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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 views

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 ...