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Chapter 1 Prehistory of the Weil conjectures

Original lecture date: September 30, 2019.

In this lecture, we discuss the “prehistory” of the Weil conjectures from Gauss/Jacobi and Riemann/Dirichlet to Artin to Weil.

Readings 1.0.1.

The primary source for this lecture is Weil's 1949 paper [132]. We will assume some familiarity with basic facts about algebraic number theory; there are many references for this, but we generally will follow Neukirch [105].

Since this topic is old and well-studied, many other expositions of it are available. A particularly detailed one has been given by Milne [98].

Section 1.1 The number field setting

For context, let's start by formulating the Riemann hypothesis for Dedekind zeta functions.

Definition 1.1.1.

Let \(K\) be a number field, i.e., a finite-degree field extension of the field of rational numbers \(\QQ\text{.}\) Let \(\calO_K\) be the ring of integers of \(K\text{,}\) which is to say the integral closure of \(\ZZ\) in \(K\) (more concretely, the elements of \(K\) which are roots of monic polynomials with integer coefficients). A basic fact about \(\calO_K\) is that it is a Dedekind domain, and so every nonzero ideal can be written uniquely as a product of powers of maximal ideals. (Note: we say “maximal ideals” rather than “prime ideals” only to exclude the zero ideal.)

The Dedekind zeta function of \(K\) is defined initially as the formal expression

\begin{equation*} \zeta_K(s) \colonequals \prod\limits_{P \subset \calO_K}\frac{1}{1-\Norm_{K/\QQ}(\frakp)^{-s}}, \end{equation*}

where the product is over all the maximal ideals of \(\calO_K\) and \(\Norm_{K/\QQ}(\frakp)\) is the cardinality of the quotient ring \(\calO_K/\frakp\text{.}\) For \(s \in \CC\) with \(\Real(s) > 1\text{,}\) the product converges absolutely and so defines a holomorphic function without zeroes in that region. By unique factorization, we can rewrite the product as a sum

\begin{equation*} \zeta_K(s) = \sum\limits_{I \subseteq \calO_K} \Norm_{K/\QQ}(I)^{-s} \end{equation*}

where \(I\) now runs over all nonzero ideals of \(\calO_K\text{.}\)

When \(K = \QQ\text{,}\) \(\zeta_K(s)\) is the usual Riemann zeta function. Like the latter, \(\zeta_K(s)\) satisfies a functional equation relating its values at \(s\) and \(1-s\text{.}\) The cleanest way to conceptualize this is to use the language of places, as follows.

Definition 1.1.3.

Each maximal ideal \(\frakp\) of \(\calO_K\) corresponds to a dense embedding of \(K\) into a field complete with respect to a multiplicative absolute value, namely the fraction field of the \(\frakp\)-adic completion of \(\calO_K\text{.}\) These embeddings are called finite places of \(K\text{.}\) By Ostrowski's theorem, the only other dense embeddings of \(K\) into a field complete with respect to a (nontrivial) multiplicative absolute value are embeddings into \(\RR\) or \(\CC\text{,}\) of which there are only finitely many; these are called infinite places of \(K\text{.}\) (For \(K = \QQ\text{,}\) there is a unique infinite place, because \(\QQ\) maps in only one way into \(\RR\text{.}\) Each prime number \(b\) corresponds to the embedding of \(\QQ\) into the \(p\)-adic numbers \(\QQ_p\text{.}\))

One may then define a completed zeta function \(\Lambda_K(s)\) by adding to the product a suitable factor for each infinite place. This factor has the form

\begin{equation*} \pi^{-s/2} \Gamma(s/2) \qquad \mbox{or} \qquad 2(2\pi)^{-s} \Gamma(s) \end{equation*}

(where \(\Gamma\) is Gauss's meromorphic interpolation of the factorial function) depending on whether the completion of \(\QQ\) is isomorphic to \(\RR\) (a real place) or \(\CC\) (a complex place). With these factors in place, the functional equation has the form

\begin{equation*} \Lambda_K(s) = \Lambda_K(1-s). \end{equation*}

Section 1.2 The function field setting

It was suggested by Artin that there should be a close analogy between number fields and function fields. This grows out of the observation that for any finite field \(\FF_q\text{,}\) the ring of integers \(\ZZ\) and the polynomial ring \(\FF_q[t]\) are both Euclidean domains and their maximal ideals have finite residue fields. To build out this perspective, let's make the following definition.

Definition 1.2.1.

Fix a finite field \(\FF_q\text{.}\) Let \(K\) be a function field, by which I mean a finite-degree extension of the field of rational functions \(\FF_q(t)\text{.}\) We may then define the ring of integers \(\calO_K\) and the Dedekind zeta function \(\zeta_K(s)\) using exactly the same formulas as in the number field case; the analogue of Dedekind's theorem also holds (with one minor quibble; see Remark 1.2.3). However, there is a key difference: in this case, the residue fields \(\calO_K/\frakp\) all contain \(\FF_q\text{,}\) so \(\zeta_K(s)\) is a power series in \(q^{-s}\) rather than a more general Dirichlet series.

The discussion of places, and the definition of and functional equation for the completed zeta function \(\Lambda_K(s)\text{,}\) also extend to this setting, but again there is a key difference the “infinite places” in the function field setting look just like finite places after a change of coordinates, so there is no need to give a separate definition for the missing factors in the completed zeta function. We will come back to this point in Remark 1.2.3.

The analogue of the Riemann hypothesis for function fields was formulated by Artin. A proof was announced by Weil in 1940 [128], and a second proof in 1941 [129], but due to the precarious state of both Weil's life and world events in that period, the missing details from these announcements did not see print until 1948 [130], [131].

Remark 1.2.3.

The analogue of the Riemann zeta function here is the Dedekind zeta function for \(K = \FF_q(t)\text{,}\) which one may easily calculate to be

\begin{equation*} \zeta_K(s) = \frac{1}{1-q^{1{-}s}} \end{equation*}

(see for example Definition 1.3.1 below). In this case, \(\zeta_K(s)\) has no zeroes at all, so the Riemann hypothesis holds for particularly trivial reasons. Note however that \(\zeta_K(s)\) has poles not only at \(s=1\text{,}\) but also at \(s = 1 + 2 \pi i n/\log q\) for any \(n \in \ZZ\text{.}\) The completed zeta function is

\begin{equation*} \Lambda_K(s) = \frac{1}{(1-q^{-s})(1-q^{1-s})}. \end{equation*}

For a general function field \(K\text{,}\) we will have

\begin{equation*} \zeta_K(s) = \frac{\textrm{ polynomial in }q^{-s}}{1-q^{1{-}s}} \end{equation*}

and

\begin{equation*} \Lambda_K(s) = \frac{\textrm{ polynomial in }q^{-s}}{(1-q^{-s})(1-q^{1-s})}. \end{equation*}

From a modern point of view, we see that the properties of \(\zeta_K(s)\) and \(\lambda_K(s)\) amount to concrete statements about points on a certain algebraic curve over a finite field, whose proofs rely on now-standard techniques in algebraic geometry. For example, the proof of the functional equation for \(\Lambda_K(s)\) uses the Riemann–Roch theorem for curves; Weil's first proof of the Riemann hypothesis uses the embedding of a curve in its Jacobian variety; and Weil's second proof uses the Hodge index theorem on the product of a curve with itself over the base field.

What one should keep in mind here is that none of this perspective was available to Weil. At the time he began his work, the subject of algebraic geometry only included varieties over the complex numbers; many of its best results did not comport with modern standards of rigor; commutative algebra had not yet developed to the point where it could be used to plug some of the gaps; and the key insights of Zariski, Serre, and Grothendieck needed to adapt sheaf theory into the modern foundations of algebraic geometry still lay years in the future (and would take Weil's work, and the Weil conjectures, as a primary impetus). As a result, the completion of Weil's announcements was delayed not just by geopolitical events, but also by Weil's need to build interim foundations on which to base his work. While these foundations are no longer in widespread use, and modern accounts of Weil's work typically reformulate his arguments using the theory of schemes, these reformulations are considered translations rather than completions.

Section 1.3 Reformulation in geometric language

In light of Remark 1.2.3, we now take the next step and reformulate the previous discussion in the language of algebraic geometry.

Definition 1.3.1.

For \(K\) a function field, let \(X\) be the normalization of \(\AAA_{\FF_q}^1\) in \(K\) and let \(X^\circ\) be the set of closed points in \(K\text{.}\) Then we have

\begin{equation} \zeta_K(s) = \prod\limits_{P \in X^\circ}\frac{1}{1-\#\kappa(P)^{-s}} = \prod\limits_{P \in X^\circ}\frac{1}{1-q^{-d_Ps}}\tag{1.3.1} \end{equation}

where \(\kappa(P)\) denotes the residue field of \(P\) and \(d_P = [\kappa(P):\FF_q]\text{.}\) This can be rearranged to

\begin{equation*} \zeta_K(s) = \exp\Big{(}\sum\limits_{n=1}^\infty \frac{q^{-ns}}{n}\#X(\FF_{q^n})\Big{)} \end{equation*}

For example, for \(K = \FF_q(t)\text{,}\) \(X = \AAA_{\FF_q}^1\) and so \(X(\FF_{q^n}) = q^n\) for all \(n\text{;}\) this recovers our earlier formula for \(\zeta_K(s)\) in this case.

Remark 1.3.2.

In the language of schemes, the previous discussion also applies in the case where \(K\) is a number field, taking \(X = \Spec(\calO_K)\) to be the normalization of \(\Spec( \ZZ)\) in \(K\text{.}\) In particular, (1.3.1) carries over.

Definition 1.3.3.

Following Weil, we now let \(X\) be an algebraic variety over \(\FF_q\) (or in modern language, a scheme of finite type over \(\FF_q\)) and define the Hasse–Weil zeta function \(\zeta_X(s)\) as in (1.3.1):

\begin{equation*} \zeta_X(s) \colonequals \prod\limits_{P \in X^\circ}\frac{1}{1-\#\kappa(P)^{-s}} = \prod\limits_{P \in X^\circ}\frac{1}{1-q^{-d_Ps}} = \exp\Big{(}\sum\limits_{n=1}^\infty \frac{q^{-ns}}{n}\#X(\FF_{q^n})\Big{)}. \end{equation*}

We then ask whether \(\zeta_X(s)\) shares any of the previously observed properties when \(\dim(X) > 1\text{.}\) To get some clarity on this question, we consider some examples.

Example 1.3.4.

Let \(X = \PP_{\FF_q}^n\text{.}\) Then

\begin{equation*} \zeta_X(s) = \frac{1}{(1-\tau)(1-q\tau)\cdots(1-q^n\tau)}, \quad \tau = q^{-s} \end{equation*}

(see Exercise 19.2.2).

Section 1.4 A key example

We now consider a key example of Weil. At this point, our chain of inquiry, which so far has flowed naturally from the Riemann zeta function, links up with another thread from elementary number theory.

Example 1.4.1.

Consider the diagonal hypersurface (or Fermat hypersurface)

\begin{equation*} a_0x_0^{n_0} + a_1x_1^{n_1}+\cdots + a_rx_r^{n_r} = b \qquad (n_i > 0, a_i,b \in \FF_q). \end{equation*}

Over \(\QQ\text{,}\) rational points on varieties of this form were considered by Fermat, Euler, and others. The finite field case was considered first by Gauss in the setting where \(q = p\) is prime, \(r = 2\text{,}\) and \(n_0,n_1,n_2\) are small. For example, Gauss proved that for \(p \neq 2\text{,}\) the equation \(x^2-y^2 = 1\) has \(p-1\) solutions in \(\FF_p\text{.}\)

Definition 1.4.2.

Let \(p\) be the characteristic of the finite field \(\FF_q\text{.}\) Let \(\chi\colon\FF_q^\times \to \CC^\times\) be a nontrivial multiplicative character and \(\psi\colon \FF_q\stackrel{\Trace}{\longrightarrow}\FF_p \to \CC^\times\) be an additive character (where \(\FF_p \to \CC^\times\) is the map \(x \mapsto e^{2 \pi i x/p})\text{.}\) The Gauss sum \(g(\chi)=g(\chi,\psi)\) associated to \(\chi\) is given by

\begin{equation*} g(\chi) \colonequals \sum\limits_{x \in \FF_q}\chi(x)\psi(x) \in \overline{\ZZ} \subset \CC. \end{equation*}

(Here \(\overline{\ZZ}\) denotes the integral closure of \(\ZZ\) in \(\CC\text{,}\) i.e., the ring of algebraic integers. We have \(g(\chi) \in \overline{\ZZ}\) because \(g(\chi)\) is a sum of roots of unity.)

Remark 1.4.3.

In Definition 1.4.2, if we were to allow \(\chi\) to be the trivial character, we would have \(g(\chi) = 0\text{.}\) (There exists at least one \(y \in \FF_q^\times\) for which \(\psi(y) \neq 1\text{,}\) and by a simple variable subsitution we have \(g(\chi) = \psi(y) g(\chi)\text{.}\))

Note that the conjugates of \(g(\chi)\) are all themselves Gauss sums for other characters for the same \(q\text{;}\) consequently, \(g(\chi)\) is an algebraic integer all of whose conjugates in \(\CC\) have absolute value \(\sqrt{q}\text{.}\)

By combining this with the Davenport–Hasse relation, we see that if we fix the hypersurface and count points over \(\FF_q^v\) for varying \(v\text{,}\) the answer is of the form \(\sum\limits_{i}\pm \alpha_i^v\text{.}\) This forms a prototype for the Weil conjectures, to be introduced in Chapter 2.

Remark 1.4.6.

There is a similar-looking but different construction that also carries the name zeta function. Namely, for \(X\) a scheme of finite type over \(\ZZ\) and \(p\) a prime, the Igusa zeta function is defined by counting the points of \(X\) valued not in finite fields, but in the rings \(\ZZ/p^n \ZZ\text{.}\) It was introduced by Igusa [64], [65]; see also [66].