The relative class number one problem for function fields, I

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.


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 F 2 , and to reduce that case to a feasible finite computation. This extends work of numerous authors [2,12,19,20,23] but our arguments are independent of these.
For comparison, the relative class number one problem for number fields was formulated by Stark [38] 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 [31] 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)(F qF ) = h F and #J(C ′ )(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 ·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.
Date: August 23, 2022. Thanks to Xander Faber for providing an early draft of [8], 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 DMS-1802161, DMS-2053473) and UC San Diego (Warschawski Professorship).
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 ′ ∈ {0, 1}. Moreover, when g F ∈ {0, 1}, for any fixed pair of isomorphism classes of F and F ′ , the existence of a single finite morphism F → F ′ implies the existence of infinitely many more. It is thus natural to separate the cases g F ≤ 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 ≤ 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 [21,40,27,35]. (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 ≤ 7 and g F ′ ≤ 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 / ∈ {6, 7}, and at least 3 with g F ∈ {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 L-functions of their abelian covers [4].
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, We now summarize the techniques used to prove Theorem 1.1, Theorem 1.2, and Theorem 1.3. The extension F ′ /F induces an injective morphism f from J(C) to the Weil restriction of J(C ′ ) from F q F ′ to F qF , and h F ′ /F can be interpreted as the order of the group A(F qF ) where A is the cokernel of f ; we call A the Prym variety of the covering C ′ → C. We restrict options for C and C ′ using the structure of simple abelian varieties of order 1 over F q : for q ≥ 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 [24] and Robinson [32].
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 ≫ 0 will violate a "linear programming" bound [34,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 [13], and later used in LMFDB as per [7]). There is a loose parallel here with the Serre-Lauter method for refining upper bounds on rational points on curves over finite fields [17].
To complete the proofs, we identify candidates for C with a given zeta function using data from LMFDB [22], which includes a table of genus-4 curves by Xarles [41], plus a similar table of genus-5 curves computed by Dragutinović [6]. 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 [15,16].
• 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 8.2). • 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 [29,30]; 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 [21]. This would require adapting Lemma 5.6 to abelian varieties over F 2 of order m. For each m it is known that there are infinitely many simple abelian varieties of order m over F 2 [14], 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 [37] for recent progress in this direction).
All computations in SageMath [33] and Magma [25] are documented in Jupyter notebooks available from a GitHub repository [18]; the computations take under 2 hours on a single CPU (Intel i5-1135G7@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.

Abelian varieties of order 1
We say that an abelian variety A over a finite field F q has order 1 if we have #A(F q ) = 1; that is, the group of F q -rational points of A is trivial. Recall that #A(F q ) = P (1) where P (T ) ∈ 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 F q .
(c) If q = 2, then each root α of the Weil polynomial of A satisfies for some root of unity η. The roots of unity η 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 p-rank 0.
We deduce some consequences for the Frobenius traces of abelian varieties of order 1; for q = 2 we establish a stronger result later (Lemma 5.6). For A an abelian variety over a finite field 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 2.3. Let A be a simple abelian variety of order 1 over F 2 . Choose α, η as in (2.2) and assume that the order of η is not in {1, 2, 7, 30}. For Proof. Our assumption on n ensures that Q(α) is a quadratic extension of Q(η). From (2.2), we see that Similarly, from (2.2) we deduce that Lemma 2.5. Let A be an abelian variety of order 1 and dimension g over Proof. Parts (a) and (b) are apparent from Lemma 2.1. To check (c), we check for g ≤ 6 using LMFDB; 1 see Table 2 for the detailed results. For g > 6, Lemma 2.3 and (2.4) yield T A,2 + T A,4 = 2g + t 1 + t 2 ≥ 2g − 1 − 2 ≥ 2, as desired.

Constant extensions
In this section, we prove Theorem 1.1. We recall a point from the introduction: for any abelian variety A over F q and any positive integer d, the Weil restriction of A from F q d to F q is isogenous to the product of A with the "Prym variety" A ′ .
Lemma 3.1. Let A be an abelian variety over F q such that #A(F q ) = #A(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, is an integer, the hypothesis that #A(F q ) = #A(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 , and hence has roots α 1 , . . . , α d−1 such that either appears in one of the parametric solutions in [5, (7.2.1)] or is a sporadic solution fitting a pattern listed in [5, Table 2].
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 ≤ 4. By Lemma 3.1 applied with Assume now that d = 2. Then the Prym variety A ′ is the quadratic twist of with the last inequality strict unless A ′ is simple of dimension at most 3. Lemma 3.3. Let C be a curve over F q such that #J(C)(F q ) = #J(C)(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.

Bounds on rational points on curves
We next compile some explicit upper bounds for the number of rational points on a curve over F q . For g ≤ 10, we reproduce in Table 1 some data from [26] (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.)  Table 1. Upper bounds on #C(F q ) for a genus-g curve C from [26].
By construction, f (θ) ≥ 0 for all θ ∈ R and c n ≥ 0 for all n (that is, f is doubly positive in the sense of Serre). Define ψ(t) = ∞ n=1 c n t n ; then by [34,Theorem 5.3.3].
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.

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 ′ → C; then A has order 1, so Lemma 2.1 implies q ≤ 4. By Riemann-Hurwitz, for each positive integer i, and hence Lemma 5.4. If q > 2, then g ≤ 6.
Proof hence g ≤ 7 if q = 3 and g ≤ 10 if q = 4. Replacing the right-hand side of (5.5) with the explicit bounds given in Table 1, we may eliminate the case g = 7.
For q = 2, it is not enough to control #C(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 F 2 , define its excess as Lemma 5.6. For A an abelian variety of order 1 and dimension g over 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 ≤ 6 using LMFDB; see Table 2. For g ≥ 7, we have g = φ(n); per Lemma 2.3 we can write the excess as 0.112g − 0.1012t 1 − 0.1656t 2 + 0.1011t 3 + 0.0537t 4 .
Taking d = 2 and using the bound on #C(F 2 ) from Lemma 4.1 yields g ≤ 40. For d ≥ 3 we obtain g ≤ 8; we then use Table 1 to obtain the remaining bounds.

Exhaustion over Weil polynomials
We next describe an exhaustive search over Weil polynomials which rules out some additional pairs (g, g ′ ); compare [34, 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 We next identify candidate Weil polynomials for C for which the resulting values of #C(F q ) (and #C(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 isogeny class 6.2.ad c a a m abg is ruled out by [8,Proposition 5.2], whose proof we summarize. By the resultant-2 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 (F 2 ) = 1, #C ′ (F 4 ) = 0. • The isogeny class 6.2.ad c a f am q occurs for a cyclicétale quintic cover of a genus-2 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 [15,Lemma 10.2].

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 δ = 0, and t ≤ 2δ in general. If q is even, then e P can never equal 2, so t ≤ δ; in particular, because the unique ramification point of C ′ is F q -rational, and similarly If C ′ → C is cyclic of prime degree d = p | q, then the Deuring-Shafarevich formula holds (e.g., see [36]): for γ C , γ C ′ the p-ranks of C, C ′ , If δ = 0 and C ′ → C is cyclic (e.g., if d = 2), then by class field theory, For small d, we have the following additional constraints (building on [10,Lemma 8]).
• When d = 2, every F q i -rational point of C lifts to either an F q i -rational ramification point or two F q 2i -rational points of C ′ . Hence by (5.2) and (5.3), this yields For i = 2j − 1 odd, every degree i-place of C ′ projects to a degree-i place of C. If t ≤ 2, then for i > 1 these points occur in pairs in fibers, and so • When d = 3, every F q i -rational point of C lifts to either at least one • When d = 4, every F q i -rational point of C lifts to at least one F q i -rational point, two F q 2i -rational ramification points, or four F q 4i -rational points of C ′ . Hence #C ′ (F q 4i ) ≥ 4(#C(F q i ) − #C ′ (F q i )) − 2t; by (5.3), this yields (7.10) Remark 7.11. For d = 2, the compositum F ′ · F q 2 contains another purely geometric quadratic extension F ′′ /F . We call the corresponding cover C ′′ → C the relative quadratic twist of C ′ → C; it also obeys the conditions listed in §6.
Proof. Suppose first that C ′ → C is a non-Galois cover which becomes Galois after a quadratic constant field extension. By Lemma 2.5, the quadratic twist C of C admits a cyclic cubicétale coverC ′ whose Prym has Weil polynomial (T 2 + qT + q) 2 . Since each F q -point ofC lifts to at most three F q -points ofC ′ , we have #C(F q ) + 2q =C ′ (F q ) ≤ 3#C(F q ) and soC(F q ) ≥ q. However, #C(F q ) ≥ 2q by (5.3), yielding the impossibility 2q + 2 = #C(F q ) + #C(F q ) ≥ 3q.
Suppose next that C ′ → C is geometrically non-Galois. 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) × A 2 × E for some elliptic curve E, so this yields #C(F q ) ≥ 3q, which is inconsistent with Table 1.
We now know that in all cases C ′ → C is cyclic, so we may proceed as follows.

A refined resultant criterion
In preparation for the case q = 2, we next introduce a refinement of the resultant criteria, modeled on [11, Proposition 2.8] (applicable over any finite base field).
where the map J(C) → J(C ′ ) is f * and ∆ is a finite flat group scheme killed by d.
Proof. The composition J(C) consequently, f * is surjective (as a morphism of group schemes) and J(C) is surjective, as then is ker(f * ) → coker(f * ); since the target is connected and reduced, Let S be an arbitrary k-scheme and suppose x ∈ (J(C) × k A)(S) maps to zero to J(C ′ ). Write x = (x 1 , x 2 ) with x 1 ∈ J(C)(S) and x 2 ∈ A(S). By definition,  To conclude, we establish parts (b) and (c) of Theorem 1.3.
We obtain Theorem 1.3(b) by a similar calculation which also accounts for (7.8) (taking j = 2), Remark 7.11, and the following Remark 10.3. Remark 10.3. If C ′ → C isétale and geometrically cyclic (i.e., cyclic after base extension from F 2 to an algebraic closure), we can upgrade Lemma 9.1 to say that ∆ has exponent exactly d (because ker(f * ) isétale and cyclic of order d; compare (7.5)), and Corollary 9.3 to say that res(h 1 , h 2 ) must be divisible by d.
If we drop these conditions on C ′ → C, we can still say something when gcd(d, res(h 1 , h 2 )) = 2: as in [11,Theorem 2.2] there must be a degree-2 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.
Remark 10.4. When d = 2 and δ ≤ 1, A admits a principal polarization; over C this is classical [3,Theorem 12.3.3], and a characteristic-free argument will appear in [1]. 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.
Completeness of the list is confirmed above the double line and conjectural below it (Conjecture 1.5). For g F = 7, J(C) does not appear in LMFDB, so we list J(C)(F 2 i ) for i = 1, . . . , 7.