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• Tags:
  • #projects/notes/reading #projects/my-talks #arithmetic-geometry
• Refs:
  • The Zeta book http://www-personal.umich.edu/~mmustata/zeta_book.pdf #resources/books
  • https://youtu.be/wEz7fCvK6sM?t=293 #resources/videos
  • Explanation of exponential appearing
  • https://arxiv.org/pdf/1807.10812.pdf
  • http://www.math.canterbury.ac.nz/~j.booher/expos/weil_conjectures.pdf
  • Weil’s Paper
• Links:
  • Weil Conjectures
  • Weil Conjectures Talks

Notes from Daniel’s Office Hours

• Definition of Zeta functions
• Statement of the conjectures
• Easy examples: \({\mathbf{P}}^n_{{\mathbf{k}}},{\operatorname{Gr}}_{{\mathbf{k}}}(k, n) = \operatorname{GL}(n, {\mathbf{k}}) / P\) the stabilizer of an \({\mathbf{k}}{\hbox{-}}\)point in \({\mathbf{C}}^n, { \mathbf{F} }_{p^n}\).
• Medium example: \(E/{\mathbf{k}}\) an elliptic curve.
• Work out a harder example as in Weil

Definition of Zeta Function

Fix \(q\) a prime and \({ \mathbf{F} }\coloneqq{ \mathbf{F} }_q\) the finite field with \(q\) elements, along with its unique degree \(n\) extensions \begin{align*} { \mathbf{F} }_n \coloneqq{ \mathbf{F} }_{q^n} = \left\{{x\in \mkern 1.5mu\overline{\mkern-1.5mu{ \mathbf{F} }\mkern-1.5mu}\mkern 1.5mu_p {~\mathrel{\Big\vert}~}x^{q^n} - x = 0}\right\} \quad \forall~ n\in {\mathbf{Z}}^{\geq 2} \end{align*}

Definition Let
\begin{align*} J = \left\langle{f_1, \cdots, f_M}\right\rangle {~\trianglelefteq~}k[x_0, \cdots, x_n] \end{align*} be an ideal, then a projective algebraic variety \(X\subset {\mathbf{P}}^N_{ \mathbf{F} }\) can be given by \begin{align*} X = V(J) = \left\{{\mathbf{x} \in {\mathbf{P}}^\infty_{ \mathbf{F} }{~\mathrel{\Big\vert}~}f_1(\mathbf{x}) = \cdots = f_M(\mathbf{x}) = \mathbf{0}}\right\} \end{align*} where an ideal generated by homogeneous polynomials in \(n+1\) variables, i.e. there is some fixed \(d\in {\mathbf{Z}}^{\geq 1}\) such that

\begin{align*} f(\mathbf{x}) = \sum_{\substack{\mathbf{I} = (i_1, \cdots, i_n) \\ \sum_j i_j = d}} \alpha_{\mathbf{I}} \cdot x_0^{i_1}\cdots x_n^{i_n} {\quad \operatorname{ and } \quad} f(\lambda \cdot \mathbf{x}) = \lambda^d f(\mathbf{x}) .\end{align*}

For the experts: we can take a reduced (possibly reducible) scheme of finite type over a field \({ \mathbf{F} }_p\). We will be thinking of \(K{\hbox{-}}\)valued points for \(K/{ \mathbf{F} }_p\) algebraic extensions. From the audience: what condition do we need to put on such a scheme to guarantee an embedding into \({\mathbf{P}}^\infty\)?

Examples:

• Dimension 1: Curves
• Dimension 2: Surfaces
• Codimension 1: Hypersurfaces

Fix \(X/{ \mathbf{F} }\subset {\mathbf{P}}\) an \(N{\hbox{-}}\)dimensional projective algebraic variety, and say it’s cut out by the equations \(f_1, \cdots f_M \in { \mathbf{F} }[x_0, \cdots, x_n]\). 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:

\begin{align*} Z_X(z) &= \exp\qty{ \sum_{n=1}^\infty \alpha_n {z^n \over n} } \in {\mathbf{Q}}[z](z) {\quad \operatorname{where} \quad} \alpha_n \coloneqq{\sharp}X({ \mathbf{F} }_n) .\end{align*}

Concretely, for \(X\subset {\mathbf{P}}^M\) a variety cut out by \(\left\{{f_i}\right\} \subset { \mathbf{F} }[x_0, \cdots, x_M]\) we are measuring the sizes of the sets \begin{align*} \alpha_n \coloneqq{\sharp}\left\{{\mathbf{x} \in {\mathbf{P}}^M_{{ \mathbf{F} }_{q^n}} {~\mathrel{\Big\vert}~}f_i(\mathbf{x}) = \mathbf{0} ~\forall i }\right\} .\end{align*}

Compare to the Poincare polynomials: \(P_{{\mathbf{RP}}^m}(x) = 1 + x + x^2 + \cdots + x^m\) and \(P_{{\mathbf{CP}}^m}(x) = 1 + x^2 + \cdots + x^{2m}\)

Statement of Weil Conjectures

(Weil 1949)

Let \(X\) be a smooth projective variety of dimension \(N\) over \({ \mathbf{F} }_{q}\) for \(q\) a prime, let \(Z_X(z)\) be its zeta function, and define \(\zeta_X(s) = Z_X(q^{-s})\).

• (Rationality)

  \(Z_X(z)\) is a rational function:

  \begin{align*} 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 {\mathbf{Q}}(z),\quad\text{i.e. }\quad p_i(z) \in {\mathbf{Z}}[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} {\quad \operatorname{for some reciprocal roots} \quad} a_{j, k} \in {\mathbf{C}} \end{align*} 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.

• (Functional Equation and Poincare Duality)

  Let \(\chi(X)\) be the Euler characteristic of \(X\), i.e. the self-intersection number of the diagonal embedding \(\Delta \hookrightarrow X\times X\); then \(Z_X(z)\) satisfies the following functional equation:

  \begin{align*} Z_X\qty{1 \over q^N z} = \pm \qty{q^{N \over 2} z}^{\chi(X)} ~~Z_X(z) .\end{align*}

  Equivalently, \begin{align*} \zeta_X(N-s) = \pm \qty{q^{\frac N 2 - s}}^{\chi(X)} ~\zeta_X(s) \\ .\end{align*}

  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 \begin{align*} z &\iff {q^N \over z} \\ \alpha_{j, k} &\iff \alpha_{2N-j, k} \end{align*} which sends roots of \(p_j\) to roots of \(p_{2N-j}\).

• (Riemann Hypothesis)

The reciprocal roots \(a_{j,k}\) are algebraic integers (roots of some monic \(p\in {\mathbf{Z}}[x]\)) which satisfy \begin{align*} {\left\lvert {a_{j,k}} \right\rvert}_{\mathbf{C}}= q^{j \over 2} \quad\quad \forall 1 \leq j \leq 2N-1,~ \forall k .\end{align*}

• (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 \(\tilde X({\mathbf{C}})\).

Why is (3) called the “Riemann Hypothesis”?

We can use the facts that

• \({\left\lvert {\exp\qty{z}} \right\rvert} = \exp\qty{\Re(z)}\) and
• \(a^z \coloneqq\exp\qty{z \operatorname{Log}(a)}\),

to replace the polynomials \(P_i\) with \begin{align*} L_j(s) \coloneqq\zeta_X(q^{-s}) = \prod_{k=1}^{\beta_j} \qty{1 - \alpha_{j, k} q^{-s}} .\end{align*}

Now consider the roots of \(L_j(s)\): we have \begin{align*} L_j(s_0) &= 0 \\ \iff q^{-s_0} &= {1 \over \alpha_{j, k}} {\quad \operatorname{for some} \quad} k \\ \implies {\left\lvert {q^{-s_0}} \right\rvert} &= {\left\lvert {1 \over \alpha_{j, k}} \right\rvert} \quad\quad \stackrel{\text{\tiny by assumption}}{=} q^{ -{j \over 2}} \\ \implies q^{-\frac j 2} \stackrel{(a)}= \exp\qty{- \frac j 2 \cdot \operatorname{Log}(q)} &= {\left\lvert { \exp\qty{-s_0 \cdot \operatorname{Log}(q)} } \right\rvert} \\ &\stackrel{(b)}= {\left\lvert { \exp\qty{-\qty{\Re(s_0) + i\cdot \Im(s_0)} \cdot \operatorname{Log}(q)} } \right\rvert} \\ &\stackrel{(a)}= \exp\qty{-\qty{\Re(s_0)} \cdot \operatorname{Log}(q)} \\ \implies - \frac j 2 \cdot \operatorname{Log}(q) &= -\Re(s_0) \cdot \operatorname{Log}(q) {\quad \operatorname{by injectivity} \quad} \\ \implies \Re(s_0) = \frac j 2 .\end{align*}

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 \({ \mathbf{F} }_q\)
• On \(L_X\):
  • Conjectured for smooth varieties over \({\mathbf{Q}}\) (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 (Taylor-Wiles c/o the Taniyama-Shimura conjecture), implies \(L_X\) is an \(L\) function coming from a modular form.

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: \begin{align*} {\mathbf{A}}_{\mathbf{Z}}^n \supseteq X = V(\left\langle{f_1, \cdots, f_d}\right\rangle) {\quad \operatorname{where} \quad} f_i \in {\mathbf{Z}}[x_0, \cdots, x_{n-1}] .\end{align*}

Then for every prime \(q\), we can reduce the equations mod \(p\) and consider \begin{align*} {\mathbf{A}}_{{ \mathbf{F} }_q}^n \supseteq X_q \coloneqq V(\left\langle{f_1 \operatorname{mod}q, \cdots ,f_d \operatorname{mod}q}\right\rangle) {\quad \operatorname{where} \quad} f_1 \operatorname{mod}q \in { \mathbf{F} }_q[x_0, \cdots, x_{n-1}] \end{align*}

Then define the Unsorted/Hasse-Weil L function

\begin{align*} L_X(s) = \prod_{p\text{ prime}} \zeta_{X_p}\qty{p^{-s}} .\end{align*}

Take \(X = \operatorname{Spec}{\mathbf{Q}}\) and \(X_p = \operatorname{Spec}{ \mathbf{F} }_p\), which is a single point since \({ \mathbf{F} }_p\) is a field. The previous example shows that \begin{align*} \zeta_{X_p}(z) = {1 \over 1-z} ,\end{align*}

We then find that \begin{align*} L_X(s) &= \prod_{p\text{ prime}} \zeta_{X_p}(p^{-s}) \\ &= \prod_{p\text{ prime}} \qty{ 1 \over 1 - p^{-s}} \\ &= \zeta(s) ,\end{align*}

which is the Euler product expansion of the classical Unsorted/Riemann Zeta function.

Moreover, it is a theorem (difficult, not proved here!) that for any variety \(X/{ \mathbf{F} }_p\), we have \begin{align*} \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} } ,\end{align*}

which we can think of as attaching a “weight” to each closed point, \({\left\lvert {x} \right\rvert} \coloneqq p^{\deg(x)}\), and the usual Riemann Zeta corresponds to assigning a weight of 1 to each point.

Note that this immediately implies that \(\zeta_X(t) \in {\mathbf{Z}}[[t]]\) is a rational function.

Recall the Riemann zeta function is given by \begin{align*} \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \prod_{p\text{ prime}} {1 \over 1 - p^{-s}} .\end{align*}

After modifying \(\zeta\) to make it symmetric about \(\Re(s) = \frac 1 2\) and eliminate the trivial zeros at \(-2{\mathbf{Z}}\) to obtain \(\widehat{\zeta}(s)\), there are three relevant properties

• “Rationality”: \(\widehat{\zeta}(s)\) has a meromorphic continuation to \({\mathbf{C}}\) with simple poles at \(s=0, 1\).
• “Functional equation”: \(\widehat{\zeta}(1-s) = \widehat{\zeta}(s)\)
• “Riemann Hypothesis”: The only zeros of \(\widehat{\zeta}\) have \(\Re(s) = \frac 1 2\).

More Examples

Example (Affine Space):

Set \(X = {\mathbf{A}}^m/{ \mathbf{F} }\), affine \(m{\hbox{-}}\)space over \({ \mathbf{F} }\), so we can just repeat with now \(m\) coordinates \begin{align*} {\mathbf{A}}^1({ \mathbf{F} }_n) = \left\{{\mathbf{x} = [x_1, \cdots, x_m] {~\mathrel{\Big\vert}~}x_i \in { \mathbf{F} }_n}\right\} \end{align*} Counting yields \begin{align*} X({ \mathbf{F} }) &= q^m \\ X({ \mathbf{F} }_2) &= (q^2)^m \\ &\vdots \\ X({ \mathbf{F} }_n) &= (q^n)^m .\end{align*}

Thus \begin{align*} \zeta_X(z) = \exp\qty{\sum_{n=1}^\infty {q^{nm} \over n} z^n } = \frac 1 {1 - q^m z} .\end{align*}

Example (Projective Line):

\(X = {\mathbf{P}}^1/{ \mathbf{F} }\) the projective line over \({ \mathbf{F} }\), then we can write use some geometry to write \begin{align*} {\mathbf{P}}^1_{ \mathbf{F} }= {\mathbf{A}}^1_{ \mathbf{F} }{\textstyle\coprod}\left\{{\infty}\right\} \end{align*} as the affine line with a point added at infinity.

We can then count by enumerating coordinates: \begin{align*} {\mathbf{P}}^1({ \mathbf{F} }_n) &= \left\{{[x_1, x_2] {~\mathrel{\Big\vert}~}x_1, x_2 \neq 0 \in { \mathbf{F} }_n}\right\}/\sim \\ &= \left\{{[x_1, 1] {~\mathrel{\Big\vert}~}x_1 \in { \mathbf{F} }_n}\right\} {\textstyle\coprod}\left\{{[1, 0]}\right\} .\end{align*}

Thus \begin{align*} X({ \mathbf{F} }) &= q + 1\\ X({ \mathbf{F} }_2) &= q^2 + 1 \\ &\vdots \\ X({ \mathbf{F} }_n) &= q^n + 1\\ .\end{align*}

Thus \begin{align*} \zeta_X(z) = {1 \over (1-z)(1-qz)} \\ .\end{align*}

Example (Projective Space):

Take \(X = {\mathbf{P}}^n_{{ \mathbf{F} }}\),

Example image of \({\mathbf{P}}^2_{{\mathbf{GF}}(3)}\):

Note that we can identify \(X = {\operatorname{Gr}}_{{ \mathbf{F} }}(1, n)\) as the space of lines in \({\mathbf{A}}^n_{ \mathbf{F} }\).

Proposition

The number of \(k{\hbox{-}}\)dimensional subspaces of \({\mathbf{A}}^m_{ \mathbf{F} }\) is the \(q{\hbox{-}}\)binomial coefficient: \begin{align*} \genfrac{[}{]}{0pt}{}{m}{k}_q \coloneqq\frac{(q^m - 1)(q^{m-1}-1) \cdots (q^{m - (k-1)} - 1)}{(q^k-1)(q^{k-1} - 1) \cdots (q-1)} .\end{align*}

Proof

To choose a \(k{\hbox{-}}\)dimensional subspace,

• Choose a nonzero vector \(\mathbf{v}_1 \in {\mathbf{A}}^n_{ \mathbf{F} }\) in \begin{align*}q^m - 1\end{align*} ways.
  • Identify \({\sharp}\mathop{\mathrm{span}}\left\{{\mathbf{v}_1}\right\} = {\sharp}\left\{{\lambda \mathbf{v}_1 {~\mathrel{\Big\vert}~}\lambda \in { \mathbf{F} }}\right\} = {\sharp}{ \mathbf{F} }= q\).
• Choose a nonzero vector \(\mathbf{v}_2\) not in the span of \(\mathbf{v}_1\) in \begin{align*}q^m - q\end{align*} ways.
  • Identify \({\sharp}\mathop{\mathrm{span}}\left\{{\mathbf{v}_1, \mathbf{v}_2}\right\} = {\sharp}\left\{{\lambda_1 \mathbf{v}_1 + \lambda_2 \mathbf{v}_2 {~\mathrel{\Big\vert}~}\lambda_i \in { \mathbf{F} }}\right\} = q\cdot q = q^2\).
• Choose a nonzero vector \(\mathbf{v}_3\) not in the span of \(\mathbf{v}_1, \mathbf{v}_2\) in \begin{align*}q^m -q^2\end{align*} ways.
• \(\cdots\) until \(\mathbf{v}_k\) is chosen in \begin{align*}(q^m-1)(q^m-q) \cdots (q^m - q^{k-1})\end{align*} ways.
  • This yields a \(k{\hbox{-}}\)tuple of linearly independent vectors spanning a \(k{\hbox{-}}\)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 \begin{align*}(q^k-1)(q^k-q) \cdots (q^k - q^{k-1})\end{align*}

Thus \begin{align*} {\sharp}\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}} .\end{align*}

We obtain a nice simplification for the number of lines corresponding to setting \(k=1\): \begin{align*} \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 .\end{align*}

Thus \begin{align*} X({ \mathbf{F} }) &= \sum_{j=0}^{m-1} q^j \\ X({ \mathbf{F} }_2) &= \sum_{j=0}^{m-1} \qty{q^2}^j \\ &\vdots \\ X({ \mathbf{F} }_n) &= \sum_{j=0}^{m-1} \qty{q^n}^j .\end{align*}

So \begin{align*} \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} \\ ,\end{align*}

Note that geometry can help us here: we have a “cell decomposition” \({\mathbf{P}}^n = {\mathbf{P}}^{n-1} {\textstyle\coprod}{\mathbf{A}}^n\), and so inductively \begin{align*} {\mathbf{P}}^n = {\mathbf{A}}^0 {\textstyle\coprod}{\mathbf{A}}^1 {\textstyle\coprod}\cdots {\textstyle\coprod}{\mathbf{A}}^n ,\end{align*}

and it’s straightforward to prove that \begin{align*} \zeta_{X{\textstyle\coprod}Y}(z) = \zeta_X(z) \cdot \zeta_Y(z) \end{align*}

and recalling that \(\zeta_{{\mathbf{A}}^j}(z) = {1 \over 1 - q^j z}\) we have \begin{align*} \zeta_{{\mathbf{P}}^m}(z) = \prod_{j=0}^m \zeta_{{\mathbf{A}}^j}(z) = \prod_{j=0}^n {1 \over 1 - q^j z} .\end{align*}

Example: Take \(X = {\operatorname{Gr}}_{{ \mathbf{F} }}(k, n)\), then ????? so \begin{align*} \zeta_X(t) = ? .\end{align*}

Hard Example: An Elliptic Curve

The Weyl conjectures take on a particularly nice form for curves. Let \(X/{ \mathbf{F} }\) be a smooth projective curve of genus \(g\), then

• (Rationality) \begin{align*}\zeta_X(z) = {p(z) \over (1-z)(1-qz)}\end{align*}
• (Functional Equation) \begin{align*}\zeta_X\qty{1 \over qz} = q^{1-g} z^{2-2g} \zeta_X(z)\end{align*}
• (Riemann Hypothesis) \begin{align*}p(t) = \prod_{i=1}^{2g} (q - a_i z) {\quad \operatorname{where} \quad} {\left\lvert {a_i} \right\rvert} = {1 \over \sqrt q}\end{align*}

Take \(X = E/{ \mathbf{F} }\).

Then \begin{align*} \zeta_X(t) = {(1-aq^{-t}) (1 - \mkern 1.5mu\overline{\mkern-1.5mua\mkern-1.5mu}\mkern 1.5mu q^{-t}) \over (1 - q^{-t})(1 - q^{1-s}) } .\end{align*}

The betti numbers are \([1,2,1, 0, \cdots]\).

The number of points are \begin{align*} X({ \mathbf{F} }_n) = (q^n + 1) - ( \alpha^n + {\mkern 1.5mu\overline{\mkern-1.5mu\alpha\mkern-1.5mu}\mkern 1.5mu}^n ) {\quad \operatorname{where} \quad} {\left\lvert {\alpha} \right\rvert} = {\left\lvert {\mkern 1.5mu\overline{\mkern-1.5mu\alpha\mkern-1.5mu}\mkern 1.5mu} \right\rvert} = \sqrt{q} \end{align*}

Rough outline of proof:

• ??

The (complex?) dimension of \(X\) is \(N=1\), The WC say we should be able to write this as \begin{align*} {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)} .\end{align*}

Since we know the number of points, we can compute \begin{align*} \zeta_X(z) &= \exp \sum_{n=1}^\infty \sizeX({ \mathbf{F} }_n) {z^n \over n} \\ &= \exp \sum_{n=1}^\infty \qty{q^n + 1 - \qty{\alpha^n + \mkern 1.5mu\overline{\mkern-1.5mu\alpha\mkern-1.5mu}\mkern 1.5mu^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 -\mkern 1.5mu\overline{\mkern-1.5mu\alpha\mkern-1.5mu}\mkern 1.5mu^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 - \mkern 1.5mu\overline{\mkern-1.5mu\alpha \mkern-1.5mu}\mkern 1.5muz} } \\ \\ &= {(1-\alpha z)(1-\mkern 1.5mu\overline{\mkern-1.5mu\alpha \mkern-1.5mu}\mkern 1.5muz) \over (1-z)(1-qz)} \in {\mathbf{Q}}(z) ,\end{align*} 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 (rationality using p-adic analysis), Artin, Grothendieck (Unsorted/etale cohomology), with completion by Deligne in 1970s (RH)

Very Hard Example: A Diagonal Hypersurface

Reference

Proof of rationality of \(Z_X(T)\) for \(X\) a diagonal hypersurface.

• Set \(q\) to be a prime power and consider \(X/{ \mathbf{F} }_q\) defined by \begin{align*}X = V(a_0x_0^{n_0} + \cdots + a_r x_r^{n_r}) \subset { \mathbf{F} }_q^{r+1}.\end{align*}

• We want to compute \(N = {\sharp}X\).

• Set \(d_i = \gcd(n_i, q-1)\).

• Define the character \begin{align*} \psi_q: { \mathbf{F} }_q & \to {\mathbf{C}}^{\times}\\ a &\mapsto \exp\qty{2\pi i ~\operatorname{Tr}_{{ \mathbf{F} }_q/{ \mathbf{F} }_p}(a) \over p} .\end{align*}

  • By Artin’s theorem for linear independence of characters, \(\psi_q \not \equiv 1\) and every additive character of \({ \mathbf{F} }_q\) is of the form \(a \mapsto \psi_q(ca)\) for some \(c\in { \mathbf{F} }_q\).

• Fix an injective multiplicative map \begin{align*} \psi: \mkern 1.5mu\overline{\mkern-1.5mu{ \mathbf{F} }\mkern-1.5mu}\mkern 1.5mu_q^{\times}\to {\mathbf{C}}^{\times} .\end{align*}

• Define \begin{align*} \chi_{\alpha, n}: { \mathbf{F} }_{q^n}^{\times}&\to {\mathbf{C}}^{\times}\\ x & \mapsto \phi(x)^{\alpha\qty{q^n-1}} \\ \\ \quad {\quad \operatorname{for} \quad} \alpha \in {\mathbf{Q}}/{\mathbf{Z}}, n\in {\mathbf{Z}}, & \quad \alpha\qty{q^n-1} \equiv 0 \operatorname{mod}1 .\end{align*}

  • Extend this to \({ \mathbf{F} }_{q^n}\) by \begin{align*} \begin{cases} 1 & \alpha \equiv0 \operatorname{mod}1 \\ 0 & \text{else} \end{cases} .\end{align*}
  • Set \(\chi_\alpha = \chi_{\alpha, 1}\).

• Shorthand notation: say \(a\sim 0 \iff a \equiv 0 \operatorname{mod}1\).

• Proposition: \begin{align*}\alpha(q-1) \equiv 0 \operatorname{mod}1 \implies \chi_{\alpha, n}(x) = \chi_\alpha(\mathrm{Nm}_{{ \mathbf{F} }_{q^n} / { \mathbf{F} }_q }(x) )\end{align*}

• Proposition: \begin{align*}d \coloneqq\gcd(n, q-1), u \in { \mathbf{F} }_q \implies {\sharp}\left\{{x\in { \mathbf{F} }_1 {~\mathrel{\Big\vert}~}x^n = u}\right\} = \sum_{d\alpha \sim 0} \chi_\alpha(u)\end{align*}

• This implies \begin{align*} N &= \sum_{\substack{\alpha = [\alpha_0, \cdots, \alpha_r] \\ d_i \alpha_i \sim 0}} \quad \sum_{\substack{\mathbf{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 ^{-1}) \sum_{\Sigma~ u_i=0} \quad \prod_{j=0}^r \chi_{\alpha_j}(u_j) } .\end{align*} 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 \(v_0 \coloneqq 1\): \begin{align*} \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) & {\quad \operatorname{if} \quad} \sum \alpha_i \sim 0 \\ 0 & {\quad \operatorname{else} \quad} \end{cases} .\end{align*}

• Define the Jacobi sum for \(\alpha\) where \(\sum \alpha_i \sim 0\): \begin{align*} J(\alpha) \coloneqq\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) \end{align*}

• Express \(N\) in terms of Jacobi sums as \begin{align*} 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^{-1}) J(\alpha) .\end{align*}

• Evaluate \(J(\alpha)\) using a Gauss sum : for \(\chi: { \mathbf{F} }_q \to {\mathbf{C}}\) a multiplicative character, define \begin{align*} G(\chi) &\coloneqq\sum_{x\in { \mathbf{F} }_q} \chi(x) \psi_q(x) .\end{align*}

• Proposition: for any \(\chi \neq \chi_0\),

  • \({\left\lvert {G(\chi)} \right\rvert} = q^{1 \over 2}\)
  • \(G(\chi) G(\mkern 1.5mu\overline{\mkern-1.5mu\chi\mkern-1.5mu}\mkern 1.5mu) = q \chi(-1)\)
  • \(G(\chi_0) = 0\) \begin{align*} \chi(t) = {G(\chi) \over q} \sum_{x\in { \mathbf{F} }_q} \mkern 1.5mu\overline{\mkern-1.5mu\chi\mkern-1.5mu}\mkern 1.5mu(x) \psi_q(tx) .\end{align*}

• Proposition: if \(\sum \alpha_i \sim 0\), then \(J(\alpha) = {1 \over q} \prod_{k=1}^r G(\chi_{\alpha_k})\) and \({\left\lvert {J(\alpha)} \right\rvert} = q^{r - 1\over 2}\).

• We thus obtain \begin{align*} 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^{-1}) G(\chi_{\alpha_j}) .\end{align*}

• We now ask for number of points in \({ \mathbf{F} }_{q^\nu}\)

• 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 \begin{align*} \mkern 1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu_\nu = {\sharp}\left\{{[x_0: \cdots : x_r] \in {\mathbf{P}}^r_{{ \mathbf{F} }_q^\nu} {~\mathrel{\Big\vert}~}\sum_{i=0}^r a_i x_i^n = 0}\right\} .\end{align*}

• We have a relation \(\qty{q^\nu - 1} \mkern 1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu_\nu = N_\nu\).

• This lets us write \begin{align*} \mkern 1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu_\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 \mkern 1.5mu\overline{\mkern-1.5mu\chi\mkern-1.5mu}\mkern 1.5mu_{\alpha_{j, \nu}}(a_i) J_\nu(\alpha) .\end{align*}

• Set \begin{align*} \mu(\alpha) &= \min\left\{{\mu {~\mathrel{\Big\vert}~}\qty{q^\mu - 1} \alpha \sim 0}\right\} \\ C(\alpha) &= (-1)^{r+1} \prod_{j=1}^r \mkern 1.5mu\overline{\mkern-1.5mu\chi\mkern-1.5mu}\mkern 1.5mu_{\alpha_0, \mu(\alpha)}(a_j) \cdot J_{\mu(\alpha)}(\alpha) .\end{align*}

• Plugging into the zeta function \(Z\) yields \begin{align*} \exp\qty{\sum_{\nu = 1}^\infty \mkern 1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu_\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)} ,\end{align*} which is evidently a rational function.