Readings 7.0.1.
Our presentation of the Honda–Tate theorem follows [127]. For a discussion of numbers of points on curves over finite fields extending Remark 7.1.12, see [126].
Chapter 7 Inverse problems for zeta functions
Original lecture date: October 21, 2019.
For any fixed \(g\) and \(q\text{,}\) the Weil conjectures imply that there are only finitely many rational functions that can occur as the zeta function of a curve of genus \(g\) over \(\FF_q\text{,}\) or of an abelian variety of dimension \(g\) over \(\FF_q\text{.}\) (See Exercise 19.6.5.)
In this lecture, we discuss the “inverse problems” of which zeta functions actually occur. One can give a relatively complete answer for abelian varieties; for curves this is much more subtle.
Section 7.1
Definition 7.1.1.
To alleviate confusion, let us fix a bit of terminology here. Let \(A\) be an abelian variety over a finite field \(k\text{,}\) and let \(F\colon A \to A\) be the Frobenius map. We refer to the characteristic polynomial \(\det(T-F, T_\ell(A))\) of \(F\) (for any prime \(\ell\) nonzero in \(k\)) as the Weil polynomial of \(A\text{,}\) and the reverse characteristic polynomial \(\det(1-FT, T_\ell(A))\) as the \(L\)-polynomial of \(A\text{.}\) The latter coincides with the factor \(P_1(A, T)\) of the zeta function \(Z(A, T)\text{.}\)
Definition 7.1.2.
An isogeny of abelian varieties \(A_1, A_2\) over a field \(k\) is a finite \(k\)-linear morphism \(f\colon A_1 \to A_2\) which is a homomorphism of group varieties. Any such morphism is surjective with finite (scheme-theoretic) kernel.
For any prime \(\ell\) nonzero in \(k\text{,}\) the induced map \(T_\ell (A_1) \to T_\ell (A_2)\) is itself an isogeny (it becomes an isomorphism after inverting \(\ell\)). As we observed earlier, this implies that the characteristic polynomials of Frobenius of isogenous abelian varieties over a finite field coincide. Amazingly, this result has a converse; see Theorem 7.1.4 below.
Before stating the converse, we state (without further discussion) the key result that goes into its proof.
Theorem 7.1.3. Tate.
Let \(A_1, A_2\) be abelian varieties over a finite field \(k = \FF_q\text{.}\) Let \(\ell\) be a prime nonzero in \(k\text{.}\) Then the map
\begin{equation*}
\ZZ_{\ell}\otimes_{\ZZ}\Hom(A_1, A_2)\rightarrow \Hom_{\ZZ_{\ell}}(T_{\ell}(A_1),T_{\ell}(A_2))
\end{equation*}
from Theorem 6.0.4 (which is injective without any condition on \(k\)) becomes a bijection if we restrict to \(G_k\)-equivariant maps on the right-hand side.
Theorem 7.1.4. Tate.
Let \(A_1, A_2\) be abelian varieties over a finite field \(k = \FF_q\text{.}\) Then the following conditions are equivalent.
-
\(A_1\) and \(A_2\) are isogenous.
-
The Weil polynomials of \(A_1, A_2\) coincide.
-
The \(L\)-polynomials of \(A_1, A_2\) coincide.
-
For each positive integer \(n\text{,}\) \(\#A_1(\FF_{q^n}) = \#A_2(\FF_{q^n})\text{.}\) (As noted earlier, this does not guarantee that the groups \(A_1(\FF_{q^n})\) and \(A_2(\FF_{q^n})\) are in fact isomorphic.)
Proof.
Note that the last condition implies equality of the Weil polynomials thanks to Exercise 19.6.2. Given this, the only issue is to prove that if the \(L\)-polynomials coincide, then \(A_1\) and \(A_2\) are isogenous. Recall that the action of Frobenius on \(T_\ell (A_i) \otimes_{\ZZ} \QQ\) is semisimple, and so is determined up to isomorphism by its characteristic polynomial. Consequently, \(\Hom_{\ZZ_{\ell}[G_k]}(T_{\ell}(A_1),T_{\ell}(A_2))\) contains a map of full rank; by Tate’s theorem, \(\Hom(A_1, A_2)\) must do so also.
Let us spell out this last point in more detail. If we start with an element of \(\Hom_{\ZZ_{\ell}[G_k]}(T_{\ell}(A_1),T_{\ell}(A_2))\) of full rank, any sufficiently close \(\ell\)-adic approximation \(\varphi \in \Hom(A_1, A_2)\) also has full rank as an element of \(\Hom_{\ZZ_{\ell}[G_k]}(T_{\ell}(A_1),T_{\ell}(A_2))\text{.}\) Now note that the connected part of \(\ker(\varphi)\) has zero rational Tate module, and hence must be the zero abelian variety; similarly, \(\coker(\varphi)\) must be the zero abelian variety. We conclude that \(\varphi\) is an isogeny.
Remark 7.1.5.
When Theorem 7.1.4 implies the existence of an isogeny, it does not give much useful information about how to find the isogeny, or what its degree might be. Indeed, finding an explicit isogeny (or one of a specific form) is a sufficiently (apparently) hard computational problem that it has been proposed as the basis of cryptographic protocols; see for example [41].
Remark 7.1.6.
It is not immediately obvious why Theorem 7.1.3 should depend on \(k\) being finite. Tate’s proof ultimately comes down to the fact that there are only finitely many isomorphism classes of abelian varieties of a given dimension over a fixed finite field.
One can hope to prove a similar theorem for other base fields using more refined finiteness statements. Indeed, one such statement (which takes into account primes of bad reduction) was proved by Faltings, giving his isogeny theorem which extends Theorem 7.1.3 to the case where \(k\) is a number field (see Theorem 9.4.2). This in turn yields an analogue of Theorem 7.1.4 asserting that two abelian varieties over a number field are isogenous if and only if their \(L\)-functions coincide.
At this point, we know that abelian varieties over a finite field are characterized up to isogeny by their Weil polynomials or their \(L\)-polynomials. We next formulate the Honda–Tate theorem which pins down exactly which polynomials can occur. This theorem almost says that every polynomial consistent with the Weil conjectures occurs, but not quite: there are some multiplicity conditions that have to be enforced also.
Theorem 7.1.7. Honda–Tate theorem, part 1.
Fix a positive integer \(g\) and a prime power \(q\text{.}\) Then for every irreducible polynomial \(P(T)\in \ZZ[T]\) such that:
-
\(P(0)=1\text{;}\)
-
\(\deg(P) = 2g\text{;}\)
-
\(P(\frac{1}{qT})=q^{-g}T^{-2g}P(T)\text{;}\)
-
all roots of \(P\) in \(\CC\) have absolute value \(q^{-1/2}\text{;}\)
there exist a positive integer \(e\) and a simple abelian variety \(A\) such that \(P_1(A, T) = P(T)^e\text{.}\)
The integer \(e\) is determined by the rational endomorphism algebra of \(A\text{,}\) which can itself be described explicitly in terms of \(P\text{.}\)
Theorem 7.1.8. Honda–Tate theorem, part 2.
With notation as in Theorem 7.1.7, \(E := \End(A)_\QQ\) is a central simple algebra of degree \(e\) over \(\Phi = \QQ(\pi_A)\text{,}\) where \(\pi_A^{-1}\) is a root of \(P\text{.}\) The invariant of \(E\) at a place \(v\) of \(\Phi\) is given by
\begin{equation*}
\begin{cases}
\frac{1}{2} & \mbox{if is real} \\
[\Phi_v: \QQ_p] \frac{\ord_v(\pi_A)}{\ord_v(q)} & \mbox{if lies over } \\
0 & \mbox{otherwise.}
\end{cases}
\end{equation*}
Note that \(e\) is then the least common denominator of the local invariants.
Corollary 7.1.9.
If the coefficient of \(T^g\) in \(P\) is nonzero modulo \(p\text{,}\) then \(e=1\text{.}\) (This corresponds to the case of an ordinary abelian variety.)
Remark 7.1.10.
We limit ourselves to a few words about the proof of Theorem 7.1.7. The construction starts by using analytic methods to construct a complex torus whose endomorphism ring contains \(\ZZ[\pi_A]\text{.}\) One then introduces a polarization to give this torus the structure of an abelian variety over \(\CC\text{.}\) Since abelian varieties with complex multiplication occur as isolated points in moduli, they are forced to descend to some number field. Reducing modulo a suitable prime, we then get the desired abelian variety except that it might be defined not over \(\FF_q\) but over a finite extension. Finally, we use Tate’s theorem to descend down to \(\FF_q\text{.}\)
It is possible to supplant the use of complex analysis with more arithmetic methods; see [20]. However, the use of characteristic 0 methods to prove this statement, which is formulated exclusively in positive characteristic, remains unavoidable with current techniques.
Remark 7.1.11.
Using the Honda–Tate theorem, one can tabulate isogeny classes of abelian varieties of dimension \(g\) over \(\FF_q\) for small values of \(g\) and \(q\text{.}\) This has been done in the LMFDB [89], using Sage to enumerate Weil polynomials. The theory behind this enumeration is described in [76] and [79].
Remark 7.1.12.
The constraint imposed by the Honda–Tate theorem may be restricted to Jacobians, so it also implies a nontrivial restriction on the zeta function of a curve of genus \(g\) over \(\FF_q\text{.}\) However, for dimensional reasons there are expected to be many fewer zeta functions of curves than zeta function of abelian varieties (once \(g\) is large enough), so it is natural to look for other constraints on zeta functions that are exclusive to curves.
One important set of constraints arises from positivity conditions. For any curve \(X\) over \(\FF_q\) (or indeed any algebraic variety at all), the following conditions obviously hold.
-
We have \(\# X(\FF_q)\geq 0\text{.}\)
-
For all positive integers \(m\) and \(n\text{,}\) \(\# X(\FF_{q^{nm}}) \geq \# X(\FF_{q^m})\text{.}\)
These conditions impose certain “linear programming” constraints which have unexpectedly strong consequences. For example, suppose we want to know the maximum value of \(\#X(\FF_q)\) for \(X\) a curve of a given genus \(g\text{.}\) This question is more than theoretical in light of the Goppa construction: one can construct interesting error-correcting codes by fixing a rational point \(\infty\) and a positive integer \(d\text{,}\) and considering the vectors
\begin{equation*}
\{(f(x)_{x \in X(\FF_q) \setminus \{\infty\}}: f \in K(X), \divis(f) + d \infty \geq 0 \}.
\end{equation*}
The corrective capacity of this code is limited by \(g\) (thanks to Riemann–Roch), so the quality of the code depends on \(\#X(\FF_q)\) being large relative to \(g\text{.}\)
Much is known about optimizing \(\#X(\FF_q)\) for fixed \(q,g\text{;}\) see [93]. However, let us consider instead the asymptotic situation where \(q\) is fixed and \(g \to \infty\text{.}\) For fixed \(q\text{,}\) the Weil conjectures imply
\begin{equation*}
\limsup_{g\to \infty} \frac{\# X(\FF_q)}{g}\leq 2\sqrt{q}
\end{equation*}
In the other direction, it is known that the bound \(\sqrt{q}-1\) is best possible when \(q\) is a square. For \(q\) not a square, it is known that
\begin{equation*}
\limsup_{g\to \infty} \frac{\# X(\FF_q)}{g} \geq c(q) > 0
\end{equation*}
but the optimal value is unknown.
We end with one concrete result with a curious history: the class number 1 problem for function fields.
Theorem 7.1.13.
There are exactly \(8\) isomorphism classes of curves of any genus \(g\) over any finite field \(\FF_q\) for which \(\Jac(X)(\FF_q) = 1\text{.}\) The pairs \((g,q)\) that occur are
\begin{equation*}
(1,2), (1,3), (1,4), (2,2), (2,2), (3,2), (3,2), (4,2).
\end{equation*}
Proof.
It was originally “proved” by Leitzel–Madan–Queen [85] that there are only \(7\) such isomorphism classes, with the case \((g,q) = (4,2)\) omitted; the list of these had previously been obtained by Madan–Queen [91]. Much later, it was discovered by Stirpe [118] that the case \((g,q) = (4,2)\) actually does occur. The completeness of the list as given above is due independently to Mercuri–Stirpe [95] and Shen–Shi [115].
