3
votes
1answer
349 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!
16
votes
1answer
718 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 ...
5
votes
2answers
858 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 ...