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 the residue of the binomial coefficient $\binom{\frac{p-1}{2}}{\frac{p-1}{4}}$ modulo $p$ of smallest absolute value.

Therefore, Hasse's bound implies that $a \leq 2\sqrt{p}$, which for large $p$ is quite a strong statement about a number you might otherwise expect to be anything mod $p$.

My question is:

Is there a direct simple proof of this fact about $\binom{\frac{p-1}{2}}{\frac{p-1}{4}}$ being small mod $p$? Could one try to unwind the proof of Hasse's theorem or more generally a proof of the Riemann hypothesis for curves to get such a proof?

On a related sidenote: of course the $(\frac{p-1}{2})!$ occuring here is a primitive fourth root of unity. The other relevant term is $(\frac{p-1}{4})!$. This leaves me to wonder if there is any useful connection with the $p$-adic $\Gamma$ function evaluated at $\frac{1}{4}$, and hence with some kind of $p$-adic analogue of the Chowla-Selberg formula...? (I don't know anything about this, so I could be clutching at straws here.) If there is anything interesting to say about this part, I should perhaps make it a separate question.

notthe fact that it relates to the number of points on an elliptic curve (which you can prove by an elementary summation trick as Charles Matthews has mentioned.) ... $\endgroup$ – ndkrempel Dec 2 '10 at 5:31