All Questions
Tagged with weil-conjectures arithmetic-geometry
13 questions
6
votes
3
answers
2k
views
Direct proof of special case of Hasse's theorem for elliptic curves
Consider the elliptic curve $y^2 = x^3 + x$ over $\mathbb{F}_p$, where $p \equiv 1 \pmod 4$.
If memory serves correctly, the number of points (excluding the point at infinity) is $p - a$ where $a$ is ...
3
votes
0
answers
288
views
Is the weight-monodromy conjecture known for unramified representations?
Let $X$ be a smooth proper variety over a number field $K$, $v$ a place of $K$ lying over a prime number $p \neq \ell$, and $V := H^n(X_{\overline{K}};\mathbb{Q}_{\ell})$. Suppose $V$ is unramified at ...
2
votes
1
answer
432
views
Why geometric generic point (in abstract algebraic geometry) replace general points in the unit disk?
In section 4.1, chapter 4 of Pierre Deligne's paper La conjecture de Weil : I (french version, translation to English) he states:
On $\mathbb{C}$ Lefshietz local results are as follows. Let $X$ be a ...
6
votes
2
answers
664
views
Could the Weil zeroes of curves be evenly distributed?
If $X$ is a smooth, geometrically connected, projective curve of genus $g$
over $\mathbb{F}_q$, then the zeta function of $X$ is of the form $P(s)/(1 - s)(1 - qs)$, where $P(s)$ is a polynomial of ...
3
votes
0
answers
377
views
Meaning of "the" general fiber in the paper "La conjecture de Weil : I"
In section 4.1, chapter 4 of Pierre Deligne's paper La conjecture de Weil : I (french version, translation to English) he states:
Let $X$ be a non singular analytic space and purely of dimension $n+1$....
11
votes
1
answer
2k
views
Effective weight-monodromy conjecture
$\DeclareMathOperator\Gr{Gr}$Let $G$ be the absolute Galois group of a finite extension of $\mathbb{Q}_p$ with inertia subgroup $I$, and let $V$ be an $\ell$-adic representation of $G$. Grothendieck's ...
5
votes
1
answer
406
views
Which $p$-adic valuations of Weil numbers (that is, eigenvalues of Frobenius) are possible?
Let $C$ be a smooth projective curve over a finite field $\mathbb F_q$, $q$ is a power of the characteristic $p$. It is well-known that if $\alpha$ is an eigenvalue of Frobenius acting on $H^1_{et}(C,\...
16
votes
1
answer
641
views
What is the automorphic interpretation of the Weil conjectures over finite fields
I am very much a beginner in the theory of automorphic forms and I might (will?) make mistakes in what follows. Please correct me.
A loose interpretation of the Langland's philosophy is that to any ...
25
votes
8
answers
3k
views
Relatively concise English expositions of the proofs of the various Weil conjectures
Where can I find relatively concise (i.e. not excessively wordy and waxing poetic about history and intuitions and such, doesn't spend an eternity carefully developing various parts of the theory of ...
2
votes
0
answers
242
views
Is there any generalization of Weil conjecture for non-smooth variety?
Is there any generalization of Weil conjecture for any non-smooth geometric-connected variety? For example, for more general curve, or at least some numerical evindence (example)?
24
votes
1
answer
2k
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When is "independence of l" known?
My question is for which varieties over local fields is "independence of l" known for
etale cohomology. Say $X/{\mathbb Q}_p$ is a complete non-singular variety and $W_l$ is the (complex) Weil-...
6
votes
1
answer
1k
views
weight monodromy conjecture for curves?
Hi,
Is there a simple proof of the weight monodromy conjecture in the case of a curve over a mixed characteristic local field?
Thanks!
17
votes
1
answer
1k
views
On the Hasse-Weil L-function of $P^n$
So let us start with the "simplest" scheme over $Spec(\mathbf{Z})$ namely $X_0=Spec(\mathbf{Z})$. Then the (reciprocal) Weil zeta function of $X_0$ at a prime $p$ is given by $Z_p(T)=1-T$ (a ...