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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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    $\begingroup$ 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. $\endgroup$ Commented Jul 21, 2014 at 3:05
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    $\begingroup$ 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. $\endgroup$
    – KConrad
    Commented Jul 21, 2014 at 14:35
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    $\begingroup$ 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. $\endgroup$ Commented Jul 22, 2014 at 16:00

3 Answers 3

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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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    $\begingroup$ Thanks! Do we know what $c_1(d)$ is, or do we have a bound in terms of $d$? $\endgroup$ Commented Jul 21, 2014 at 11:44
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    $\begingroup$ @Dustin: yes, see my answer. $\endgroup$ Commented Jul 22, 2014 at 15:52
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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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  • $\begingroup$ This is just what I am looking for. So anyone can understand the precise meaning even without knowing much about algebraic geometry. I also find the paper by Leep and Yeomans cited by you. Indeed they proved this elementary version in terms of polynomials in the paper. That's perfect for me although I didn't ask the question. Thanks a lot! $\endgroup$
    – Joy-Joy
    Commented May 30, 2016 at 23:15
  • $\begingroup$ @Joy-Joy: I'm happy this was useful for you, thanks for the feedback. You might want to be aware that the literature also contains many incorrect versions of these results; so you should proceed with caution when using other versions. Some of these mistakes are spelled out in the remark after Lemma 2.2 in my paper with Peter Mueller "Low-degree planar monomials in characteristic two" (J.Algebraic Combinatorics 42 (2015), 695-699). $\endgroup$ Commented Jun 13, 2016 at 0:47
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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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  • $\begingroup$ Don’t you mean the first $g$ numbers rather than the first $2g$? $\endgroup$
    – Lubin
    Commented Jul 21, 2014 at 16:01
  • $\begingroup$ @Lubin: it looks like only the rationality (and degree of numerator) is being used here, not also the functional equation. $\endgroup$
    – KConrad
    Commented Jul 22, 2014 at 12:13
  • $\begingroup$ @KConrad, ah yes, right you are. $\endgroup$
    – Lubin
    Commented Jul 22, 2014 at 17:30

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