Abstract
We reduce the classification of finite extensions of function fields (of curves over finite fields) with the same class number to a finite computation; complete this computation in all cases except when both curves have base field \(\mathbb {F}_2\) and genus \(>1\); and give a conjectural answer in the remaining cases. The conjecture will be resolved in subsequent papers.
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1 Introduction
The relative class number one problem for function fields (of curves over finite fields) is to classify finite extensions for which the relative class number equals 1, or equivalently the class numbers of the two function fields coincide. In this paper, we solve this problem in all cases except where both function fields have base field \(\mathbb {F}_2\), and to reduce that case to a feasible finite computation. This extends work of numerous authors [2,3,4,5,6] but our arguments are independent of these.
For comparison, the relative class number one problem for number fields was formulated by Stark [7] only for CM fields, viewed as totally imaginary quadratic extensions of totally real fields. This restriction is quite natural: outside of this case, the relative unit rank is nonzero and the relative class number behaves erratically (e.g., it is not generally integral). Odlyzko [8] established conditionally on GRH that there are only finitely many CM fields with relative class number one. The complete set of normal CM fields with relative class number one has been determined recently by Hoffman–Sircana [9].
Before continuing, we introduce some terminology and notation. By a function field, we mean the field of rational functions on a curve over some finite field. Given a finite extension \(F'/F\) of function fields, we write \(C, C'\) for the curves corresponding to \(F,F'\); \(q_F, q_{F'}\) for the orders of the base fields of \(C,C'\); \(g_F, g_{F'}\) for the genera of \(C, C'\); and \(h_F, h_{F'}\) for the class numbers of \(F, F'\). We write \(J(C), J(C')\) for the Jacobians of \(C,C'\), so that \(\#J(C)(\mathbb {F}_{q_F}) = h_F\) and \(\#J(C')(\mathbb {F}_{q_{F'}}) = h_{F'}\).
The relative class number \(h_{F'/F}\) is the ratio \(h_{F'}/h_F\); this can be interpreted as the order of a certain finite group (see below), and hence is an integer. This implies the following reduction: for \(E = F \cdot \mathbb {F}_{q_{F'}}\), \(h_{F'/F} = 1\) if and only if \(h_{E/F} = h_{F'/E} = 1\). We may thus focus on the cases where \(F' = E\), in which case we say the extension \(F'/F\) is constant, and where \(E = F\), in which case we say \(F'/F\) is purely geometric.
In the case of a constant extension, the equality \(h_{F'/F} = 1\) holds for trivial reasons when \(F' = F\) and when \(g_F = g_{F'} = 0\) (as in this case \(h_F = h_{F'} = 1\)). Excluding these, we have the following result; see §3 for the proof.
Theorem 1.1
Let \(F'/F\) be a constant extension of degree \(d>1\) of function fields with \(g_{F} > 0\), \(q_{F'} > q_F \), and \(h_{F'/F} = 1\). Then \((q_F, d, g_F, J(C))\) is one of
where J(C) is specified up to isogeny by an LMFDB label.
In the case of a purely geometric extension, the equality \(h_{F'/F} = 1\) holds for trivial reasons when \(F' = F\) and when \(g_F = g_{F'} \in \{0,1\}\). Moreover, when \(g_F \in \{0,1\}\), for any fixed pair of isomorphism classes of F and \(F'\), the existence of a single finite morphism \(F \rightarrow F'\) implies the existence of infinitely many more. It is thus natural to separate the cases \(g_F \le 1\) and \(g_F > 1\); see §6 and §8 for the proofs.
Theorem 1.2
Let \(F'/F\) be a purely geometric extension of degree d of function fields with \(g_F \le 1\), \(g_{F'} > g_F\), and \(h_{F'/F} = 1\). Then \((q_F,g_F, g_F',J(C),J(C'))\) appears in Table 3. (Note that the tuple does not always uniquely determine \(F'\).)
When \(g_F = 0\), Theorem 1.2 recovers the solution of the absolute class number one problem for function fields [10,11,12,13].
Theorem 1.3
Let \(F'/F\) be a purely geometric extension of degree d of function fields with \(g_{F'}> g_F > 1\) and \(h_{F'/F} = 1\).

(a)
If \(q_F > 2\), then \(q_F \in \{3,4\}\), \((g_F, g_F') \in \{(2,3), (2,4), (3,5)\}\), \(F'/F\) is (Galois) cyclic, and \((q_F, g_F, g_{F'}, F)\) appears in Table 4. In each listed case, the tuple uniquely determines \(F'\).

(b)
If \(q_F = 2\), then \(g_F \le 7\) and \(g_{F'} \le 13\). The isogeny classes of J(C) and the Prym variety A (see below) form one of 208 pairs listed in Table 7.

(c)
If \(q_F = 2\), then assuming that \(F'/F\) is cyclic, there are exactly 61 tuples \((d,g_F, g_{F'}, F)\) with \(g_F \notin \{6,7\}\), and at least 3 with \(g_F \in \{6,7\}\); see Tables 5 and 6. In each listed case, the tuple uniquely determines \(F'\).
In Theorem 1.3(c), there are only two cases (3.2.ab_a_c and 5.2.b_c_e_i_i) where F is not uniquely specified by \(d, g_F, g_{F'}, J(C)\). The scarcity of such examples reflects that curves with isogenous Jacobians can typically be distinguished by the Lfunctions of their abelian covers [14, 15].
By our earlier reduction, we recover the following corollary.
Corollary 1.4
Let \(F'/F\) be an extension of degree d of function fields with \(g_{F'} > g_F\) and \(h_{F'/F} = 1\) which is neither constant nor purely geometric. Then \(q_F = 2\), \(q_{F'} = 4\), and \((g_F, g_{F'}, J(C), J(C')) \in \{(0,1,0, 1.4.ae), (1,2,1.2.c, 2.4.ae\_i)\}\).
We now summarize the techniques used to prove Theorems 1.1, 1.2, and 1.3. The extension \(F'/F\) induces an injective morphism f from J(C) to the Weil restriction of \(J(C')\) from \(\mathbb {F}_{q_{F'}}\) to \(\mathbb {F}_{q_F}\), and \(h_{F'/F}\) can be interpreted as the order of the group \(A(\mathbb {F}_{q_F})\) where A is the cokernel of f; we call A the Prym variety of the covering \(C' \rightarrow C\). We restrict options for C and \(C'\) using the structure of simple abelian varieties of order 1 over \(\mathbb {F}_q\): for \(q \ge 5\) there are none; for \(q = 3,4\) there are only elliptic curves; for \(q=2\) there is an infinite series described in work of Madan–Pal [16] and Robinson [17].
The severe restrictions on A impose constraints in turn on the number of rational points on C and \(C'\) over various finite extensions of their base fields. In the constant case, the restrictions lead quickly to Theorem 1.1 because the zeta function of \(C'\) is uniquely determined by the zeta function of C and the degree of the extension. By contrast, in the purely geometric case there is no obvious way to predict the zeta function of \(C'\) from that of C; we instead argue that C is forced to have many rational points, which for \(g_F \gg 0\) will violate a “linear programming” bound [18, Part II]. This yields effective upper bounds on \(g_F\) and \(g_{F'}\); we then obtain a list of candidates for the Weil polynomials of F and \(F'\) by an exhaustion in SageMath (as described in [19], and later used in LMFDB as per [20]). There is a loose parallel here with the Serre–Lauter method for refining upper bounds on rational points on curves over finite fields [21].
To complete the proofs, we identify candidates for C with a given zeta function using data from LMFDB [22], which includes a table of genus4 curves by Xarles [23], plus a similar table of genus5 curves computed by Dragutinović [24]. We then make a computation of abelian extensions of function fields in Magma.
The relative class number one problem is now reduced to the following.
Conjecture 1.5
Let \(F'/F\) be a purely geometric extension of degree \(d>1\) with \(q_F = 2\), \(g_F > 1\), and \(h_{F'/F} = 1\). Then F appears in one of Tables 5 or 6
By Theorem 1.3, this further reduces to the following two logically independent statements, which will be addressed in subsequent work [25, 26].

Any extension as in Conjecture 1.5 is cyclic. This will follow from Theorem 1.3(b) by extending the argument for \(q_F > 2\) (see Lemma 8.1).

Table 6 is complete in genera 6 and 7. This will follow from a limited census based on Mukai’s descriptions of canonical curves of these genera [27, 28]; the entries in Table 6 come from a preliminary version of this census.
We have not considered the relative class number m problem for \(m > 1\), as in [10]. This would require adapting Lemma 5.2 to abelian varieties over \(\mathbb {F}_2\) of order m. For each m it is known that there are infinitely many simple abelian varieties of order m over \(\mathbb {F}_2\) [29], but it seems hopeless to give a complete classification; a better approach might be modeled on the use of resultants to prove statements about small algebraic integers (see [30] for recent progress in this direction).
All computations in SageMath [31] and Magma [32] are documented in Jupyter notebooks available from a GitHub repository [33]; the computations take under 2 hours on a single CPU (Intel i51135G7@2.40GHz) and generate an Excel spreadsheet of the 208 pairs of Weil polynomials in Theorem 1.3(b). We use LMFDB labels for isogeny classes of abelian varieties over finite fields, formatted as links into the site.
2 Abelian varieties of order 1
We say that an abelian variety A over a finite field \(\mathbb {F}_q\) has order 1 if we have \(\#A(\mathbb {F}_q) = 1\); that is, the group of \(\mathbb {F}_q\)rational points of A is trivial. Recall that \(\#A(\mathbb {F}_q) = P(1)\) where \(P(T) \in \mathbb {Z}[T]\) is the Weil polynomial associated to A.
Lemma 2.1
Let A be a simple abelian variety of order 1 over some finite field \(\mathbb {F}_q\).

(a)
We must have \(q \le 4\).

(b)
If \(q \in \{3,4\}\), then A is an elliptic curve with Weil polynomial \(T^2  qT + q\).

(c)
If \(q=2\), then each root \(\alpha \) of the Weil polynomial of A satisfies
$$\begin{aligned} \alpha ^2 + (\eta  1)\alpha  2 \eta = 0 \end{aligned}$$(2.2)for some root of unity \(\eta \). The roots of unity \(\eta \) of order n give rise to two irreducible Weil polynomials if \(n = 7, 30\) and one otherwise. The resulting A is ordinary unless n is a power of 2, in which case it has prank 0.
Proof
This follows from [16, Theorem 4], [17] (for the second assertion of (c)), [34, Lemma 5.1] (for the description in (2.2)), and [34, Lemma 4.3] (for the prank). \(\square \)
We deduce some consequences for the Frobenius traces of abelian varieties of order 1; for \(q=2\) we establish a stronger result later (Lemma 5.2). For A an abelian variety over a finite field \(\mathbb {F}_q\) and n a positive integer, let \(T_{A,q^n}\) be the trace of the \(q^n\)power Frobenius on A; we also write \(T_{C,q^n}\) in case \(A = J(C)\).
Lemma 2.2
Let A be a simple abelian variety of order 1 over \(\mathbb {F}_2\). Choose \(\alpha , \eta \) as in (2.2) and assume that the order of \(\eta \) is not in \(\{1,2,7,30\}\). For
(where \(\mu \) is the Möbius function), we have
Proof
Our assumption on n ensures that \(\mathbb {Q}(\alpha )\) is a quadratic extension of \(\mathbb {Q}(\eta )\). From (2.2), we see that
Similarly, from (2.2) we deduce that
from which we read off the expressions for \(T_{A,4}, T_{A,8}, T_{A,16}\). \(\square \)
Lemma 2.3
Let A be an abelian variety of order 1 and dimension g over \(\mathbb {F}_q\).

(a)
If \(q=4\), then \(T_{A,q} = 4g\), \(T_{A,q^2} =8g\).

(b)
If \(q=3\), then \(T_{A,q} = 3g\), \(T_{A,q^2} = 3g\).

(c)
If \(q=2\) and A is simple, then \(T_{A,2} + T_{A,4} \ge 2\). This is strict if \(g \ge 4\).
Proof
Parts (a) and (b) are apparent from Lemma 2.1. To check (c), we check for \(g \le 6\) using LMFDB;^{Footnote 1} see Table 2 for the detailed results. For \(g > 6\), Lemma 2.2 and (2.3) yield \(T_{A,2} + T_{A,4} = 2g + t_1 + t_2 \ge 2g  12 \ge 2\), as desired. \(\square \)
3 Constant extensions
In this section, we prove Theorem 1.1. We recall a point from the introduction: for any abelian variety A over \(\mathbb {F}_q\) and any positive integer d, the Weil restriction of A from \(\mathbb {F}_{q^d}\) to \(\mathbb {F}_q\) is isogenous to the product of A with the “Prym variety” \(A'\).
Lemma 3.1
Let A be an abelian variety over \(\mathbb {F}_q\) such that \(\#A(\mathbb {F}_q) = \#A(\mathbb {F}_{q^d})\) for some prime \(d>2\). Then \(q=2\), \(d=3\), and the Weil polynomial of every simple isogeny factor of A belongs to \(\{T^2 + T + 2, T^2 + 2T + 2\}\).
Proof
Since \([A(\mathbb {F}_{q^d}):A(\mathbb {F}_q)] = \#A'(\mathbb {F}_q)\) is an integer, the hypothesis that \(\#A(\mathbb {F}_q) = \#A(\mathbb {F}_{q^d})\) implies the same for the isogeny factors of A; we may thus assume that A is simple. Let P(T) be the Weil polynomial of A. Then the Weil polynomial of \(A'\) is \(\prod _{i=1}^{d1} P(\zeta _d^i T)\), and hence has roots \(\alpha _1,\dots ,\alpha _{d1}\) such that
by Lemma 2.1, this is impossible if \(q > 2\). If \(q=2\), then by (2.2) there must exist roots of unity \(\eta _1,\dots ,\eta _{d1}\) with
For \(1\le i < j \le d1\), applying [34, Lemma 5.2, Lemma 7.2] to the equation \(\alpha _i = \alpha _j \zeta _d^{ji}\) shows that \((\eta _i, \eta _j, \zeta _d^{ji})\) either appears in one of the parametric solutions in [34, (7.2.1)] or is a sporadic solution fitting a pattern listed in [34, Table 2].
If only parametric solutions occur, then from [34, (7.2.1)] we have \(\eta _1 = \cdots = \eta _{d1}\), leaving only two distinct values for \(\alpha _1,\dots ,\alpha _{d1}\). Hence \(d= 3\); from [34, (7.2.1)] again, \(\eta _1 = \eta _2 = \zeta _3\) has order 6. This yields the Weil polynomial \(T^2+T+2\).
If we get a sporadic solution for some i, j, then [34, Table 2] indicates that \(\zeta _d^{ji}\) has order dividing 21, 24, or 30; this forces \(d \le 7\). For \(d \in \{5,7\}\), the \(\eta _i\) must all have order 30 or 7, respectively; however, if \(\alpha \) satisfies (2.2) for some root of unity \(\eta \) of this order, then at most two of the quantities \(\{\alpha \zeta _d^i: i=1,\dots ,d1\}\) do likewise, and this leaves no options for \(A'\). Hence \(d=3\); from [34, Table 2] (taking \(\eta _3 = \zeta _3\)), \(\eta _1=\eta _2\) has order 4. This yields the Weil polynomial \(T^2+2T+2\). \(\square \)
Lemma 3.2
Let C be an algebraic curve of genus \(g>0\) over \(\mathbb {F}_q\) such that \(\#J(C)(\mathbb {F}_q) = \#J(C)(\mathbb {F}_{q^d})\) for some integer \(d>1\). Then
Proof
It suffices to prove the claim when d is prime, as the result will then rule out composite values of d. By Lemma 2.1, \(q \le 4\). By Lemma 3.1 applied with \(A = J(C)\), if \(d > 2\) then \((q,d,g) = (2,3,1)\).
Assume now that \(d=2\). Then the Prym variety \(A'\) is the quadratic twist of J(C), so \(T_{A',q^i} = (1)^i T_{C,q^i}\). If \(q \in \{3,4\}\), then by Lemma 2.3,
and so \(g \le 1\). If \(q=2\), then
with the last inequality strict unless \(A'\) is simple of dimension at most 3. \(\square \)
Lemma 3.3
Let C be a curve over \(\mathbb {F}_q\) such that \(\#J(C)(\mathbb {F}_q) = \#J(C)(\mathbb {F}_{q^d})\) for some \(d>1\). Then C appears in Theorem 1.1.
Proof
As this property only depends on the isogeny class of J(C), it suffices to search over the isogeny classes in LMFDB permitted by Lemma 3.2. \(\square \)
4 Bounds on rational points on curves
We next compile some explicit upper bounds for the number of rational points on a curve over \(\mathbb {F}_q\). For \(g \le 10\), we reproduce in Table 1 some data from [35] (see therein for underlying references). For larger g, we use the “linear programming” method of Oesterlé. (All decimal expansions herein refer to exact rational numbers.)
Lemma 4.1
Let C be a curve of genus g over \(\mathbb {F}_q\) with \(q \in \{2,3,4\}\). Then
Proof
For \(q=2\), this is the “third choice” bound of [18, (7.1.4)]. For \(q=3,4\), we adapt the proof of the “first choice” bound of [18, (7.1.1)]). For \(x_1, x_2, \ldots \ge 0\), define \(c = 1 + 2x_1^2 + 2x_2^2 + \cdots \) and consider the function
By construction, \(f(\theta ) \ge 0\) for all \(\theta \in \mathbb {R}\) and \(c_n \ge 0\) for all n (that is, f is doubly positive in the sense of Serre). Define \(\psi (t) = \sum _{n=1}^\infty c_n t^n\); then by [18, Theorem 5.3.3].
or in other words
For \(x_1 = 1, x_2 = 0.7, x_3 = 0.2, x_4 = \cdots = 0\), this yields the indicated results. \(\square \)
The bounds produced by linear programming also include some correction terms counting points over extension fields. We make one such bound explicit for \(q=2\).
Lemma 4.2
Let C be a curve of genus g over \(\mathbb {F}_2\). For \(d=1,2,\dots \), let \(a_d\) be the number of closed points of degree d on C. Then
Proof
With notation as in the proof of Lemma 4.1, define \(\psi _d(t) = \sum _{n=1}^\infty c_{dn} t^{dn}\). Then by [18, Theorem 5.3.3] again,
or in other words
We apply this with \(x_1 = 1, x_2 = 0.85, x_3 = 0.25, x_4 = \cdots = 0\). This yields (4.1) by discarding the terms \(d \ge 5\) in (4.2). \(\square \)
5 Numerical estimates
We next apply the bounds on rational points to bound the genera of function fields occurring in a purely geometric extension with relative class number 1. We will later take a closer account of the degree of the extension; see §7.
For the remainder of the paper, let \(F'/F\) be a purely geometric extension of degree d such that \(g_{F'} > g_F\) and \(h_{F'/F} = 1\). For brevity, we write \(q,g,g'\) in place of \(q_F, g_F, g_{F'}\). Let A be the Prym variety of \(C' \rightarrow C\); then A has order 1, so Lemma 2.1 implies \(q \le 4\). By Riemann–Hurwitz,
with equality if and only if \(C' \rightarrow C\) is étale. Since \(T_{C',q^i} = T_{C,q^i} + T_{A,q^i}\), we have
for each positive integer i, and hence
Lemma 5.1
If \(q > 2\), then \(g \le 6\).
Proof
By combining Lemma 2.3, Lemma 4.1, (5.1), and (5.3), we obtain
Comparing the ends of this equation yields
hence \(g \le 7\) if \(q=3\) and \(g \le 10\) if \(q = 4\). Replacing the righthand side of (5.4) with the explicit bounds given in Table 1, we may eliminate the case \(g = 7\). \(\square \)
For \(q=2\), it is not enough to control \(\#C(\mathbb {F}_2)\) because there exists a simple abelian variety of order 1 with trace 0 (namely 2.2.a_ae). Instead, we use a bound modeled on Lemma 4.2. For A an abelian variety over \(\mathbb {F}_2\), define its excess as
Lemma 5.2
For A an abelian variety of order 1 and dimension g over \(\mathbb {F}_2\), the excess of A is nonnegative.
Proof
We may assume that A is simple; define n as in Lemma 2.1. We again treat the case \(g \le 6\) using LMFDB; see Table 2. For \(g \ge 7\), we have \(g = \phi (n)\); per Lemma 2.2 we can write the excess as
For \(g \in \{7,8\}\), we have \(n \in \{15,16,20,24,30\}\); we compute the excess using (2.3) to obtain a lower bound of 0.4807. For \(g \ge 9\), we apply (2.3) to deduce that \(t_d \le d\) and then obtain a lower bound of \(0.112g  0.9505 \ge 0.112\cdot 9  0.9505 \ge 0.0575\). \(\square \)
Lemma 5.3
For \(q=2\), we have
Proof
We combine Lemma 4.2, (5.3), and Lemma 5.2 to obtain
which yields the claimed inequality. \(\square \)
Corollary 5.4
For \(q=2\), we have \(g \le 40\). Moreover, if \(d \ge 3\) then \(g \le 6\); if \(d \ge 4\) then \(g \le 4\); if \(d \ge 5\) then \(g \le 3\); and if \(d \ge 6\) then \(g \le 2\).
Proof
Taking \(d = 2\) and using the bound on \(\#C(\mathbb {F}_2)\) from Lemma 4.1 yields \(g \le 40\). For \(d \ge 3\) we obtain \(g \le 8\); we then use Table 1 to obtain the remaining bounds. \(\square \)
6 Exhaustion over Weil polynomials
We next describe an exhaustive search over Weil polynomials which rules out some additional pairs \((g,g')\); compare [18, Theorem 7.2.1] for an example in the context of bounding rational points on curves. This will yield Theorem 1.2; for \(g > 1\), we will do better with constraints depending on d (see §7).
We first make a list of candidate Weil polynomials for A. For \(q > 2\) this consists of the single polynomial \((T^2  qT + q)^{g'g}\). For \(q=2\), we identify isogeny classes of simple abelian varieties A of order 1 such that for \(i=1,2\), \(T_{A,2^i}\) and \(T_{A,4}\) is at most the value listed in Table 1 for the pair \((g, 2^i)\), and moreover
these are all necessary conditions by Lemma 4.2 and (5.3).
We next identify candidate Weil polynomials for C for which the resulting values of \(\#C(\mathbb {F}_q)\) (and \(\#C(\mathbb {F}_{q^2})\) for \(q=2\)) are consistent with at least one choice of A, and eliminate those that are ruled out by any of the following.

Bounds on point counts from Table 1.

The positivity condition: the number of degreei places on C must be nonnegative for all \(i \ge 1\).

Data from LMFDB (genus \(\le 3\)), [23] (genus 4), and [24] (genus 5) indicating which curves have a particular Weil polynomial.

The resultant1 and resultant2 criteria of Serre [18, Theorem 2.4.1] as extended by Howe–Lauter [36, Proposition 2.8]. The resultant2 criterion forces C to occur as a double cover of another curve, whose Weil polynomial can sometimes be ruled out. (Compare Corollary 9.2.)
Finally, we exhaust over pairs of candidate Weil polynomials for C and A to confirm that the resulting Weil polynomial for \(C'\) is not ruled out. This yields the following.
Lemma 6.1
For \(q= 2\), for \(g = 0,\dots ,6\) we have \(g' \le 4,6,8,10,12,14,16\), respectively. Hence by (5.1), if \(d \ge 4\) then \(g \le 3\); and if \(d \ge 5\) then \(g \le 2\).
Proof
From Lemma 5.3, for \(g=0,\dots ,6\) we obtain \(g' \le 4, 7, 9, 11, 13, 15, 17\), respectively. We rule out the pairs \((g,g') \in \{(1, 7), (2, 9), (3, 11), (4, 13), (5, 15), (6, 17) \}\) by exhausting over Weil polynomials as described above. \(\square \)
We can now prove Theorem 1.2 as follows. By (5.4), Lemma 6.1, and Table 1, for \((q,g) = (2,0), (2,1), (3,0), (3,1), (4,0), (4,1)\) we have respectively \(g' \le 4, 6, 1, 3, 1, 3\).
We may settle all cases by table lookups except \((q,g,g') = (2,1,6)\), which we settle as follows.

The isogeny class 6.2.ad_c_a_a_m_abg is ruled out by [1, Proposition 5.2], whose proof we summarize. By the resultant2 criterion (compare Remark 10.3), \(C'\) is a double cover of a curve \(C_0\) with real Weil polynomial \(T^2  2T  2\); this is inconsistent with \(\#C_0(\mathbb {F}_2) = 1, \#C'(\mathbb {F}_4) = 0\).

The isogeny class 6.2.ad_c_a_f_am_q occurs for a cyclic étale quintic cover of a genus2 curve listed in Table 5 (see also Remark 6.2).
Remark 6.2
Table 3 includes a column counting Jacobians in the isogeny class of \(J(C')\). This can be obtained by table lookups except for 6.2.ad_c_a_f_am_q, for which Table 3 reports a unique Jacobian; this will be proved in [25, Lemma 10.2].
7 Additional constraints on Weil polynomials
We assume hereafter that \(g > 1\) and introduce constraints on the Weil polynomials of C and \(C'\) based on d. Note that none of these presumes \(h_{F'/F} = 1\), and so may be applicable in other cases of interest.
We start with the full form of Riemann–Hurwitz:
where P runs over geometric points of \(C'\) and \(e_P\) is the ramification index at P.
Let t denote the number of geometric ramification points, i.e., the number of P for which \(e_P > 1\). Then \(t=0\) iff \(\delta = 0\), and \(t \le 2\delta \) in general. If q is even, then \(e_P\) can never equal 2, so \(t \le \delta \); in particular,
because the unique ramification point of \(C'\) is \(\mathbb {F}_q\)rational, and similarly
If \(C' \rightarrow C\) is cyclic of prime degree \(d = p \mid q\), then the Deuring–Shafarevich formula holds (e.g., see [37]): for \(\gamma _C, \gamma _{C'}\) the pranks of \(C, C'\),
If \(\delta =0\) and \(C' \rightarrow C\) is cyclic (e.g., if \(d=2\)), then by class field theory,
For small d, we have the following additional constraints (building on [38, Lemma 8]).

When \(d=2\), every \(\mathbb {F}_{q^i}\)rational point of C lifts to either an \(\mathbb {F}_{q^i}\)rational ramification point or two \(\mathbb {F}_{q^{2i}}\)rational points of \(C'\). Hence
$$\begin{aligned} \#C'(\mathbb {F}_{q^{2i}}) \ge 2\#C(\mathbb {F}_{q^i})  t; \end{aligned}$$(7.6)by (5.2) and (5.3), this yields
$$\begin{aligned} 2T_{A,q^i} + T_{A,q^{2i}}  t \le 2\#C(\mathbb {F}_{q^i}) + T_{A,q^{2i}}  t \le \#C(\mathbb {F}_{q^{2i}}). \end{aligned}$$(7.7)For \(i = 2j1\) odd, every degree iplace of \(C'\) projects to a degreei place of C. If \(t \le 2\), then for \(i>1\) these points occur in pairs in fibers, and so
$$\begin{aligned} t \le 2 \Longrightarrow \#C'(\mathbb {F}_{q^{2j1}}) \equiv \#C'(\mathbb {F}_q) \pmod {2} \qquad (j > 0). \end{aligned}$$(7.8) 
When \(d=3\), every \(\mathbb {F}_{q^i}\)rational point of C lifts to either at least one \(\mathbb {F}_{q^i}\)rational point or three \(\mathbb {F}_{q^{3i}}\)rational points of \(C'\). Hence \(\#C'(\mathbb {F}_{q^{3i}})  \#C'(\mathbb {F}_{q^i}) \ge 3(\#C(\mathbb {F}_{q^i})  \#C'(\mathbb {F}_{q^i}))\); by (5.3), this yields
$$\begin{aligned} \#C(\mathbb {F}_{q^i}) + 2T_{A,q^i} + T_{A,q^{3i}} \le \#C(\mathbb {F}_{q^{3i}}). \end{aligned}$$(7.9) 
When \(d=4\), every \(\mathbb {F}_{q^i}\)rational point of C lifts to at least one \(\mathbb {F}_{q^i}\)rational point, two \(\mathbb {F}_{q^{2i}}\)rational ramification points, or four \(\mathbb {F}_{q^{4i}}\)rational points of \(C'\). Hence \(\#C'(\mathbb {F}_{q^{4i}}) \ge 4(\#C(\mathbb {F}_{q^i})  \#C'(\mathbb {F}_{q^i}))  2t\); by (5.3), this yields
$$\begin{aligned} 4T_{A,q^i} + T_{A,q^{4i}} 2\delta \le 4T_{A,q^i} + T_{A,q^{4i}} 2t \le \#C(\mathbb {F}_{q^{4i}}). \end{aligned}$$(7.10)
Remark 7.1
For \(d=2\), the compositum \(F' \cdot \mathbb {F}_{q^2}\) contains another purely geometric quadratic extension \(F''/F\). We call the corresponding cover \(C'' \rightarrow C\) the relative quadratic twist of \(C' \rightarrow C\); it also obeys the conditions listed in §6.
8 Purely geometric extensions: q>2
We settle Theorem 1.3(a) as follows. For \(q> 2\), Lemma 5.1 implies \(g \le 6\). If \(d=2\), then by Lemma 2.3 plus (7.7),
Combining (8.1) with Table 1, we deduce that
If \(d > 2\), then by upgrading (5.4) using Table 1, we deduce that \((d,g,g') = (3,2,4)\). We also have the following.
Lemma 8.1
If \(q>2\), \(g=2\), and \(d=3\), then \(C' \rightarrow C\) is cyclic.
Proof
Suppose first that \(C' \rightarrow C\) is a nonGalois cover which becomes Galois after a quadratic constant field extension. By Lemma 2.3, the quadratic twist \(\tilde{C}\) of C admits a cyclic cubic étale cover \(\tilde{C}'\) whose Prym has Weil polynomial \((T^2 + qT + q)^2\). Since each \(\mathbb {F}_q\)point of \(\tilde{C}\) lifts to at most three \(\mathbb {F}_q\)points of \(\tilde{C}'\), we have \(\#\tilde{C}(\mathbb {F}_q) + 2q = \tilde{C}'(\mathbb {F}_q) \le 3 \#\tilde{C}(\mathbb {F}_q)\) and so \(\tilde{C}(\mathbb {F}_q) \ge q\). However, \(\#C(\mathbb {F}_q) \ge 2q\) by (5.3), yielding the impossibility
Suppose next that \(C' \rightarrow C\) is geometrically nonGalois. In this case, the Galois closure \(F''\) of \(F'/F\) is itself the function field of a curve \(C''\) with \(q_{F''} = q_F\). The abelian variety \(J(C'')\) is isogenous to \(J(C) \times A^2 \times E\) for some elliptic curve E, so
this yields \(\#C(\mathbb {F}_q) \ge 3q\), which is inconsistent with Table 1. \(\square \)
We now know that in all cases \(C' \rightarrow C\) is cyclic, so we may proceed as follows.

We again exhaust over Weil polynomials for C and A, but this time accounting for (7.4), (7.5), (7.7), (7.9), (8.1). At this point the cases \((q,d,g,g') = (3,2,3,6), (4,2,2,4)\) drop out.

For each candidate Weil polynomial for C, we consult LMFDB to find all candidates for C. At this point the case \((q,d,g,g') = (3,2,4,7)\) drops out: the only isogeny class for J(C) is 4.3.f_v_ca_eg, which contains no Jacobian.

We then use Magma to compute all cyclic extensions of F with the desired degree and ramification behavior and check the resulting Weil polynomial for A. At this point the case \((q,d,g,g') = (4,2,3,5)\) drops out.
This yields Theorem 1.3(a).
9 A refined resultant criterion
In preparation for the case \(q=2\), we next introduce a refinement of the resultant criteria, modeled on [36, Proposition 2.8] (applicable over any finite base field).
Lemma 9.1
Let \(f: C' \rightarrow C\) be a finite flat morphism of degree d between smooth projective curves over an arbitrary field k. Let \(f^*: J(C) \rightarrow J(C')\) denote the pullback map and let \(f_*: J(C') \rightarrow J(C)\) denote the pushforward map. Let A be the Prym variety of f, defined as the reduced closed subscheme of the identity component of \(\ker (f_*)\). Then there is an exact sequence
where the map \(J(C) \rightarrow J(C')\) is \(f^*\) and \(\Delta \) is a finite flat group scheme killed by d.
Proof
The composition \(J(C) {\mathop {\rightarrow }\limits ^{f^*}} J(C') {\mathop {\rightarrow }\limits ^{f_*}} J(C)\) equals the isogeny [d]; consequently, \(f_*\) is surjective (as a morphism of group schemes) and \(J(C) {\mathop {\rightarrow }\limits ^{f^*}} J(C') \rightarrow J(C')/\ker (f_*)\) is surjective. The latter implies that \(J(C) \times _k \ker (f_*) \rightarrow J(C')\) is surjective, as then is \(\ker (f_*) \rightarrow {{\,\textrm{coker}\,}}(f^*)\); since the target is connected and reduced, \(A \rightarrow {{\,\textrm{coker}\,}}(f^*)\) is surjective, as then is \(J(C) \times _k A \rightarrow J(C')\).
Let S be an arbitrary kscheme and suppose \(x \in (J(C) \times _k A)(S)\) maps to zero to \(J(C')\). Write \(x = (x_1, x_2)\) with \(x_1 \in J(C)(S)\) and \(x_2 \in A(S)\). By definition, \(x_1\) and \(x_2\) have the same image in \(J(C')(S)\); that is, \(f^*(x_1) = x_2\). Applying \(f_*\), we deduce that \(f_* f^* (x_1) = 0\), and so \([d](x_1) = 0\); it follows that \([d](x) = ([d](x_1), [d](x_2)) = (0, [d](x_2))\) maps to zero in \(J(C')\), and hence \([d](x_2) = 0\). \(\square \)
Corollary 9.2
In Lemma 9.1, let \(h_1\) and \(h_2\) be the radicals of the real Weil polynomials associated to J(C) and A. Let \(\widetilde{{{\,\textrm{res}\,}}}(h_1,h_2)\) be the modified reduced resultant of \(h_1\) and \(h_2\) in the sense of [36, Proposition 2.8]. Then
Proof
In (9.1), \(\Delta \) cannot be trivial: otherwise, \(J(C')\) would be decomposable as a principally polarized abelian variety, violating Torelli [39, Theorem 12.1]. The exponent of \(\Delta \) divides d by Lemma 9.1 and \(\widetilde{{{\,\textrm{res}\,}}}(h_1,h_2)\) by [36, Proposition 2.8]. \(\square \)
10 Purely geometric extensions: \(q=2\)
To conclude, we establish parts (b) and (c) of Theorem 1.3.
Lemma 10.1
If \(q=2\) and \(d=2\), then \(g \le 9\). Moreover, for \(g=2,\dots ,9\) we have respectively \(g' \le 7, 9, 10, 11, 13, 14, 15, 17\).
Proof
Combining (5.1), Lemma 5.2,and (7.7) yields
This yields the claimed results. \(\square \)
Lemma 10.2
Suppose that \(q=2\) and \(g>1\).

(a)
If \(d=2\), then
$$\begin{aligned} (g,g') \in \{(2,3), (2,4),(2,5), (3,5),(3,6), (4,7), (4,8), (5,9), (6, 11), (7,13)\}. \end{aligned}$$ 
(b)
If \(d=3\), then \((g,g') \in \{(2,4), (2,6), (3,7), (4, 10)\}\).

(c)
If \(d=4\), then \((g,g') \in \{(2,5), (2,6), (3,9)\}\).

(d)
If \(d > 4\), then \(g=2\) and \((d, g') \in \{(5, 6), (6, 7), (7, 8)\}\).
Proof
We run an exhaustive search over Weil polynomials as in §6, but also accounting for (7.2), (7.3), (7.5) (for \(d=2\)), (7.6) (taking \(i=1,2,3\)), (7.9) (taking \(i=1,2\)), (7.10) (taking \(i=1\)), and (9.2). This rules out
(The runtime is dominated by the cases \((d,g,g') = (2,8,15), (2,9,17)\).) We may thus deduce (a) from (5.1) and Lemma 10.1, (b) from Lemma 5.4 and Lemma 6.1, and (c) and (d) from Lemma 6.1. \(\square \)
We obtain Theorem 1.3(b) by a similar calculation which also accounts for (7.8) (taking \(j = 2\)), Remark 7.1, and the following Remark 10.3.
Remark 10.3
If \(C' \rightarrow C\) is étale and geometrically cyclic (i.e., cyclic after base extension from \(\mathbb {F}_2\) to an algebraic closure), we can upgrade Lemma 9.1 to say that \(\Delta \) has exponent exactly d (because \(\ker (f^*)\) is étale and cyclic of order d; compare (7.5)), and Corollary 9.2 to say that \(\widetilde{{{\,\textrm{res}\,}}}(h_1,h_2)\) must be divisible by d.
If we drop these conditions on \(C' \rightarrow C\), we can still say something when \(\gcd (d, \widetilde{{{\,\textrm{res}\,}}}(h_1,h_2)) = 2\): as in [36, Theorem 2.2] there must be a degree2 map from \(C'\) to another curve D whose Jacobian is isogenous to J(C) or A. By (5.1), the second option cannot occur if \(g' > 2g+1\); in characteristic 2, (7.4) also applies.
In the context of Theorem 1.3(b), the condition that \(\gcd (d, \widetilde{{{\,\textrm{res}\,}}}(h_1,h_2)) = 2\) rules out some cases with \((d,g,g') \in \{(4,2,6), (4,3,9), (6,2,7)\}\): there would have to be a double cover \(C' \rightarrow D\) with J(D) isogenous to J(C), but this is forbidden by Lemma 10.2(a). Similarly, if \((d,g,g') = (4,2,5)\), then J(D) cannot be isogenous to A: otherwise D would admit an étale double cover while \(\#J(D)(\mathbb {F}_2) = 1\). Hence \(J(C), J(C')\) must occur in Theorem 1.3(c) with \((d,g,g') = (2,2,5)\).
Remark 10.4
When \(d=2\) and \(\delta \le 1\), A admits a principal polarization; over \(\mathbb {C}\) this is classical [40, Theorem 12.3.3], and a characteristicfree argument will appear in [41]. Our formulation of Theorem 1.3(b) does not account for this constraint; it would rule out a further 16 pairs, which are marked with stars in Table 7.
To obtain Theorem 1.3(c), we use table lookups to find candidates for C with a given Weil polynomial (see §6), then use Magma to enumerate cyclic extensions. As a consistency check, for each triple \((d,g,g')\) listed in Lemma 10.2 with \(g \le 5\), we enumerated cyclic extensions for all curves C of genus g; this took about 14 h and yielded no new results.
Data Availibility Statement
All data generated during this project is fully reproducible using the code available from [33].
Notes
On an LMDFB page, the entry “Point counts of the curve” lists \(q^i+1T_{A,q^i}\) for \(i=1,\dots ,10\).
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Thanks to Xander Faber for providing an early draft of [1], Thomas Grubb for bringing the work of Dragutinović to our attention, and Drew Sutherland for help with computing abelian extensions of function fields in Magma. The author was supported by NSF (grants DMS1802161, DMS2053473) and UC San Diego (Warschawski Professorship).
Appendix A: Extensions of relative class number 1
Appendix A: Extensions of relative class number 1
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Kedlaya, K.S. The relative class number one problem for function fields, I. Res. number theory 8, 79 (2022). https://doi.org/10.1007/s4099302200369y
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DOI: https://doi.org/10.1007/s4099302200369y