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Chapter 6 RH for abelian varieties

Original lecture date: October 16, 2019.

In this lecture, we discuss Weil's first proof of the Riemann hypothesis for curves, and the Weil conjectures for abelian varieties.

Readings 6.0.1.

We follow the presentation of Weil's proof given in [98]. For background on abelian varieties, see [103].

We begin by discussing Weil's construction of the Jacobian of a curve.

Definition 6.0.2.

Let \(X\) be a smooth, projective, geometrically irreducible curve of genus \(g\text{.}\) Consider the symmetric product \(\Sym^g X\text{.}\) We want to obtain an abelian variety from the symmetric product. One begins by fixing a point on \(\Sym^g X\) and constructing rational maps \(m:\Sym^g X\times \Sym^g X\dashrightarrow \Sym^g X\) and \(i:\Sym^g X\dashrightarrow \Sym^g X\text{,}\) which serve as a birational multiplication law and inverse law, respectively. Weil proved that any such set up as above is birational to a genuine group variety, the Jacobian \(\Jac(X)\text{.}\) The construction of the multiplication law above, for example, comes from Riemann–Roch. A more modern way to view Jacobians is as the moduli space of degree \(0\) line bundles on a curve (as discussed in a previous lecture).

We want to study the action of Frobenius on the Jacobian. It makes sense to more generally discuss endomorphisms on abelian varieties. We will want to study the action of Frobenius, or more generally any endomorphism, on the Tate module of an abelian variety.

Definition 6.0.3.

Let \(A\) be an abelian variety over a field \(k\text{.}\) Let \(\ell\) be a prime nonzero in \(k\text{.}\) The \(\ell\)-adic Tate module \(T_\ell(A)\) is defined as

\begin{equation*} T_\ell(A):=\varprojlim A(\overline{k})[\ell^n]. \end{equation*}

The rational \(\ell\)-adic Tate module \(V_\ell(A)\) is the base extension \(T_\ell(A) \otimes_{\ZZ_\ell} \QQ_\ell\text{.}\)

The Tate module is a free rank-\(2g\) module over the \(\ell\)-adic integers \(\ZZ_\ell\) (this will follow from Example 6.0.8 below), and it comes equipped with an action of the absolute Galois group of \(k\text{.}\) It records information about endomorphisms faithfully, in the following sense.

See [103], \S 18, Theorem~3.

For \(\alpha\in \End(A)\text{,}\) the characteristic polynomial of \(\alpha\) acting on the Tate module \(T_{\ell}(A)\) is of degree \(2g\) as a polynomial over \(\ZZ_\ell\text{.}\)

By the Cayley–Hamilton theorem, the minimal polynomial of \(\alpha\) kills \(\alpha\) in \(\ZZ_{\ell}\otimes_{\ZZ}\End(A)\text{.}\) From the injection

\begin{equation*} \End(A)\hookrightarrow \ZZ_{\ell}\otimes \End(A), \end{equation*}

one sees that the linear dependence relation between \(1,\alpha,\dots,\alpha^{2g-1}\) coming from the Cayley–Hamilton theorem provides a linear dependence relation in \(\End(A)\text{.}\) Because \(\End(A)\) is a finite free \(\ZZ\)-module, the minimal polynomial and characteristic polynomial of \(\alpha\) on the Tate module \(T_{\ell}(A)\) are both actually polynomials defined over the integers.

We will now use this result to obtain information about the number of rational points on an abelian variety over a finite field.

Definition 6.0.6.

Let \(A\) be an abelian variety over a finite field \(k = \FF_q\text{,}\) and let \(F:A\rightarrow A\) be the Frobenius endomorphism over \(\FF_q\text{.}\) Then one can see easily that \(A(k)=A[1-F]\text{,}\) where the notation \(A[\alpha]\) denotes the kernel of the endomorphism \(\alpha\) (the key point is that \(A[1-F]\) is reduced, which is an easy local calculation).

Hence to count \(A(k)\text{,}\) we want to understand the kernel of \(1-F\text{.}\) We do this using the degree function.

Definition 6.0.7.

Let \(\calL\) be a symmetric ample line bundle on \(A\text{.}\) The degree map \(\deg\colon \End(A)\rightarrow \QQ\) is given by the formula

\begin{equation*} \alpha\mapsto c_1(\alpha^*\calL)^g/c_1(\calL)^g, \end{equation*}

where \(c_1\) denotes the first Chern class.

If \(\alpha\) is an endomorphism with finite kernel (i.e., an isogeny from \(A\) to itself), then \(\alpha\) defines a finite morphism \(A \to A\text{,}\) and the degree of this morphism is the same as the quantity \(\deg(\alpha)\) defined above. It is also equal to the \(k\)-length of the scheme-theoretic kernel \(A[\alpha]\text{,}\) which agrees with the number of geometric points of the kernel if \(\alpha\) is separable. (For example, \(\alpha = 1-F\) is always separable.)

On the other hand, if \(\alpha\) does not have finite kernel, then from dimensional considerations one may see that \(\deg(\alpha) = 0\text{.}\)

Example 6.0.8.

For any positive integer \(n\text{,}\) using the theorem of the cube we may see that \([n]^* \calL \cong \calL^{n^2}\text{.}\) Therefore \(\deg([n]) = n^{2g}\text{.}\)

As in Example 6.0.8, we see that \(\deg(n\alpha) = n^{2g} \deg(\alpha)\) for any \(\alpha \in \End(A)_\QQ\text{.}\) Hence to prove that \(\deg\) is a polynomial of degree \(2g\text{,}\) it will suffice to show that for any fixed \(\alpha, \beta \in \End(A)\text{,}\) \(\deg(\alpha + n\beta)\) is a polynomial in \(n\) of degree at most \(2g\text{.}\) This again can be deduced using the theorem of the cube; we omit the details.

We now know that both quantities are polynomials of degree \(2g\) on \(\End(A)\text{.}\) To compare them, we first examine what happens on \(\ell\)-power torsion to see that

\begin{equation*} \left| \det(\alpha, T_\ell(A)) \right|_\ell = \left| \deg(\alpha) \right|_\ell \qquad (\alpha \in \End(A)). \end{equation*}

In particular, for any fixed \(\alpha \in \End(A)\text{,}\)

\begin{equation*} \left| \det(F(\alpha), T_\ell(A)) \right|_\ell = \left| \deg(F(\alpha)) \right|_\ell \qquad (F \in \ZZ[T]). \end{equation*}

By an elementary argument (see Exercise 19.6.4), this is enough to deduce that \(\det(\alpha, T_\ell A) = \deg(\alpha)\text{.}\)

In particular, \(\#A(\FF_{q^n})=(1-r_1^n)\dots(1-r_{2g}^{n})\) where \(r_1,\dots,r_{2g}\) are the roots of the characteristic polynomial of Frobenius acting on \(T_{\ell}(A)\) for any \(\ell\text{.}\) It follows (see Exercise 19.3.3) that \(Z(A,T)\) is of the form

\begin{equation*} \frac{P_1(T)\dots P_{2g-1}(T)}{P_0(T)\dots P_{2g}(T)}, \end{equation*}

where \(P_1(T)=(1-r_1 T)\dots(1-r_{2g}T)\) and \(P_i(T)=\wedge^i P_1(T)\text{.}\)

Example 6.0.11.

If \(A\) is the product of the abelian varieties \(A_1\) and \(A_2\text{,}\) then \(\#A(\FF_{q^n}) = \#A_1(\FF_{q^n}) \#A_2(\FF_{q^n})\) for all \(n\text{.}\) From this, it follows that the polynomial \(P_1\) for \(A\) is the product of the polynomials \(P_1\) for \(A_1\) and \(A_2\text{.}\)

Remark 6.0.12.

From the interpretation of \(\#A(\FF_{q^n})\) as \(\deg(1-F^n)\text{,}\) we see that it is invariant under isogeny, as then is the whole zeta function. Beware however that the structure of the abelian group \(A(\FF_{q^n})\) is not an isogeny invariant.

This leaves the matter of the Riemann hypothesis. As in Weil's second proof of the Riemann hypothesis for curves, the key is a positivity assertion, here given in terms of the Rosati involution.

Definition 6.0.13.

Let \(\lambda: A\rightarrow A^{\vee}\) be an isogeny induced by a polarization of \(A\text{,}\) and define \(\End(A)_\QQ \colonequals \End(A)\otimes_{\ZZ} \QQ\text{.}\) The Rosati involution is the map \(\dagger\colon \End(A)_\QQ \rightarrow \End(A)_\QQ\) given by

\begin{equation*} \alpha\mapsto \lambda^{-1}\circ \alpha^{\vee}\circ \lambda\colonequals\alpha^{\dagger}. \end{equation*}

For \(\calL\) an ample line bundle defining the polarization, we obtain

\begin{equation*} \Trace(\alpha\circ\alpha^{\dagger})=\frac{2g}{\calL^g}(\calL^{g-1}\cdot \alpha^{*}\calL). \end{equation*}

This number is positive for \(\alpha \neq 0\) because \(\calL\) is ample.

Let \(F: A \to A\) again be the Frobenius endomorphism of an abelian variety \(A\) over \(k = \FF_q\text{.}\) Using the Weil pairing (or a more direct calculation), one may calculate that \(F^\dagger \circ F = [q]\text{.}\) In particular the ring \(\QQ[F] \subset \End(A)\) is stable under \(\dagger\text{,}\) and hence must be semisimple. Hence \(\QQ[F] \otimes_\QQ \RR\) must also be semisimple, meaning that it is a product of copies of \(\RR\) and \(\CC\text{;}\) the action of \(\dagger\) extends to \(\QQ[F] \otimes_\QQ \RR\) and (in order to obey positivity) must fix each copy of \(\RR\) and conjugate each copy of \(\CC\text{.}\) Each eigenvalue \(r\) of \(\CC\) appears as the image of \(F\) in one of the factors, and the image of \(F^\dagger\) in the same factor is \(\overline{r}\text{;}\) hence \(\left| r^2 \right| = q\text{.}\)

As a corollary, we may now recover the Riemann hypothesis for a curve over \(k\text{,}\) by applying the previous theorem to its Jacobian.

Remark 6.0.16.

Alternatively, one can go the other way and deduce the Riemann hypothesis for abelian varieties from the corresponding statement for curves. This seems not obvious at first, because an arbitrary abelian variety \(A\) over \(k\) is not isomorphic, or even isogenous, to a Jacobian. However, if we take \(X\) to be a transverse intersection of \(\dim(A)-1\) ample divisors (see Remark 6.0.17), then \(\Jac(X)\) maps surjectively onto \(A\text{,}\) and is in fact isogenous to the product of \(A\) with the kernel \(A'\) of the map. As per Example 6.0.11, the polynomial \(P_1\) for \(\Jac(X)\) factors as the product of the \(P_1\) polynomials for \(A\) and \(A'\text{,}\) so the Riemann hypothesis for \(X\) does imply the Riemann hypothesis for \(A\text{.}\)

Remark 6.0.17.

In the previous remark, it is not immediately obvious that a transverse intersection exists because we are working over a finite field; the assertion of the Bertini smoothness theorem that a “generic” intersection is transverse is of no value. However, one may apply Bertini over \(\overline{k}\) and then just make a suitable base extension; the latter does not affect the proof of the Riemann hypothesis. Alternatively, one can find such an intersection defined over \(k\) by using a probabilistic adaptation of the Bertini smoothness theorem to finite fields given by Poonen [106], or more directly a further adaptation by Bucur–Kedlaya [18] that directly address complete intersections rather than single hypersurface sections. The latter can be used to give a good bound on the genus of the curve \(X\) in terms of the dimension of \(A\text{;}\) see [14].