can be quite difficult to ...
- 44.7k
6
votes
Accepted
Cancellation in a particular sum
We can actually get an explicit formula for the whole sum (I will assume $p \ne 2,3$ throughout). We start with the Salié sum:
$$\sum_{d=1}^{p-1}\bigg(\frac{d}{p}\bigg)w^{-d-(a^2+b^2+c^2)/d} = \bigg(\...
- 8,688
6
votes
Accepted
Is the Hilbert–Pólya intuition vindicated in the function field case?
In the function field setting, the most natural spectral explanation for the Riemann hypothesis might be expressing the eigenvalues of Frobenius as the eigenvalues of a unitary operator on a finite-...
- 149k
5
votes
Accepted
Purity of Frobenius on cohomology of a projective variety over $\mathbb F_q$ with isolated singularities
Following Piotr Achinger's suggestion, if we let $z$ be the inclusion of a smooth hyperplane $H$ and $j$ its affine open complement $U$, the distinguished triangle $z_* z^! \mathbb Q_\ell \to \mathbb ...
- 149k
5
votes
Accepted
Which $p$-adic valuations of Weil numbers (that is, eigenvalues of Frobenius) are possible?
Every value $c \in \big[0,\tfrac{1}{2}\big] \cap \mathbf Q$ can occur as the smallest slope of an abelian variety over $\mathbf F_q$; see the corollary below.
What Honda actually proves [Hon68] (see [...
- 28.7k
5
votes
Accepted
Geometric (or at least non-cohomological) proof of Lefschetz trace formula for curves
This was proved by Weil using his intersection theory. For a modern exposition, see 11.2 of Milne, J. S. Jacobian varieties. Arithmetic geometry (Storrs, Conn., 1984), 167--212, Springer, New York, ...
- 66
5
votes
Relatively concise English expositions of the proofs of the various Weil conjectures
Sorry for a bit more of self-promotion but I want to share it somewhere (when I was searching for such an article, it was not available). Maybe I will try to publish this as an expository paper (not ...
- 131
4
votes
Relatively concise English expositions of the proofs of the various Weil conjectures
Contributing here for completeness. I have translated Deligne's original paper into English (it is only 35 pages) - https://drive.google.com/file/d/1wxVbCrm_0D4jInvG1pd1vhnI7YnvSk0S/view?usp=sharing ...
- 131
4
votes
Why geometric generic point (in abstract algebraic geometry) replace general points in the unit disk?
If you want a canonical description of the homotopy type of the general fiber, you can take the homotopy type of the space $X\times_{\mathbb{D}}\mathbb{H}$ where $\mathbb{H}:=\{z\in\mathbb{C}\mid \Im ...
- 16.5k
4
votes
Purity of vanishing cycle for proper scheme over DVR with smooth generic fiber
Here's a typical example of what can go wrong in high dimension:
Let $\mathcal E$ be an elliptic curve degenerating to a nodal cubic. Let $X = \mathcal E^n$.
Then $H^* ( X_{\eta}, \mathbb Q_\ell ) = ...
- 149k
2
votes
Accepted
Pure varieties which are neither smooth nor projective
I don't know about interesting, but here are a few easy examples. Start with a cone $V\subset \mathbb{A}^n_k$. This is easily seen to be $\mathbb{A}^1$-homotopy equivalent to a point, so $H^i(\...
- 35.2k
2
votes
Accepted
Computing weights of $\bar{\mathbb{Q}}_l(1)$ from the definition
First, there is no point in including $X$ in the definition. We're interested in the stalk at $x$. We can also view $x$ as a geometric point of $X_0$. Pulling back from $X_0$ to $X$ and then taking ...
- 149k
1
vote
The numbers of isomorphism classes of abelian variety over finite fields
I did a little search work on this problem, and it seems that I found the following article.
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The Lefschetz trace formula for algebraic stacks
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The result is that: If $\mathcal{X}$ is an algebraic ...
- 547
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