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:


*

*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.

*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.
 A: 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).
A: 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.
A: 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 $$\Bigl|\sum_{x\in{\bf F}_q}\chi(f(x))\Bigr|\le(m-1)\sqrt q$$
