# Weil's Riemann Hypothesis for dummies?

Weil's Riemann Hypothesis is a deep result that I don't fully understand, but it has understandable corollaries which interest me. For example:

(a) For any projective curve $X$ satisfying certain conditions, the number $N$ of points in $X$ with coordinates in $\mathrm{GF}(q)$ satisfies $|N-(q+1)|\leq\mathrm{const}\cdot\sqrt{q}$. (The deviation is $0$ when $X$ is a projective line.)

(b) For any nontrivial multiplicative character $\chi$ on $\mathrm{GF}(q)$ and any polynomial $f$ of degree $n$ satisfying certain conditions, we have

$$\bigg|\sum_{x\in\mathrm{GF}(q)}\chi(f(x))\bigg|\leq(n-1)\sqrt{q}.$$

Questions:

1. Is there a reference (legible to an English-speaking non-expert in the field) which gives the rigorous statements of these corollaries? In particular, I would like conditions which one can verify without a background in algebraic geometry.

2. Are there other corollaries of Weil's Riemann Hypothesis which are also widely understandable? EDIT: I'm mostly interested in the Riemann Hypothesis, but I'm also happy to learn understandable consequences of the other Weil conjectures and related results.

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Statements are given in Wolfgang Schmidt's book, Equations over Finite Fields, Springer Lecture Notes in Mathematics 536. (a) is on page 2, (b) is on page 43. –  Gerry Myerson Jul 21 '14 at 3:05
Are you interested specifically in consequences of the Riemann hypothesis for curves, or in any aspects of the Weil conjectures for curves (of which RH is the hardest part, but also includes rationality and a functional equation for the zeta-function of the curve)? What about for higher-dimensional varieties? I ask because one of the answers below is an application of the rationality of the zeta-function for higher-dimensional varieties, not involving RH for that zeta-function. Is that a kind of consequence you're interested in too? It'd be good to clarify this in your question. –  KConrad Jul 21 '14 at 14:35
You might like to look into Stichtenoth's book "Algebraic function fields and codes", which gives a completely self-contained exposition of the Riemann hypothesis for curves and all material leading up to it. It is completely algebraic and non-geometric, but it does get to Weil's Riemann hypothesis very quickly starting from nothing. –  Michael Zieve Jul 22 '14 at 16:00

Here are the statements from Schmidt's book (as pointed to in my comment).

(a) Suppose $f(x,y)$ is a polynomial of total degree $d$, with coefficients in the field of $q$ elements and with $N$ zeros with coordinates in that field. Suppose $f(x,y)$ is absolutely irreducible, that is, irreducible not only over the field of $q$ elements, but also over every algebraic extension thereof. Then $$|N-q|\le2g\sqrt q+c_1(d)$$ where $g$ is the genus of the curve $f(x,y)=0$.

I am not up to explaining "genus" without algebraic geometry, but it is known that $g\le(d-1)(d-2)/2$, so if you are willing to settle for $$|N-q|\le(d-1)(d-2)\sqrt q+c_1(d)$$ then I think you have what you are after.

(b) Let $\chi$ be a multiplicative character of order $d>1$. Suppose that $f(x)$, a polynomial in one variable over the field of $q$ elements, has $m$ distinct zeros, and is not a $d$th power. Then $$\Bigl|\sum_{x\in{\bf F}_q}\chi(f(x))\Bigr|\le(m-1)\sqrt q$$

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Thanks! Do we know what $c_1(d)$ is, or do we have a bound in terms of $d$? –  Dustin G. Mixon Jul 21 '14 at 11:44
@Dustin: yes, see my answer. –  Michael Zieve Jul 22 '14 at 15:52

For your second question, here is a corollary of the Weil conjectures that can be widely understood and verified:

Let $\{f_i(T_1,...,T_n)\}_i$ be a "good" system of homogeneous polynomials ("good" means they define a smooth connected projective variety) defined over a finite field $\mathbb{F}_q$, and denote by $N_s$ the number of their solutions in $\mathbb{F}_{q^s}$. Then the rationality part of the Weil conjectures implies that all the $\{N_s\}_s$ are determined by the first few finitely many $N_s$ (for example, for a curve of genus $g$, the $N_1,...,N_{2g}$ determines all $N_s$).

This can be seen by expanding the rational form of zeta function into series and compare the coefficients by linear recursion relation.

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Don’t you mean the first $g$ numbers rather than the first $2g$? –  Lubin Jul 21 '14 at 16:01
@Lubin: it looks like only the rationality (and degree of numerator) is being used here, not also the functional equation. –  KConrad Jul 22 '14 at 12:13
@KConrad, ah yes, right you are. –  Lubin Jul 22 '14 at 17:30

This is an explicit version of part (a) of Gerry Myerson's answer. If $f(x,y)$ is an absolutely irreducible polynomial in $\mathbf{F}_q[x,y]$ of total degree $d>0$, and $N$ is the number of zeroes of $f(x,y)$ in $\mathbf{F}_q\times\mathbf{F}_q$, then $$q+1-(d-1)(d-2)\sqrt{q}-d\le N\le q+1+(d-1)(d-2)\sqrt{q}.$$ Likewise, if $f(x,y,z)$ is an absolutely irreducible homogeneous polynomial in $\mathbf{F}_q[x,y,z]$ of total degree $d>0$, and $N$ is the number of zeroes of $f(x,y,z)$ in $\mathbf{P}^2(\mathbf{F}_q)$, then $$\lvert N-(q+1)\rvert\le (d-1)(d-2)\sqrt{q}.$$ These results comprise Corollary 2 in the paper "The number of points on a singular curve over a finite field" by Leep and Yeomans (Arch. Math. 63 (1994), 420-426).

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