--- date: 2021-10-27 19:35 modification date: Tuesday 29th March 2022 17:11:48 title: "2021-04-28_Weil_Conjectures_4" aliases: [2021-04-28_Weil_Conjectures_4] --- Last modified date: <%+ tp.file.last_modified_date() %> --- - Tags: - #projects/notes/reading #projects/my-talks #arithmetic-geometry - Refs: - The Zeta book #resources/books - #resources/videos - [Explanation of exponential appearing](https://mathoverflow.net/questions/325186/motivation-for-zeta-function-of-an-algebraic-variety) - [https://arxiv.org/pdf/1807.10812.pdf](https://arxiv.org/pdf/1807.10812.pdf) - [http://www.math.canterbury.ac.nz/~j.booher/expos/weil_conjectures.pdf](http://www.math.canterbury.ac.nz/~j.booher/expos/weil_conjectures.pdf) - [Weil's Paper](https://www.ams.org/journals/bull/1949-55-05/S0002-9904-1949-09219-4/S0002-9904-1949-09219-4.pdf) - Links: - [Weil Conjectures](Weil%20Conjectures.md) - [Weil Conjectures Talks](Weil%20Conjectures%20Talks.md) --- # Notes from Daniel's Office Hours 0. Definition of Zeta functions 0. Statement of the conjectures 2. Easy examples: $\PP^n_{\kk},\Gr_{\kk}(k, n) = \GL(n, \kk) / P$ the stabilizer of an $\kk\dash$point in $\CC^n, \FF_{p^n}$. 3. Medium example: $E/\kk$ an elliptic curve. 4. Work out a harder example as in Weil ## Definition of Zeta Function Fix $q$ a prime and $\FF \definedas \FF_q$ the finite field with $q$ elements, along with its unique degree $n$ extensions $$ \FF_n \definedas \FF_{q^n} = \theset{x\in \bar \FF_p \suchthat x^{q^n} - x = 0} \quad \forall~ n\in \ZZ^{\geq 2} $$ **Definition** Let $$ J = \gens{f_1, \cdots, f_M} \normal k[x_0, \cdots, x_n] $$ be an ideal, then a *projective algebraic* variety $X\subset \PP^N_\FF$ can be given by $$ X = V(J) = \theset{\vector x \in \PP^\infty_\FF \suchthat f_1(\vector x) = \cdots = f_M(\vector x) = \vector 0} $$ where an ideal generated by *homogeneous* polynomials in $n+1$ variables, i.e. there is some fixed $d\in \ZZ^{\geq 1}$ such that \[ f(\vector x) = \sum_{\substack{\vector I = (i_1, \cdots, i_n) \\ \sum_j i_j = d}} \alpha_{\vector I} \cdot x_0^{i_1}\cdots x_n^{i_n} \qtext{ and } f(\lambda \cdot \vector x) = \lambda^d f(\vector x) .\] > For the experts: we can take a reduced (possibly reducible) scheme of finite type over a field $\FF_p$. > We will be thinking of $K\dash$valued points for $K/\FF_p$ algebraic extensions. > From the audience: what condition do we need to put on such a scheme to guarantee an embedding into $\PP^\infty$? Examples: - Dimension 1: [Curves](Curves.md) - Dimension 2: Surfaces - Codimension 1: Hypersurfaces Fix $X/\FF \subset \PP$ an $N\dash$dimensional projective algebraic variety, and say it's cut out by the equations $f_1, \cdots f_M \in \FF[x_0, \cdots, x_n]$. Note that it then has points in any finite extension $L/K$. **Definition** Define the *local zeta function* of $X$ the following formal power series: \[ Z_X(z) &= \exp\qty{ \sum_{n=1}^\infty \alpha_n {z^n \over n} } \in \QQ[z](z) \qtext{where} \alpha_n \definedas \size X(\FF_n) .\] Concretely, for $X\subset \PP^M$ a variety cut out by $\theset{f_i} \subset \FF[x_0, \cdots, x_M]$ we are measuring the sizes of the sets \[ \alpha_n \definedas \size \theset{\vector x \in \PP^M_{\FF_{q^n}} \suchthat f_i(\vector x) = \vector 0 ~\forall i } .\] > Compare to the Poincare polynomials: $P_{\RP^m}(x) = 1 + x + x^2 + \cdots + x^m$ and $P_{\CP^m}(x) = 1 + x^2 + \cdots + x^{2m}$ ## Statement of Weil Conjectures (Weil 1949) Let $X$ be a smooth projective variety of dimension $N$ over $\FF_{q}$ for $q$ a prime, let $Z_X(z)$ be its zeta function, and define $\zeta_X(s) = Z_X(q^{-s})$. 1. (Rationality) $Z_X(z)$ is a rational function: \[ Z_X(z) &= {p_1(z) \cdot p_3(z) \cdots p_{2N-1}(z) \over p_0(z) \cdot p_2(z) \cdots p_{2N}(z)} \in \QQ(z),\quad\text{i.e. }\quad p_i(z) \in \ZZ[z] \\ \\ P_0(z) &= 1-z \\ P_{2N}(z) &= 1 - q^N z \\ P_j(z) &= \prod_{j=1}^{\beta_i} \qty{1 - a_{j, k} z} \qtext{for some reciprocal roots} a_{j, k} \in \CC \] where we've factored each $P_i$ using its reciprocal roots $a_{ij}$. In particular, this implies the existence of a meromorphic continuation of the associated function $\zeta_X(s)$, which a priori only converges for $\Re(s)\gg 0$. This also implies that for $n$ large enough, $N_n$ satisfies a linear recurrence relation. 2. (Functional Equation and Poincare Duality) Let $\chi(X)$ be the Euler characteristic of $X$, i.e. the self-intersection number of the diagonal embedding $\Delta \injects X\cross X$; then $Z_X(z)$ satisfies the following *functional equation*: \[ Z_X\qty{1 \over q^N z} = \pm \qty{q^{N \over 2} z}^{\chi(X)} ~~Z_X(z) .\] Equivalently, \[ \zeta_X(N-s) = \pm \qty{q^{\frac N 2 - s}}^{\chi(X)} ~\zeta_X(s) \\ .\] > Note that when $N=1$, e.g. for a curve, this relates $\zeta_X(s)$ to $\zeta_X(1-s)$. Equivalently, there is an involutive map on the (reciprocal) roots \[ z &\iff {q^N \over z} \\ \alpha_{j, k} &\iff \alpha_{2N-j, k} \] which sends roots of $p_j$ to roots of $p_{2N-j}$. 3. ([Riemann Hypothesis](Riemann%20Hypothesis.md)) The reciprocal roots $a_{j,k}$ are *algebraic* integers (roots of some monic $p\in \ZZ[x]$) which satisfy \[ \abs{a_{j,k}}_\CC = q^{j \over 2} \quad\quad \forall 1 \leq j \leq 2N-1,~ \forall k .\] 4. (Betti Numbers) If $X$ is a "good reduction mod $q$" of a nonsingular projective variety $\tilde X$ in characteristic zero, then the $\beta_i = \deg p_i(z)$ are the Betti numbers of the topological space $\tilde X(\CC)$. Why is (3) called the "Riemann Hypothesis"? We can use the facts that a. $\abs{\exp\qty{z}} = \exp\qty{\Re(z)}$ and b. $a^z \definedas \exp\qty{z \Log(a)}$, to replace the polynomials $P_i$ with \[ L_j(s) \definedas \zeta_X(q^{-s}) = \prod_{k=1}^{\beta_j} \qty{1 - \alpha_{j, k} q^{-s}} .\] Now consider the roots of $L_j(s)$: we have \[ L_j(s_0) &= 0 \\ \iff q^{-s_0} &= {1 \over \alpha_{j, k}} \qtext{for some} k \\ \implies \abs{q^{-s_0}} &= \abs{1 \over \alpha_{j, k}} \quad\quad \stackrel{\text{\tiny by assumption}}{=} q^{ -{j \over 2}} \\ \implies q^{-\frac j 2} \stackrel{(a)}= \exp\qty{- \frac j 2 \cdot \Log(q)} &= \abs{ \exp\qty{-s_0 \cdot \Log(q)} } \\ &\stackrel{(b)}= \abs{ \exp\qty{-\qty{\Re(s_0) + i\cdot \Im(s_0)} \cdot \Log(q)} } \\ &\stackrel{(a)}= \exp\qty{-\qty{\Re(s_0)} \cdot \Log(q)} \\ \implies - \frac j 2 \cdot \Log(q) &= -\Re(s_0) \cdot \Log(q) \qtext{by injectivity} \\ \implies \Re(s_0) = \frac j 2 .\] Roughly speaking, realizing that we would need to apply a logarithm (a conformal map) to send the $\alpha_{j, k}$ to zeros of the $L_j$, this says that the zeros all must lie on the "critical lines" $\frac{j}{2}$. In particular, the zeros of $L_1$ have real part $\frac 1 2$, analogous to the classical Riemann hypothesis. > Moral: the Diophantine properties of a variety's zeta function are governed by its (algebraic) topology. > Conversely, the analytic properties of encode a lot of geometric/topological/algebraic information. > Plug for Langland's: it similarly asks for every $L$ function arising from an automorphic representation that (essentially) satisfy Weil 2 and 3. Historical note - Desire for a "cohomology theory of varieties" drove 25 years of progress in AG Remarks: - Resolved for varieties over $\FF_q$ - On $L_X$: - Conjectured for smooth varieties over $\QQ$ (rationality $\sim$ analytically continues to a meromorphic function, some functional equation), little is known. - Resolved for [Projects/2022 Advanced Qual Projects/Elliptic Curves/Elliptic Curves](Projects/2022%20Advanced%20Qual%20Projects/Elliptic%20Curves/Elliptic%20Curves.md) (Taylor-Wiles c/o the [Taniyama-Shimura conjecture](Taniyama-Shimura%20conjecture)), implies $L_X$ is an $L$ function coming from a [modular form](modular%20form.md). ### Aside: Why call it a Zeta function? Knowing the zeta function of a point, we can now make a precise analogy. Suppose we have an algebraic variety cut out by equations: \[ \AA_\ZZ^n \supseteq X = V(\gens{f_1, \cdots, f_d}) \qtext{where} f_i \in \ZZ[x_0, \cdots, x_{n-1}] .\] Then for every prime $q$, we can reduce the equations mod $p$ and consider $$ \AA_{\FF_q}^n \supseteq X_q \definedas V(\gens{f_1 \mod q, \cdots ,f_d \mod q}) \qtext{where} f_1 \mod q \in \FF_q[x_0, \cdots, x_{n-1}] $$ Then define the [Unsorted/Hasse-Weil L function](Unsorted/Hasse-Weil%20L%20function.md) \[ L_X(s) = \prod_{p\text{ prime}} \zeta_{X_p}\qty{p^{-s}} .\] Take $X = \spec \QQ$ and $X_p = \spec \FF_p$, which is a single point since $\FF_p$ is a field. The previous example shows that \[ \zeta_{X_p}(z) = {1 \over 1-z} ,\] We then find that \[ L_X(s) &= \prod_{p\text{ prime}} \zeta_{X_p}(p^{-s}) \\ &= \prod_{p\text{ prime}} \qty{ 1 \over 1 - p^{-s}} \\ &= \zeta(s) ,\] which is the [Euler product](Euler%20product) expansion of the classical [Unsorted/Riemann Zeta](Unsorted/Riemann%20Zeta.md) function. Moreover, it is a theorem (difficult, not proved here!) that for any variety $X/\FF_p$, we have \[ \zeta_X(t) = \prod_{x\in X_{\text{cl}}} \qty{1 \over 1 - t^{\deg(x)}} \quad \overset{t = p^{-s}}{\implies} \quad \zeta_X(s) = \prod_{x\in X_{\text{cl}}} \qty{1 \over 1 - \qty{p^{\deg(x)}}^{-s} } ,\] which we can think of as attaching a "weight" to each closed point, $\abs{x} \definedas p^{\deg(x)}$, and the usual Riemann Zeta corresponds to assigning a weight of 1 to each point. > Note that this immediately implies that $\zeta_X(t) \in \ZZ[[t]]$ is a *rational* function. Recall the [Riemann zeta function](Unsorted/Riemann%20zeta%20function.md) is given by \[ \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \prod_{p\text{ prime}} {1 \over 1 - p^{-s}} .\] After modifying $\zeta$ to make it symmetric about $\Re(s) = \frac 1 2$ and eliminate the trivial zeros at $-2\ZZ$ to obtain $\hat \zeta(s)$, there are three relevant properties - "Rationality": $\hat \zeta(s)$ has a meromorphic continuation to $\CC$ with simple poles at $s=0, 1$. - "Functional equation": $\hat \zeta(1-s) = \hat \zeta(s)$ - "Riemann Hypothesis": The only zeros of $\hat \zeta$ have $\Re(s) = \frac 1 2$. ### More Examples #### Example (Affine Space): Set $X = \AA^m/\FF$, affine $m\dash$space over $\FF$, so we can just repeat with now $m$ coordinates $$ \AA^1(\FF_n) = \theset{\vector x = [x_1, \cdots, x_m] \suchthat x_i \in \FF_n} $$ Counting yields \[ X(\FF) &= q^m \\ X(\FF_2) &= (q^2)^m \\ &\vdots \\ X(\FF_n) &= (q^n)^m .\] Thus \[ \zeta_X(z) = \exp\qty{\sum_{n=1}^\infty {q^{nm} \over n} z^n } = \frac 1 {1 - q^m z} .\] #### Example (Projective Line): $X = \PP^1/\FF$ the projective line over $\FF$, then we can write use some geometry to write $$ \PP^1_\FF = \AA^1_\FF \disjoint \theset{\infty} $$ as the affine line with a point added at infinity. We can then count by enumerating coordinates: \[ \PP^1(\FF_n) &= \theset{[x_1, x_2] \suchthat x_1, x_2 \neq 0 \in \FF_n}/\sim \\ &= \theset{[x_1, 1] \suchthat x_1 \in \FF_n} \disjoint \theset{[1, 0]} .\] Thus \[ X(\FF) &= q + 1\\ X(\FF_2) &= q^2 + 1 \\ &\vdots \\ X(\FF_n) &= q^n + 1\\ .\] Thus \[ \zeta_X(z) = {1 \over (1-z)(1-qz)} \\ .\] #### Example (Projective Space): Take $X = \PP^n_{\FF}$, Example image of $\PP^2_{\GF(3)}$: Note that we can identify $X = \Gr_{\FF}(1, n)$ as the space of lines in $\AA^n_\FF$. Proposition : The number of $k\dash$dimensional subspaces of $\AA^m_\FF$ is the $q\dash$binomial coefficient: \[ \genfrac{[}{]}{0pt}{}{m}{k}_q \definedas \frac{(q^m - 1)(q^{m-1}-1) \cdots (q^{m - (k-1)} - 1)}{(q^k-1)(q^{k-1} - 1) \cdots (q-1)} .\] Proof : To choose a $k\dash$dimensional subspace, - Choose a nonzero vector $\vector v_1 \in \AA^n_\FF$ in $$q^m - 1$$ ways. - Identify $\size\spanof\theset{\vector v_1} = \size\theset{\lambda \vector v_1 \suchthat \lambda \in \FF} = \size \FF = q$. - Choose a nonzero vector $\vector v_2$ *not* in the span of $\vector v_1$ in $$q^m - q$$ ways. - Identify $\size \spanof\theset{\vector v_1, \vector v_2} = \size \theset{\lambda_1 \vector v_1 + \lambda_2 \vector v_2 \suchthat \lambda_i \in \FF} = q\cdot q = q^2$. - Choose a nonzero vector $\vector v_3$ not in the span of $\vector v_1, \vector v_2$ in $$q^m -q^2$$ ways. - $\cdots$ until $\vector v_k$ is chosen in $$(q^m-1)(q^m-q) \cdots (q^m - q^{k-1})$$ ways. - This yields a $k\dash$tuple of linearly independent vectors spanning a $k\dash$dimensional subspace $V_k$ - This overcounts because many linearly independent sets span $V_k$, we need to divide out by the number of choose a basis inside of $V_k$. - By the same argument, this is given by $$(q^k-1)(q^k-q) \cdots (q^k - q^{k-1})$$ Thus \[ \size \text{subspaces} &= \frac{ (q^m-1)(q^m-q)(q^m - q^2) \cdots (q^m - q^{k-1}) }{ (q^k-1)(q^k-q)(q^k-q^2) \cdots (q^k - q^{k-1})}\\ &= {q^m - 1 \over q^k - 1} \cdot \qty{q \over q} {q^{m-1} - 1 \over q^{k-1} - 1} \cdot \qty{q^2 \over q^2}{q^{m-2} - 1 \over q^{k-2} - 1} \cdots \qty{q^{k-1} \over q^{k-1}}{q^{m - (k-1)} - 1 \over q^{k - (k-1) - 1}} .\] We obtain a nice simplification for the number of lines corresponding to setting $k=1$: \[ \genfrac{[}{]}{0pt}{}{m}{1}_q = {q^m-1 \over q - 1} = q^{m-1} + q^{m-2} + \cdots + q + 1 = \sum_{j=0}^{m-1} q^j .\] Thus \[ X(\FF) &= \sum_{j=0}^{m-1} q^j \\ X(\FF_2) &= \sum_{j=0}^{m-1} \qty{q^2}^j \\ &\vdots \\ X(\FF_n) &= \sum_{j=0}^{m-1} \qty{q^n}^j .\] So \[ \zeta_X(z) = \qty{1 \over 1 - z} \qty{1 \over 1 - qz} \qty{1 \over 1 - q^2 z} \cdots \qty{1 \over 1- q^m z} \\ ,\] Note that geometry can help us here: we have a "cell decomposition" $\PP^n = \PP^{n-1} \disjoint \AA^n$, and so inductively \[ \PP^n = \AA^0 \disjoint \AA^1 \disjoint \cdots \disjoint \AA^n ,\] and it's straightforward to prove that $$ \zeta_{X\disjoint Y}(z) = \zeta_X(z) \cdot \zeta_Y(z) $$ and recalling that $\zeta_{\AA^j}(z) = {1 \over 1 - q^j z}$ we have \[ \zeta_{\PP^m}(z) = \prod_{j=0}^m \zeta_{\AA^j}(z) = \prod_{j=0}^n {1 \over 1 - q^j z} .\] Example: Take $X = \Gr_{\FF}(k, n)$, then ????? so \[ \zeta_X(t) = ? .\] ## Hard Example: An Elliptic Curve The Weyl conjectures take on a particularly nice form for [curves](curves.md). Let $X/\FF$ be a smooth projective curve of genus $g$, then 1. (Rationality) $$\zeta_X(z) = {p(z) \over (1-z)(1-qz)}$$ 2. (Functional Equation) $$\zeta_X\qty{1 \over qz} = q^{1-g} z^{2-2g} \zeta_X(z)$$ 3. (Riemann Hypothesis) $$p(t) = \prod_{i=1}^{2g} (q - a_i z) \qtext{where} \abs{a_i} = {1 \over \sqrt q}$$ Take $X = E/\FF$. Then \[ \zeta_X(t) = {(1-aq^{-t}) (1 - \bar a q^{-t}) \over (1 - q^{-t})(1 - q^{1-s}) } .\] The betti numbers are $[1,2,1, 0, \cdots]$. The number of points are $$ X(\FF_n) = (q^n + 1) - ( \alpha^n + {\bar \alpha}^n ) \qtext{where} \abs{\alpha} = \abs{\bar \alpha} = \sqrt{q} $$ Rough outline of proof: - ?? The (complex?) dimension of $X$ is $N=1$, The WC say we should be able to write this as \[ {p_1(z) \over p_0(z) p_2(z)} = {p_1(z) \over (1-z) (1 - qz)} = { (1 - \alpha_{1, 1}z)(1 - \alpha_{1, 2}z) \over (1-z)(1- qz)} .\] Since we know the number of points, we can compute \[ \zeta_X(z) &= \exp \sum_{n=1}^\infty \sizeX(\FF_n) {z^n \over n} \\ &= \exp \sum_{n=1}^\infty \qty{q^n + 1 - \qty{\alpha^n + \bar\alpha^n}} {z^n \over n} \\ &= \exp \qty{ \sum_{n=1}^\infty q^n\cdot {z^n \over n} } \exp \qty{ \sum_{n=1}^\infty 1\cdot {z^n \over n} } \exp \qty{ \sum_{n=1}^\infty -\alpha^n \cdot {z^n \over n} } \exp \qty{ \sum_{n=1}^\infty -\bar\alpha^n\cdot{z^n \over n} } \\ \\ &= \exp\qty{-\log\qty{1-qz} } \,\,\exp\qty{-\log\qty{1-z} } \,\,\exp\qty{\log\qty{1- \alpha z} } \.\,\exp\qty{\log\qty{1 - \bar \alpha z} } \\ \\ &= {(1-\alpha z)(1-\bar \alpha z) \over (1-z)(1-qz)} \in \QQ(z) ,\] which is indeed a rational function. > Originally conjectured for curves by Artin > Proved by Weil in 1949, proposed generalization to projective varieties > Proof had work contributed by [Dwork](Dwork) (rationality using p-adic analysis), Artin, Grothendieck ([Unsorted/etale cohomology](Unsorted/etale%20cohomology)), with completion by Deligne in 1970s (RH) ## Very Hard Example: A Diagonal Hypersurface > [Reference](https://math.mit.edu/~notzeb/weil.pdf) Proof of rationality of $Z_X(T)$ for $X$ a diagonal hypersurface. - Set $q$ to be a prime power and consider $X/\FF_q$ defined by $$X = V(a_0x_0^{n_0} + \cdots + a_r x_r^{n_r}) \subset \FF_q^{r+1}.$$ - We want to compute $N = \size X$. - Set $d_i = \gcd(n_i, q-1)$. - Define the [character](character) \[ \psi_q: \FF_q & \to \CC\units \\ a &\mapsto \exp\qty{2\pi i ~\Tr_{\FF_q/\FF_p}(a) \over p} .\] - By Artin's theorem for linear independence of characters, $\psi_q \not \equiv 1$ and every additive character of $\FF_q$ is of the form $a \mapsto \psi_q(ca)$ for some $c\in \FF_q$. - Fix an injective multiplicative map \[ \psi: \bar{\FF}_q\units \to \CC\units .\] - Define \[ \chi_{\alpha, n}: \FF_{q^n}\units &\to \CC\units \\ x & \mapsto \phi(x)^{\alpha\qty{q^n-1}} \\ \\ \quad \qtext{for} \alpha \in \QQ/\ZZ, n\in \ZZ, & \quad \alpha\qty{q^n-1} \equiv 0 \mod 1 .\] - Extend this to $\FF_{q^n}$ by \[ \begin{cases} 1 & \alpha \equiv0 \mod 1 \\ 0 & \text{else} \end{cases} .\] - Set $\chi_\alpha = \chi_{\alpha, 1}$. - Shorthand notation: say $a\sim 0 \iff a \equiv 0 \mod 1$. - Proposition: $$\alpha(q-1) \equiv 0 \mod 1 \implies \chi_{\alpha, n}(x) = \chi_\alpha(\mathrm{Nm}_{\FF_{q^n} / \FF_q }(x) )$$ - Proposition: $$d \definedas \gcd(n, q-1), u \in \FF_q \implies \size\theset {x\in \FF_1 \suchthat x^n = u} = \sum_{d\alpha \sim 0} \chi_\alpha(u)$$ - This implies \[ N &= \sum_{\substack{\alpha = [\alpha_0, \cdots, \alpha_r] \\ d_i \alpha_i \sim 0}} \quad \sum_{\substack{\vector u = [u_0, \cdots ,u_r] \\ {\textstyle{\sum}} a_i u_i = 0}} \quad \prod_{j=0}^r \chi_{\alpha_j}(u_j) \\ \\ &= q^r + \sum_{\substack{\alpha,~ \alpha_i \in (0, 1) \\ d_i \alpha_i \sim 0}} \qty{ \prod_{j=0}^r \chi_{\alpha_j}(a_j \inv ) \sum_{\Sigma~ u_i=0} \quad \prod_{j=0}^r \chi_{\alpha_j}(u_j) } .\] since the inner sum is zero if some *but not all* of the $\alpha_i \sim 0$. - Evaluate the innermost sum by restricting to $u_0 \neq 0$ and setting $u_i = u_0 v_i$ and $v_0 \definedas 1$: \[ \sum_{\Sigma~ u_i=0} \quad \prod_{j=0}^r \chi_{\alpha_j}(u_j) &= \sum_{u_0 \neq 0} \chi_{_{\Sigma ~ \alpha_i}}(u_0) \sum_{\Sigma ~v_i = 0} ~\prod_{j=0}^r \chi_{\alpha_j} (v_j) \\ &= \begin{cases} \qty{q-1} \sum_{\Sigma~ v_i = 0} ~\prod_{j=0}^r \chi_{\alpha_j}(v_j) & \qtext{if} \sum \alpha_i \sim 0 \\ 0 & \qtext{else} \end{cases} .\] - Define the *Jacobi sum* for $\alpha$ where $\sum \alpha_i \sim 0$: \[ J(\alpha) \definedas \qty {1 \over q-1} \sum_{\Sigma~ u_i = 0} ~\prod_{j=0}^r \chi_{\alpha_j}(u_j) = \sum_{\Sigma~ v_i = 0} ~\prod_{j=1}^r \chi_{\alpha_j}(v_j) \] - Express $N$ in terms of [Jacobi sums](Jacobi%20sums) as \[ N = q^r + \qty{q-1} \sum_{\substack{\Sigma \alpha_i \sim 0 \\ d_i \alpha_i \sim 0 \\ \alpha\in (0, 1)}} \prod_{j=0}^r \chi_{\alpha_j}(a_j\inv ) J(\alpha) .\] - Evaluate $J(\alpha)$ using a [Gauss sum](Unsorted/quadratic%20reciprocity.md) : for $\chi: \FF_q \to \CC$ a [multiplicative character](multiplicative%20character), define \[ G(\chi) &\definedas \sum_{x\in \FF_q} \chi(x) \psi_q(x) .\] - **Proposition**: for any $\chi \neq \chi_0$, - $\abs{G(\chi)} = q^{1 \over 2}$ - $G(\chi) G(\bar \chi) = q \chi(-1)$ - $G(\chi_0) = 0$ \[ \chi(t) = {G(\chi) \over q} \sum_{x\in \FF_q} \bar \chi(x) \psi_q(tx) .\] - **Proposition**: if $\sum \alpha_i \sim 0$, then $J(\alpha) = {1 \over q} \prod_{k=1}^r G(\chi_{\alpha_k})$ and $\abs{J(\alpha)} = q^{r - 1\over 2}$. - We thus obtain \[ N = q^r + \qty{q-1 \over q} \sum_{\substack{\Sigma \alpha_i \sim 0 \\ d_i \alpha_i \sim 0 \\ \alpha\in (0, 1)}} ~\prod_{j=0}^r \chi_{\alpha_j}(a_j\inv ) G(\chi_{\alpha_j}) .\] - We now ask for number of points in $\FF_{q^\nu}$ - **Theorem (Davenport, Hasse)** $\qty{q-1}\alpha \sim 0 \implies -G(\chi_{\alpha, \nu}) = \qty{-G(\chi_\alpha)}^\nu$. --- - Now restrict to $n_0 = \cdots = n_r = n$ a constant, and we consider a point count \[ \bar{N}_\nu = \size \theset{[x_0: \cdots : x_r] \in \PP^r_{\FF_q^\nu} \suchthat \sum_{i=0}^r a_i x_i^n = 0} .\] - We have a relation $\qty{q^\nu - 1} \bar N_\nu = N_\nu$. - This lets us write \[ \bar N_\nu = \sum_{j=0}^{r-1} q^{j\nu} + \sum_{\substack{\sum \alpha_i ~\sim 0 \\ \gcd(n, q^\nu - 1)\alpha_i \sim 0 \\ \alpha_i \in (0, 1) }} \prod_{j=0}^r \bar \chi_{\alpha_{j, \nu}}(a_i) J_\nu(\alpha) .\] - Set \[ \mu(\alpha) &= \min\theset{\mu \suchthat \qty{q^\mu - 1} \alpha \sim 0} \\ C(\alpha) &= (-1)^{r+1} \prod_{j=1}^r \bar \chi_{\alpha_0, \mu(\alpha)}(a_j) \cdot J_{\mu(\alpha)}(\alpha) .\] - Plugging into the zeta function $Z$ yields \[ \exp\qty{\sum_{\nu = 1}^\infty \bar N_\nu {T^\nu \over \nu} } = {1 \over (1-T) (1-qT) \cdots (1-q^{r-1}T) } \prod_{\substack{\sum \alpha_i \sim 0 \\ \gcd(n, q^\nu - 1)\alpha_i \sim 0 \\ \alpha_i \in (0, 1) }} \qty{1 - C(\alpha) T^{\mu(\alpha)}}^{(-1)^r \over \mu(\alpha)} ,\] which is evidently a rational function.