# 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 the underlying intuitions.

The previous version of this question was asking for a "proof without words", which I realize now was a silly thing to ask for.

• Just to be clear, Weil gave two proofs; one uses the Jacobian, and the other $X\times X$. The second proof is given as an exercise, with hints, on page 368 of Hartshorne's AG. – Donu Arapura Oct 21 '16 at 21:59
• Milne's recent survey on the Riemann Hypothesis over finite fields contains sketches of both proofs , and much more (including detailed historical discussions) jmilne.org/math/xnotes/pRH.html – Denis Chaperon de Lauzières Oct 22 '16 at 6:42
• Write down a linear combination of H := C x {P_0}, V := {P_0} x C, \Delta [the diagonal], and \Gamma [the graph of Frobenius] as divisors in C \times C. Impose linear conditions on your coefficients to make sure your divisor has zero intersection with both H and V. The Hodge index theorem then tells you that the self intersection is nonpositive, so just calculate it and optimize the bound (the number of points appears as \Delta\cdot \Gamma). [Wait —- apparently this question was resurrected from 3 years ago by a random edit... I’ll leave this here anyway.] – alpoge Jun 12 '19 at 17:02

## 2 Answers

There is an REU paper from Chicago which seems to cover it https://math.uchicago.edu/~may/VIGRE/VIGRE2007/REUPapers/FINALFULL/Raskin.pdf

Alexander Beilinson gave four lectures on this topic (probably, during the same REU). Unfortunately, right now the link is not working, but maybe someone could provide another link?

Also, in Manin’s “Cubic Forms: Algebra, Geometry, Arithmetic” one can find beautiful and self-contained exposition of the Weil conjectures for del Pezzo surfaces (and some arithmetic applications).