# A strange logical implication in algebraic geometry

So there's an old theorem of Lang and Weil showing that the Riemann hypothesis for curves over finite fields implies a kind of quasi-riemann hypothesis for surfaces over finite fields.

I am wondering: is it possible to go in the other direction? Namely, if we knew the Riemann hypothesis for surfaces which are not products of curves (so no trivial argument with Kunneth applies), would it be possible to deduce the RH for curves?

Since this is such an odd question, let me lay my cards on the table. I am looking for possible motivation for a number field analogue; namely, if the Riemann hypothesis for $GL_2$ L-functions implies anything about the Riemann hypothesis for $GL_1$.

Edit: If $X\to C$ is an elliptic surface which is not split, there is an isomorphism $H^1(X)=H^1(C)$, which answers my original question in a lot of cases. So let me be even more persnickity and refine my question to: if we knew RH for Frobenius acting on $H^2$ of non-split surfaces over finite fields, could we show this implies RH for curves?

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I linkified the paper. – David Speyer Dec 24 '09 at 19:13
Thanks a bunch! – David Hansen Dec 24 '09 at 19:14
Did that old theorem inspire the song, It's a Lang Weil to Tipperary? – Gerry Myerson Nov 15 '10 at 11:33

Lang-Weil applies to varieties of any dimension, not just surfaces. If you allow varieties of any dimension, then just use the Jacobian. If you want to stick to surfaces, then you can sort of cheat and use a surface inside the Jacobian, either by intersecting with a linear space of suitable dimension in an embedding of the Jacobian or simply use the surface parametrizing effective divisors of degree two. The latter is birational to $C^2/S_2$, so it's almost like using $C^2$. These constructions won't make sense in the number field case, unfortunately.