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Chapter 4 Curves and abelian varieties

Original lecture date: October 9, 2019.

In this lecture, we study zeta functions for curves and abelian varieties.

Readings 4.0.1.

We follow [90], Chapters VIII–IX. For background on abelian varieties, see also [103].

Section 4.1 Divisors on curves

Definition 4.1.1.

Throughout this lecture, let \(X\) be a geometrically irreducible smooth projective curve of genus \(g\) over the finite field \(k=\FF_q\) of characteristic \(p\text{.}\) The field of rational functions \(k(X)\) is finite over \(k(t)\) for any element \(t \in k(X)\) which is not in \(k\) (or equivalently, which is not integral over \(k\text{;}\) note that the geometrically irreducible condition implies that \(k\) is integrally closed in \(k(X)\)).

Let \(\Div(X)\) be the free abelian group generated by the closed points \(X^\circ\) of \(X\text{;}\) the elements of \(\Div(X)\) are called divisors on \(X\text{.}\) We have a degree map

\begin{align*} \deg\colon \Div(X)&\longrightarrow \ZZ\\ \sum a_i[P_i]& \longmapsto \sum a_i [\kappa(P_i):k] \end{align*}

where \(\kappa(P)\) denotes the residue field of \(P\text{.}\) A divisor is called effective if it is a nonnegative linear combination of closed points; the degree of an effective divisor is also nonnegative.

Denote \(\Div^{0}(X) := \deg^{-1}(0)\text{.}\) Then for \(f\in k(X)^\times\text{,}\) the divisor

\begin{equation*} \divis(f) := \sum_{P \in X^\circ} \ord_P(f) [P] \end{equation*}

associated to \(f\) belongs to \(\Div^0(X)\text{,}\) hence

\begin{equation*} \Pic^0(X) := \coker(\divis: k(X)^\times \to \Div^0(X)) \end{equation*}

is well-defined.

Remark 4.1.2.

In what follows, it is helpful to bifurcate the discussion based on whether or not \(X(k) = \emptyset\text{.}\) For an example with \(X(k) = \emptyset\text{,}\) take the genus-2 curve

\begin{equation*} y^2 = 2x^6 - 2x^2 + 2 \end{equation*}

over \(\FF_3\text{.}\) (Note: it is impossible to have \(X(k) = \emptyset\) for a curve of genus 1 over a finite field; see Exercise 19.6.1.)

Suppose now that \(X(k) \neq \emptyset\text{;}\) then the degree map \(\deg: \Div(X) \to \ZZ\) is evidently surjective. Specifically, if we fix a choice of \(O \in X(k)\text{,}\) we can define a map

\begin{align*} \cl: \text{Effective divisors of degree } d &\longrightarrow \operatorname{Pic}^{0}(X)\\ D&\longmapsto [D-dO] \end{align*}

which is surjective. For \(d \geq 2g-1\text{,}\) each fiber has order \(\frac{q^{d-g+1}-1}{q-1}\) for \(d\geq 2g-1\text{;}\) this follows from the Riemann–Roch theorem, which implies that \(h^{0}(X, \calL)=\deg(\calL)-g+1\) for a line bundle \(\calL\) with \(\deg(\calL)\geq 2g-1\text{.}\) (We will use the full strength of Riemann–Roch a bit later.)

Now write

\begin{equation*} Z(X, T)=\prod_{x \in X^{\circ}} \frac{1}{1-T^{\deg(x)}}=\sum_{D \geq 0} T^{\deg(D)} \end{equation*}

where the last sum is over the effective divisors \(D\) on \(X\) (this is analogous to the equality between the sum and product representations of a Dedekind zeta function). Breaking this sum into two parts according to whether \(\mathrm{deg}(D)\geq 2g-1\) or \(\mathrm{deg}(D)\leq 2g-1\) leads to the following proposition.

Remark 4.1.4.

The equality \(f(1) = \# \Pic^0(X)\text{,}\) which crucially implies that \(f\) does not have a zero at \(T=1\text{,}\) is analogous to a property of Dedekind zeta functions which we did not comment on earlier. For \(K\) a number field, the residue of \(\zeta_K(s)\) at \(s=1\) (where the function has a simple pole) is given by the class number formula. It includes factors coming from the class number of \(\calO_K\) and the regulator of the unit lattice of \(K\text{.}\) In this context, there are no infinite places and so we see only a class number contribution.

Let us now see about getting rid of the condition that \(X(k) \neq \emptyset\text{.}\) Obviously \(X\) has points over some finite extension of \(k\text{,}\) so let us try passing from \(X\) to its base extension \(X_{\FF_{q^n}}\) for some positive integer \(n\) chosen so that \(X(\FF_{q^n})\neq \emptyset\text{.}\) We can then try to recover information about \(X\) using the identity

\begin{equation*} Z\left(X_{\FF_{q^n}}, T^{n}\right)=\prod_{i=0}^{n-1} Z\left(X, \zeta_n^{i} T\right) \end{equation*}

where \(\zeta_n\) is a primitive \(n\)-th root of unity.

However, there is a strict loss of information between \(Z(X,T)\) and \(Z\left(X_{\FF_{q^n}}, T\right)\text{,}\) even for curves.

Example 4.1.5.

If \(Z(X_1,T)=\frac{1-aT+qT^2}{(1-T)(1-qT)}, ~~Z(X_2,T)=\frac{1+aT+qT^2}{(1-T)(1-qT)}\) then \(Z\left(X_{1,\FF_{q^2}}, t\right)=Z\left(X_{2,\FF_{q^2}}, t\right)\text{.}\) This occurs when \(X_1\) is an elliptic curve and \(X_2\) is a quadratic twist; to make this explicit (assuming \(p > 2\)), let \(X_1\) be a curve of the form

\begin{equation*} y^2 = x^3 + a x^2 + bx + c \end{equation*}

and let \(X_2\) be the curve

\begin{equation*} dy^2 = x^3 + a x^2 + bx + c \end{equation*}

where \(d\) is a nonsquare in \(\FF_q^\times\text{.}\)

A key observation is that the previous proof in the case \(X(\FF_q)\neq \emptyset\) only relies on the surjectivity of the degree map. Hence if could show such surjectivity always hold (without assuming \(X(\FF_q)\neq \emptyset\)), then we do not have to worry about the existence of \(O\in X(\FF_q)\text{.}\) Fortunately, this is the case.

Since the degree map is clearly nonzero, we have \(\deg(\Pic(X))=e \ZZ\) for some positive integer \(e\text{.}\) Let us again compute \(Z(X,T) = \sum_{D \geq 0} T^{\deg(D)}\) by breaking the sum in two as before; the second sum then runs over \(T^{de}\) with \(d\geq d_0\text{,}\) where \(d_0\) is the smallest integer such that \(d_0e\geq 2g-1\text{.}\) As a result, we have

\begin{equation*} Z(X, T)=\frac{f\left(t^{e}\right)}{\left(1-T^{e}\right)\left(1-q^{e} T^{e}\right)} \end{equation*}

and \(f(1)=\# \Pic^{e}(X) \neq 0\text{.}\) In particular, \(Z(X, T)\) has a pole of order \(1\) at \(T=1\text{.}\)

The same logic applies also to \(X_{\FF_{q^e}}\text{,}\) so \(Z\left(X_{\FF_{q^e}}, T\right)\) has a pole of order \(1\) at \(T=1\text{.}\) As a result, \(Z\left(X_{\FF_{q^e}}, T^{e}\right)\) has a pole of order \(1\) at \(T=1\text{.}\) On the other hand,

\begin{equation*} Z\left(X_{\FF_{q^e}}, T^{e}\right)=\prod_{i=0}^{e-1} Z\left(X, \zeta_e^{i} T\right)=Z\left(X, T\right)^e. \end{equation*}

Comparing the pole orders at \(T=1\text{,}\) we deduce that \(e=1\text{,}\) which finishes the proof.

Remark 4.1.7.

Using the full strength of the Weil conjectures, one can prove more: for any fixed \(X\text{,}\) we have \(X(\FF_{q^n}) \neq \emptyset\) for every sufficiently large \(n\text{.}\) See Exercise 19.6.3.

Given Proposition 4.1.6, we can now reprise the proof of Proposition 4.1.3 to deduce the following.

Note that we currently only know that \(\deg(f) \leq 2g\text{,}\) whereas we expect equality. To resolve this, we must prove the functional equation using the Riemann–Roch theorem.

Write \((q-1) Z(X,T)\) as a sum of two terms:

\begin{align*} \alpha(T) &\colonequals \sum_{0 \leq \deg(\calL) \leq 2g-2} q^{h^0(\calL)} T^{\deg(\calL)} \\ \beta(T) &\colonequals \sum_{\deg(L) \geq 2g-1} q^{h^0(\calL)} T^{\deg(\calL)} - \sum_{\deg(\calL) \geq 0} T^{\deg(\calL)}. \end{align*}

We will prove that each of these satisfies the same functional equation that we desire for \(Z(X,T)\text{.}\) For \(\beta(T)\text{,}\) using the weak form of Riemann–Roch used earlier, we obtain

\begin{equation*} \beta(T) = \# \Pic^0(X) \left( \frac{q^g T^{2g-1}}{1-qT} - \frac{1}{1-T} \right) \end{equation*}

and the functional equation is clear. To analyze \(\alpha(T)\text{,}\) we must use Riemann–Roch at full strength: for \(\Omega\) the sheaf of Kähler differentials on \(X\) and \(\calL\) any line bundle on \(X\text{,}\)

\begin{equation*} h^0(X, \calL) = \deg(\calL) + 1 - g + h^0(\Omega \otimes \calL^{-1}). \end{equation*}

Since \(\deg(\Omega) = 2g-2\text{,}\) we may rewrite \(\alpha(T)\) by substituting \(\Omega \otimes \calL^{-1}\) for \(\calL\text{.}\) Using Riemann–Roch, we then obtain

\begin{align*} \alpha(T) &= \sum_{0 \leq \deg(\calL) \leq 2g-2} q^{h^0(\Omega \otimes \calL^{-1})} T^{\deg(\Omega \otimes \calL^{-1})} \\ &= \sum_{0 \leq \deg(\calL) \leq 2g-2} q^{h^0(\calL)-\deg(\calL)-1+g} T^{2g-2-\deg(\calL)} \end{align*}

and again read off the desired functional equation.

Remark 4.1.10.

We will show a bit later, using the Riemann–Roch theorem, that \(Z(X,T)\) satisfies the functional equation; this will also show that \(\deg(f) = 2g\text{.}\) One can also establish the Riemann hypothesis in this framework, but we postpone this to a later lecture.

Section 4.2 Between curves and abelian varieties

In the remainder of this lecture, we describe (without proofs) the relationship between curves and abelian varieties, and between the Weil conjectures in these two cases.

Definition 4.2.1.

An abelian variety over a field \(k\) is a smooth, projective, geometrically connected \(k\)-scheme equipped with a commutative group structure. It turns out that the commutativity hypothesis is superfluous; see [103].

Example 4.2.2.

Elliptic curves over \(k\) are abelian varieties of dimension 1. Products of elliptic curves give examples of higher-dimensional abelian varieties.

Definition 4.2.3.

Given a curve of genus \(g\text{,}\) there are two different constructions giving rise to a \(g\)-dimensional abelian variety.

  • The Albanese construction:

    \begin{equation*} \mbox{pointed curve $X/k$ of genus $g$} \leadsto \Alb(X) \end{equation*}
    This is a covariant functor, and comes with a (functorial) map \(X \to \Alb(X)\) sending the marked point to the identity. This map does not factor through any abelian subvariety of \(\Alb(X)\text{,}\) and induces a homomorphism
    \begin{equation*} \Div^{0}(X) \rightarrow \Alb(X)(k) \end{equation*}
    which factors through \(\Pic^{0}(X)\text{.}\)

  • The Picard construction:

    \begin{equation*} \mbox{curve $X/k$ of genus $g$} \leadsto \underline{\Pic}^0(X) \colonequals \mbox{Moduli space of degree-$0$ line bundles on $X$}. \end{equation*}
    This is a contravariant functor. The following universal property holds: maps from an abelian variety \(S\) over \(k\) to \(\underline{\Pic}^0(X)\) correspond to line bundles on \(S\times_k X\) whose restriction to every fiber \(s \times X\) has degree 0.

Remark 4.2.4.

These two construction are related by the Abel–Jacobi map:

\begin{equation*} \Alb^{\vee}(X)\cong \underline{\Pic}^0(X) \end{equation*}

where for an abelian variety \(A\text{,}\) the dual variety \(A^{\vee}\) is defined as \(\underline{\Pic}^0(A)\text{.}\) Using the Poincaré bundle, we obtain a natural isomorphism \((A^{\vee})^{\vee} \cong A\text{.}\)

For a general abelian variety \(A\) over \(k\text{,}\) \(A\) and \(A^{\vee}\) need not be isomorphic (although they are necessarily isogenous: there is a morphism between them which is surjective with finite kernel). However, one can construct a principal polarization giving rise to an isomorphism

\begin{equation*} \Alb(X)\cong \underline{\Pic}^0(X). \end{equation*}

Example 4.2.5.

Over \(\CC\text{,}\) every abelian variety arises analytically as a complex torus \(\CC^g/\Lambda\text{.}\) The dual variety is then \((\CC^g/\Lambda)^{\vee}\cong \CC^g/\Lambda^{\vee}\text{,}\) where \(\Lambda^{\vee}:=\{\mu:\Hom_{\RR}(\CC^g,\RR)|~\mu(\Lambda)\subset \ZZ\}\text{.}\)

The zeta functions of a curve \(A\) and its Jacobian \(\Jac(X):=\underline{\Pic}^0(X)\) are related as follows.