§1. Introduction. In [T1] M.J.Taylor began the study of the Galois module structure of certain Kummer orders arising from group laws of abelian varieties defined over number fields (see also [ST], [CN-S] and [CN-T]). The purpose of this paper is to study similar Kummer orders which are derived from CM abelian varieties over global function fields.

For any field F we shall write F^{c} for a separable closure of F and Ω_{F} for Gal(F^{c}/F). Let C be a smooth, geometrically irreducible curve defined over a field k ⊆ 𝔽_{p}^{c}. Set L = k(C), the function field of C over k. Let S = {v_{1}, …, v_{t}} be a fixed nonempty set of places of L and let 𝔒_{L} = 𝔒_{L,S} denote the ring of functions in L which are regular away from S. 𝔒_{L} is the function field analogue of the ring of integers of a number field. Write 𝔒^{c} for the integral closure of 𝔒_{L} in L^{c}.

Let A/L be a simple abelian variety defined over L with complex multiplication. This implies (see [M] p.220) that A is either a constant or a twisted constant variety over L. In what follows, we shall always assume that S contains all places of bad reduction of A. We shall also suppose that all endomorphisms of A that we consider are defined over L. The endomorphism ring D =End(A) of A is an order in a finitedimensional semisimple division algebra K over ℚ. If K is non-abelian, then (since A is simple) A is isogenous (possibly after an extension of L) to a supersingular elliptic curve and K is isomorphic to the quaternion division algebra over ℚ which splits at all primes l ≠ p, ∞ (see e.g. [Mu] p. 256). Choose an endomorphism λ ∈ D such that the degree of λ is both greater than one and coprime to p.

Denote by G the group of all points of A(L^{c}) which are killed by λ. The 𝔒_{L}- group scheme of λ-torsion points on A is affine and etale (since (|G|, p) = 1) and is therefore equal to Spec(𝔅(L; λ)), where 𝔅(L; λ) = Map_{Ω_{L}}(G, 𝔒^{c}), the 𝔒_{L}-Hopf algebra consisting of Ω_{L}-maps from G to 𝔒^{c}. It follows that the 𝔒_{L}-Cartier dual of 𝔅(L; λ) is 𝔄(L; λ) = (𝔒^{c}G)^{Ω_{L}} (here Ω_{L} acts on both 𝔒^{c} and G). 𝔄(L; λ) is thus the unique 𝔒_{L}-maximal order in the L-algebra A(L; λ) = (L^{c}G)^{Ω_{L}} since 𝔒^{c}G is the unique 𝔒^{c}-maximal order in the algebra L^{c}G.

Now suppose that Q ∈ A(L) and write

G_{Q}(λ) = {Q′ ∈ A(L^{c}) | λQ′ = Q}.

Define the L-algebra L_{Q}(λ) (the Kummer algebra) by L_{Q}(λ) = Map_{Ω_{L}}(G_{Q}, L^{c}), and let L(\frac{1}{λ}Q) be the field obtained by adjoining the coordinates of the points of G_{Q}(λ) to L. Let Q^{′(1)}, …, Q^{′(s)} be a set of representatives of Ω_{L}-orbits of G_{Q}(λ). Then (as an L-algebra),

L_{Q}(λ) ≃ ∏_{i=1}^{s} L[Q^{′(i)}]  (1.1)

where L[Q^{′(i)}] is the field obtained by adjoining the coordinates of Q^{′(i)} to L. (Explicitly, the isomorphism is given by f ↦ ∏_{i=1}^{s} f(Q^{′(i)}), for f ∈ L_{Q}(λ). Note also that if G ⊆ A(L), then all the fields L[Q^{′(i)}] are the same.) Hence [L_{Q}(λ) : L] = |G|. A(L; λ) acts on L_{Q}(λ) via

(f. ∑_{g} a_{g}g)(Q′) = ∑_{g} a_{g}f(Q′ + g),

for f ∈ L_{Q}(λ) and ∑_{g} a_{g}g ∈ A(L; λ)(L).

Let 𝔒_{Q}(L; λ) denote the integral closure of 𝔒_{L} in L_{Q}(λ). Then 𝔒_{Q}(L; λ) (the Kummer order) is an 𝔄(L; λ)-module. Since S contains all places of bad reduction of A and (|G|, p) = 1, it follows from (1.1) above and the criterion of Neron-Ogg-Shafarevitch that L_{Q}(λ)/L is unramified at all places of 𝔒_{L}. (This fact, and slight variants of it will be used several times in the sequel.) As 𝔄(L; λ) is the maximal order of A(L; λ), it follows that 𝔒_{Q}(L; λ) is a locally free 𝔄(L; λ) module (see e.g.

[CR] Proposition 31.2). Thus, if Cl(𝔄(L; λ)) denotes the locally free classgroup of 𝔄(L; λ), then we have a map

ψ : A(L)/λA(L) ⟶ Cl(𝔄(L; λ))

given by ψ(Q) = (𝔒_{Q}(L; λ)), where (𝔒_{Q}(L; λ)) is the class of 𝔒_{Q}(L; λ) in Cl(𝔄(L; λ)). As A has good reduction at all places of 𝔒_{L}, it follows exactly as in [T1] that ψ is a homomorphism, and so in particular the image of ψ is annihilated by |G|. Observe that since G is abelian, 𝔄(L; λ) satisfies the Eichler condition. Hence 𝔒_{Q}(L; λ) is a free 𝔄(L; λ)-module if and only if ψ(Q) = 0.

We are now able to state the first result of this paper. Theorem 1.2. Let Q ∈ A(L) be a point of infinite order. Suppose that the image of Q has order m > 1 in A(L)/(λA(L) + A(L)_{tors}), where A(L)_{tors} denotes the torsion subgroup of A(L) Then there is an integer N_{1} (which is independent of Q) such that ψ(Q) ≠ 0 if m > N_{1}. Furthermore, if A is a constant variety and if S consists of a single place of degree one, then ψ(Q) ≠ 0, i.e. we may take N_{1} = 1.

We refer the reader to [A] and [AT] for an analysis of the Galois module structure of Kummer orders arising from points of infinite order on CM elliptic curves defined over number fields.

The first three sections of this paper are devoted to the proof of Theorem 1.2.

In the fourth section, we give a geometric interpretation of the homomorphism ψ when A is a constant variety over L. We then show how the zeta function of A/k may be used to obtain precise information concerning the image of ψ in the case that λ is a cyclic endomorphism of prime degree.

The results contained in this paper formed a portion of my Ph.D. thesis (Columbia University 1991). It is with the greatest of pleasure that I thank my advisor, Professor T. Chinburg for all of his help, kindness and encouragement. I also wish to thank Professor M. J. Taylor for suggesting that I look at function fields and for much generous advice, and Professor K. Rubin for many very helpful conversations. Thanks also to Dr. R. J. Chapman, Professor R. Friedman, and Mr. J. McKernan for interesting discussions.

§2. Preliminary Remarks We shall first prove Theorem 1.2 for constant varieties and then deduce the result for twisted constant varieties. Thus, from now on until further notice, A denotes a constant abelian variety. We remark that in this case, A(L)_{tors} = A(k).

The following proposition is an extension of a result shown to me by Professor K. Rubin. Proposition 2.1. Let Q ∈ A(L) be a point of infinite order and write L′ = L ⊗_{k} k^{c}. If Q has order m′ in A(L′)/λA(L′), then Q has order dividing m′ in A(L)/(λA(L) + A(k)).

Thus, if Q is not divisible by λ in A(L)/A(k), then Q is not divisible by λ in A(L_{1})/A(k_{1}), where k_{1}/k is any finite extension and L_{1} = L ⊗_{k} k_{1}. Hence L′(\frac{1}{λ}Q)/L′ is a nontrivial finite extension. Proof. Consider the natural homomorphism η : A(L)/λA(L) → A(L′)/λA(L′) induced by the inclusion A(L) ↪ A(L′). We have the following commutative diagram

A(L)/λA(L) ⟶^{η} A(L′)/λA(L′)
↓^{i} ↓
H^{1}(Gal(L′/L), G) ⟶ H^{1}(Gal(L^{c}/L), G) ⟶_{Res} H^{1}(Gal(L^{c}/L′), G)

(2.2)

where the vertical arrows are injective and the bottom row is exact.

From the exactness of the bottom row, we deduce that Ker(Res) has order at most |H^{1}(Gal(L′/L), G)|. On the other hand, the natural homomorphism A(k)/λA(k) → A(L)/λA(L) is an injection because k is the constant subfield of L. Clearly A(k)/λA(k) ⊂ Ker(η), from which it follows that Ker(Res) has order at least |A(k)/λA(k)|.

Now Gal(L′/L) ≃ Ω_{k}, and the isomorphism respects the action of these groups on G because G is defined over k^{c}. Hence H^{1}(Gal(L′/L), G) ≃ H^{1}(Ω_{k}, G). Kummer theory on A(k^{c}) yields the exact sequence

0 → A(k)/λA(k) → H^{1}(Ω_{k}, G) → H^{1}(Ω_{k}, A(k^{c})). It is a theorem of Lang (see [L]) that a principal homogeneous space of any

algebraic group defined over a finite field is trivial. Hence H^{1}(Ω_{k}, A(k^{c})) = 0 and so |A(k)/λA(k)| = |H^{1}(Ω_{k}, G)|. Thus Ker(Res) = i(A(k)/λA(k)).

Now since m′Q = 0 in A(L′)/λA(L′) i.e. m′Q ∈ Ker(η), we have that i(m′Q) ∈ i(A(k)/λA(k)). Therefore m′Q ∈ A(k) + λA(L), and so Q has order dividing m′ in A(L)/(A(k) + λA(L)).

The final paragraph of the proposition now follows immediately.

Now let k_{1}/k be any finite extension such that G ⊆ A(k_{1}), and write L_{1} = L ⊗_{k} k_{1}. Then A(L_{1}; λ) = L_{1}G = A(L; λ) ⊗_{k} k_{1}, 𝔄(L_{1}; λ) = 𝔒_{L_{1}}G = 𝔄(L; λ) ⊗_{k} k_{1}, and 𝔒_{Q}(L_{1}; λ) = 𝔒_{Q}(L; λ) ⊗_{k} k_{1}. Hence if 𝔒_{Q}(L_{1}; λ) is not free over 𝔄(L_{1}; λ) then 𝔒_{Q}(L; λ) is not free over 𝔄(L; λ). Furthermore, it follows from Proposition 2.1 and the hypotheses of Theorem 2.2 that the order of Q in A(L_{1})/(A(k_{1}) + λA(L_{1})) is at least as large as the order of Q in A(L)/(A(k) + λA(L)) Thus, for the proof of Theorem 1.2, we may assume without loss of generality that G ⊂ A(L), A(L; λ) = LG, 𝔄(L; λ) = 𝔒_{L}G, and that L contains the |G|th roots of unity. We shall make these assumptions.

We caution the reader that where there is no danger of ambiguity, we shall feel free to drop the dependence on L and/or λ in our notation for 𝔄(L; λ), 𝔒_{Q}(L; λ), L_{Q}(λ), etc.

§3. Classgroup Theory We now recall some elementary facts from the theory of tame classgroups over global function fields. For details of these results we refer the reader to [C1] or [C3].

Let R_{G} denote the ring of virtual characters of G and write Ĝ for the group of irreducible characters of G. Since L contains the |G|th roots of unity, the maximal order 𝔒_{L}G of LG splits, i.e. 𝔒_{L}G ≃ ∏_{χ∈Ĝ} 𝔒_{L}. Hence we have

Cl(𝔒_{L}G) ≃ Map(Ĝ, Cl(𝔒_{L})) ≃ \frac{Hom(R_{G}, Div_{S}(L))}{Hom(R_{G}, D_{S}(L^{∗}))}.

Here Div_{S}(L) is the group of divisors of L with support away from S. D_{S}(L^{∗}) is the subgroup of Div_{S}(L) consisting of the image of the natural map that sends an element x ∈ L^{∗} to its associated divisor ∑_{𝔓 prime of 𝔒_{L}} v_{𝔓}(x)𝔓. We shall sometimes view elements of Div_{S}(L) as ideals of 𝔒_{L}.

A homomorphism in Hom(R_{G}, Div_{S}(L)) which represents the class (𝔒_{Q}(L)) ∈ Cl(𝔒_{L}G) = Cl(𝔄(L)) may be contructed as follows:

Suppose that a ∈ L_{Q} and χ ∈ Ĝ. Then the resolvent of a at χ is

(a|χ) = ∑_{g∈G} χ(g^{−1})a^{g} ∈ L_{Q}.

Choose d_{Q} ∈ L_{Q} such that L_{Q} = d_{Q}.LG. For each place 𝔓 of 𝔒_{L}, let 𝔒_{L,𝔓} (resp. 𝔒_{Q,𝔓}(L)) denote the semi-localisation of 𝔒_{L} (resp. 𝔒_{Q}(L)) at 𝔓. Choose a_{𝔓} ∈ 𝔒_{Q,𝔓}(L) such that 𝔒_{Q,𝔓}(L) = a_{𝔓}.𝔒_{L,𝔓}G. Then (a_{𝔓}|χ)(d_{Q}|χ)^{−1} ∈ L, and the homomorphism we require is given by

χ ↦ ∑_{𝔓 of 𝔒_{L}} v_{𝔓}((a_{𝔓}|χ)(d_{Q}|χ)^{−1})𝔓  (3.1)

extended to R_{G} via linearity.

We now make some further remarks on resolvents. Since

(d_{Q}|χ)^{g} = χ(g)(d_{Q}|χ)  (3.1(a))

and [L_{Q} : L] = |G|, we have the following direct sum decomposition of L_{Q} into G-eigenspaces:

L_{Q} = ⨁_{χ∈Ĝ} L.(d_{Q}|χ)  (3.2)

It thus follows from (1.1) and (3.2) that

L(\frac{1}{λ}Q) = L(\{(d_{Q}|χ)(Q′)\}_{χ∈Ĝ}),  (3.3)

where Q′ is any fixed element of G_{Q}. Since

[(d_{Q}|χ_{1})(d_{Q}|χ_{2})]^{g} = χ_{1}(g)χ_{2}(g)[(d_{Q}|χ_{1})(d_{Q}|χ_{2})]  (3.3(a))

for all χ_{1}, χ_{2} ∈ Ĝ, we conclude that, for all χ ∈ Ĝ

(d_{Q}|χ)^{r} = γ_{χ}(d_{Q}|χ^{r})  (3.4)

for some γ_{χ} ∈ L.

As A/L has good reduction at all places of L (since A/L is a constant variety) and (|G|, p) = 1, it follows from the criterion of Neron-Ogg-Shafarevitch and the definition of L_{Q} that L_{Q}/L is unramified at all places of L (and not just at all places of 𝔒_{L} as in §1). The following result is a simple extension of Proposition 4.3 in Ch. I of [F] from fields to Galois algebras. Proposition 3.5. Let a_{𝔓} ∈ 𝔒_{Q,𝔓}(L) be such that 𝔒_{Q,𝔓}(L) = a_{𝔓}.𝔒_{L,𝔓}G. Then for all χ ∈ Ĝ, we have (a_{𝔓}|χ) ∈ 𝔒_{Q,𝔓}^{∗}.

Let us now recall the isomorphism (1.1) and focus our attention on a fixed component L[Q^{′(j)}] say, of L_{Q}. Write 𝔒(Q^{′(j)}) for the integral closure of 𝔒_{L} in L[Q^{′(j)}]. From (3.1(a)), we deduce that (d_{Q}|χ)(Q^{′(j)})^{|G|} ∈ L^{∗} for all χ ∈ Ĝ. As L[Q^{′(j)}]/L is unramified at all places of L, it follows that the 𝔒(Q^{′(j)})-submodule (d_{Q}|χ)(Q^{′(j)}).𝔒(Q^{′(j)}) of L[Q^{′(j)}] is an ambiguous ideal, i.e. that

(d_{Q}|χ)(Q^{′(j)}).𝔒(Q^{′(j)}) = 𝔇_{χ}𝔒(Q^{′(j)})  (3.6),

where 𝔇_{χ} is a fractional 𝔒_{L} ideal. (Here, by a fractional 𝔒_{L}-ideal, we mean a finitely generated 𝔒_{L}-submodule of L. Note that the remarks immediately after (1.1) together with (3.2) and the fact that L contains the |G|th roots of unity imply that 𝔇_{χ} is independent of the choice of component L[Q^{′(j)}]/L.)

Write

𝔡_{χ} = ∑_{𝔓 of 𝔒_{L}} v_{𝔓}(𝔇_{χ})𝔓.  (3.7)

From (3.1), (3.5) and (3.6), we deduce that (𝔒_{Q}) ∈ Cl(𝔒_{L}G) is represented by the element of Map(Ĝ, Cl(𝔒_{L})) given by

χ ↦ (𝔡_{χ})^{−1}  (3.8)

where (𝔡_{χ}) denotes the class of 𝔡_{χ} in Cl(𝔒_{L}).

We remark that this map in fact lies in Hom(Ĝ, Cl(𝔒_{L})) since it follows from equation (3.4) that

(𝔡_{χ^{r}}) = (𝔡_{χ})^{r}.  (3.9)

§4. Kummer Theory. Consider the homomorphism ε′ : A(L) → Hom_{Ω_{L}}(Ĝ, L^{∗}/L^{∗|G|}) given by Q ↦ {χ ↦ (d_{Q}|χ)^{|G|}}. Suppose that Q ∈ Ker(ε′). Then (d_{Q}|χ) ∈ L^{∗} for all χ ∈ Ĝ, and so from (3.3(a)) we deduce that L(\frac{1}{λ}Q) = L i.e. that Q ∈ λA(L). Hence ε′ induces an injective homomorphism ε : A(L)/λA(L) → Hom_{Ω_{L}}(Ĝ, L^{∗}/L^{∗|G|}). The following proposition is essentially the same as Proposition 8 of [T1]. Proposition 4.1. Let U denote the group of units of 𝔒_{L}. Then Q ∈ Ker(ψ) if and only if ε(Q) ∈ Hom_{Ω_{L}}(Ĝ, U.L^{∗|G|}/L^{∗|G|}). Proof. Using the notation of §3, we have that ψ(Q) = 0 if and only if for all χ ∈ Ĝ, 𝔡_{χ} = X_{χ}𝔒_{L} for some X_{χ} ∈ L^{∗}. If this is the case, then (d_{Q}|χ)X_{χ}^{−1} is a unit of 𝔒_{Q} and so ε(Q)(χ) is represented by [(d_{Q}|χ)X_{χ}^{−1}]^{|G|} ∈ U.

Now since L(\frac{1}{λ}Q) = L(\{(d_{Q}|χ)\}_{χ∈Ĝ}), it follows from Proposition 4.1 that if ψ(Q) = 0, then L(\frac{1}{λ}Q)/L is a Kummer extension of L obtained by adjoining |G|th roots of various units of 𝔒_{L}. Moreover, from Proposition 2.1, L(\frac{1}{λ}Q)/L is a nontrivial, non-constant extension. Also, as remarked in §3, L(\frac{1}{λ}Q)/L is unramified at all places of L since A/L is a constant variety. An element of L is a unit of 𝔒_{L} if and only if it is an S-unit of L. In particular, we may observe at this point that if S consists of a single place of degree one, then 𝔒_{L}^{∗} = k^{∗}, and Theorem 1.2 in this case follows immediately from Propositions 2.1 and 4.1 above. To proceed further, we need the following result.

Proposition 4.2. Write L′ = L ⊗_{k} k^{c} and let n ∈ ℕ with n coprime to p. Let S′ denote the set of places of L′ lying above S and suppose that there is an S′-unit α of L′ such that L′(α^{1/n})/L′ is an everywhere unramified extension of degree n. Then there is an integer N depending only upon S′ and L′ such that n|N.

Proof. Write U(S′) for the group of S′-units of L′, and let Div^{0}(S′) be the group of divisors of degree zero of C/k^{c} whose support is contained in S′. Then we have an exact sequence

0 → k^{c∗} → U(S′) → Div^{0}(S′) → T → 0,  (4.3)

where T is a finite group.

Write (α) for the divisor of α, and suppose that

(α) = ∑_{i=1}^{d} a_{i}v_{i},

where a_{i} ∈ ℤ and the v_{i} are places of L′. Since L′(α^{1/n})/L′ is everywhere unramified, we have that n|a_{i} for i = 1, … d. Set

D = ∑_{i=1}^{d} \frac{a_{i}}{n} v_{i}.

As L′(α^{1/n})/L′ has degree n and k^{c} ⊂ L′, it follows that the image of D in T has order n. Hence, taking N = |T |, the proposition follows.

The next result relates the order of Q in A(L′)/λA(L′) to the degree of the extension L′(\frac{1}{λ}Q)/L′. Lemma 4.4. Let d = [L′(\frac{1}{λ}Q) : L′] and write m′ for the order of Q in A(L′)/λA(L′). Then m′|d. Proof. Write H = Gal(L′(\frac{1}{λ}Q)/L′), and fix Q′ ∈ G_{Q}. For each h ∈ H, we have Q′^{h} = Q′ + s_{h}, with s_{h} ∈ G ⊂ A(k^{c}). Hence

P = ∑_{h∈H} Q′^{h} = d.Q′ + s,  (4.5)

with P ∈ A(L′) and s ∈ A(k^{c}). Thus d.Q = λ(P −s), i.e. d.Q = 0 in A(L′)/λA(L′), and now the result follows.

Now suppose that A has dimension t, and write A_{|G|} for the group of |G|-torsion points of A(k^{c}). Then A_{|G|} ≃ (ℤ/|G|ℤ)^{2t}, and G is a subgroup of A_{|G|}. Hence there exist χ_{1}, …, χ_{s} with 1 ≤ s ≤ 2t such that Ĝ =< χ_{1}, …, χ_{s} >. From (3.3) and (3.3(a)), it follows that L′(\frac{1}{λ}Q) = L′(\{d_{Q}|χ_{i})(Q′)\}_{i=1}^{s}). Thus if for example d = [L′(\frac{1}{λ}Q) : L′] ≥ 2N^{2t}, then we must have [L′((d_{Q}|χ_{j})(Q′) : L′] > N for at least one j, 1 ≤ j ≤ s. If this is the case, then we deduce using Proposition 4.2 that (d_{Q}|χ_{j})(Q′)^{|G|} ∉ U.L^{∗}^{|G|}/L^{∗}^{|G|} (where U = 𝔒_{L}^{∗}), which (from Proposition 4.1) implies that ψ(Q) ≠ 0.

Recall that m > 1 is the order of Q in A(L)/(λA(L) + A(L)_{tors}) and that A(L)_{tors} = A(k), since A is a constant variety. From Proposition 2.1 and Lemma 4.4, we have that m ≤ m′ ≤ d. Hence if e.g. m ≥ 2N^{2t}, then ψ(Q) ≠ 0, which completes the proof of Theorem 1.2 for constant abelian varieties.

We now turn to the case in which A is a twisted constant variety. Thus, from now on until further notice, assume that A is a twisted constant variety and let M/L be a finite extension over which A becomes isomorphic to a constant variety. Write d_{1} = [M : L] and let d_{2} = |A(M)_{tors}|.

Associated to the field M and the set of places S_{M} of M lying above S is an integer N_{M} as in Proposition 4.2. Suppose e.g. that m ≥ 2d_{1}d_{2}N_{M}^{2t}. It follows via the natural injections A(L)/λA(L) → H^{1}(Gal(L^{c}/L), G) and A(M)/λA(M) → H^{1}(Gal(M^{c}/M), G), together with the inflation-restriction sequence of Galois cohomology, that the kernel of the natural map A(L)/λA(L) → A(M)/λA(M) is killed by d_{1}. Hence the image of Q in A(M)/(λA(M) + A(M)_{tors}) has order at least m/d_{1}d_{2} = 2N_{M}^{2t}. Now reasoning exactly as in the already treated constant case, we deduce that (𝔒_{Q}(M; λ)) ≠ 0 in Cl(𝔄(M; λ)). As 𝔒_{Q}(M; λ) = 𝔒_{Q}(L; λ) ⊗_{𝔒_{L}} 𝔒_{M} and 𝔄_{Q}(M; λ) = 𝔄_{Q}(L; λ)⊗_{𝔒_{L}}𝔒_{M}, this implies that (𝔒_{Q}(L; λ)) ≠ 0 in Cl(𝔄(L; λ)), as required.

Let us now return to the case of an arbitrary CM abelian variety A as in the statement of Theorem 1.2. We conclude this section with some remarks on a simple consequence of the proof of that theorem. It is well known that A(L′)/A(L′)_{tors} is a finitely generated D-module. Hence, if Q ∈ A(L) is a point of infinite order, then Q is not infinitely divisible by λ in A(L′). This implies that the degree of the extension L′(\frac{1}{λ^{n}} Q)/L′ tends to infinity with n. Theorem 4.5. Let Q ∈ A(L) be a point of infinite order. Then for all sufficiently large n, 𝔒_{Q}(L; λ^{n}) is not a free 𝔄_{Q}(L; λ^{n})-module. Proof. Let M, N_{M}, and t be as above (so in particular if A is a constant variety, then M = L). Suppose that n is sufficiently large that [M′(\frac{1}{λ^{n}} Q) : M′] > 2d_{1}N_{M}^{2t}. Then 𝔒_{Q}(M; λ^{n}) is not a free 𝔄(M; λ^{n})-module, which in turn implies that 𝔒_{Q}(L; λ^{n}) is not 𝔄(L; λ^{n})-free.

§5. Realisable Classes. In this section we give a geometric interpretation of the homomorphism ψ which enables us to relate the image of ψ to the Hasse-Weil zeta function of A when A is a constant variety. This raises interesting questions regarding the analogous situation over number fields which we hope to pursue elsewhere.

Let us begin by recalling certain basic facts regarding the theory of classgroups of sheaves (see [C2]). We identify C with its associated scheme over Spec k^{c}, and we write O for the structure sheaf of C/k^{c}. Set L′ = L ⊗_{k} k^{c}. The places of L′ may be identified with the set of closed points of C/k^{c}. Let G be a finite group with (|G|, p) = 1. Define OG to be the sheaf of rings whose sections over each open set U of C/k^{c} are given by OG(U) = O(U)G. If S is a finite collection of closed points of C and U = C − S then OG(C − S) = 𝔒_{L′,S}G, where here 𝔒_{L′,S}G denotes the ring of functions in L′ which are regular away from S.

Let 𝔐 be a sheaf of OG-modules. 𝔐 is said to be locally free of rank r if each stalk 𝔐_{P} is a free O_{P} G-module of rank r. Thus, if U = C − S is an open subset of C as above, then 𝔐(U) is a locally free OG-module. The locally free classgroup Cl(OG) of OG is defined to be Map(Ĝ, Pic(C)), where Pic(C) denotes the Picard group of C. Each locally free OG-module 𝔐 defines an element (𝔐) ∈ Cl(OG) as described in §2 of [C2]. It is shown in [C2] that for all such open subsets U of C, (𝔐(U)) ∈ Cl(OG(U)) = Cl(𝔒_{L′,S}G) is completely determined by (𝔐) ∈ Cl(OG).

Assume throughout this section that A/k is a constant variety. Suppose that λ ∈ End(A) is such that the kernel of λ : A(k^{c}) → A(k^{c}) is cyclic of prime order l with (l, p) = 1. Suppose that Q ∈ A(L). Then Q defines a k-rational section Q̃ : C → A of the structure map A → C, and for each i ∈ ℕ, we may form the pullback diagram (of varieties over k^{c}):

\begin{aligned}A & ⟵ & C_{Q}(i) \\ _{λ^{i}}↓ & & ↓^{η(i)} \\ A & ⟵_{Q̃} & C \end{aligned}  (5.1)

C_{Q}(i)/C is a possibly reducible, everywhere unramified Galois cover with Galois group G_{i} ≃ Ker(λ^{i} : A(k^{c}) → A(k^{c})) which is cyclic of order l^{i}. As C_{Q}(i)/C is Galois, all of the irreducible components of C_{Q}(i) are isomorphic to a variety B_{i}, say. Since (5.1) is a pullback diagram, it follows that the function field k^{c}(B_{i}) of B_{i}/k^{c} is isomorphic to the field L′(\frac{1}{λ^{i}} Q). We deduce from this and from (1.1) that the algebra of functions on C_{Q}(i)/k^{c} is isomorphic to Map_{Ω_{L′}} (G_{Q}(λ^{i}), L^{c}) = L′_{Q}(λ^{i}) = L′_{Q}(i), say.

Let O_{Q}(i) be the structure sheaf of the variety C_{Q}(i)/k^{c}. Since C_{Q}(i)/C is everywhere unramified, the direct image sheaf η(i)_{∗}O_{Q}(i) is a locally free OG_{i}-module, and so it yields an element (η(i)_{∗}O_{Q}(i)) ∈ Cl(OG_{i}). We have a decomposition (see e.g. [M] p.72)

η(i)_{∗}O_{Q}(i) = ⨁_{χ∈Ĝ_{i}} L_{χ}  (5.2)

where

L_{χ} = {α ∈ η(i)_{∗}O_{Q}(i) : α^{g} = χ(g)α} ∀g ∈ G_{i} (5.3)

Let (L_{χ}) denote the class of L_{χ} in Pic(C)(k^{c}). The following proposition is simply a reformulation of certain results contained in §2 of [C2]. Proposition 5.4. The class of η(i)_{∗}O_{Q}(i) in Cl(OG_{i}) is given by the map χ ↦ (L_{χ}). Proof. Write ξ for the generic point of C_{Q}(i)/C. Then (O_{G_{i}})_{ξ} = L′G_{i} and (η(i)_{∗}O_{Q}(i))_{ξ} = L′_{Q}(i). If P is a closed point of C, then (OG_{i})_{P} = O_{P} G_{i}, while the stalk (η(i)_{∗}O_{Q}(i))_{P} is equal to the semilocal ring of L_{Q} at P. Since η(i)_{∗}O_{Q}(i) is a locally free OG_{i} module of rank one, we may choose d_{Q} ∈ L′_{Q}(i) such that (η(i)_{∗}O_{Q}(i))_{ξ} = d_{Q}.(OG_{i})_{ξ}, and a_{P} ∈ L′_{Q}(i) such that (η(i)_{∗}O_{Q}(i))_{P} = a_{P} .(OG_{i})_{P} for each closed point P. This implies that the stalks of the line bundle L_{χ} are given by

L_{χ,ξ} = (d_{Q}|χ).O_{ξ}  (5.5(a))
L_{χ,P} = (a_{P} |χ).O_{P} , P closed  (5.5(b))

Hence the Weil divisor associated to L_{χ} is given by

D_{χ} = ∑_{P closed} v_{P} ((a_{P} |χ)(d_{Q}|χ)^{−1})P  (5.6)

However it is shown in §2 of [C2] (see Proposition 1 and the remarks following Lemma 4) that (η(i)_{∗}O_{Q}(i)) ∈ Cl(OG_{i}) is given by the map χ ↦ (D_{χ}), where (D_{χ}) denotes the class of (D_{χ}) in Pic(C)(k^{c}). This establishes the result.

Now if S is a non-empty set of closed points of C/k^{c} and U = C − S, then η(i)_{∗}O_{Q}(i)(U) = 𝔒_{Q}(L′; λ^{i}), and OG_{i}(U) = 𝔒_{L′}G_{i} = 𝔄(L′; λ^{i}), using the notation of §1. Thus (η(i)_{∗}O_{Q}(i)) ∈ Cl(OG_{i}) completely determines (𝔒_{Q}(L′; λ^{i})) ∈ Cl(𝔄(L′; λ^{i})) for all choices of S.

If S consists of a single place of degree one over k, then somewhat more is true. For each i, we have a Wedderburn decomposition

A(L; λ^{i}) = (L^{c}G_{i})^{Ω_{L}} = ∏_{r=1}^{t} L_{r}  (5.7).

Each L_{r}/L is a constant field extension, since the values taken by the characters in Ĝ_{i} lie in k^{c}. It follows from (5.7) that

𝔄(L; λ^{i}) = (𝔒^{c}G_{i})^{Ω_{L}} = ∏_{r=1}^{t} 𝔒_{L_{r}}  (5.8)

where 𝔒_{L_{r}} denotes the integral closure of 𝔒_{L} in L_{r}.

Since S consists of a single place of degree one over k, the natural homomorphism Cl(𝔒_{L_{r}}) → Cl(𝔒_{L′}) is injective for each r (see e.g. [C3], Lemma 2). This implies that the natural homomorphism Cl(𝔄(L; λ^{i})) → Cl(𝔄(L′; λ^{i})) is also injective. Hence, in this case (𝔒_{Q}(L; λ^{i})) ∈ Cl(𝔄(L; λ^{i})) is determined by (η(i)_{∗}O_{Q}(i)) ∈ Cl(OG_{i}). In particular, if λ^{i} does not divide Q in A(L′), then it follows from Theorem 1.2 that (η(i)_{∗}O_{Q}(i)) ≠ 0 in Cl(𝔒G_{i}).

Suppose for the rest of this section that λ ∈ End(A) is such that the kernel of λ : A(k^{c}) → A(k^{c}) is cyclic of prime order l with (l, p) = 1. We now turn our attention to the line bundles L_{χ}, χ ∈ Ĝ_{i}. Since C_{Q}(i)/C is everywhere unramified, the degree of the line bundle det(η(i)_{∗}O_{Q}(i)) is zero (see e.g. [H] Ch. IV Ex. 2.6). As C_{Q}(i)/C is cyclic and L_{χ_{1}} ⊗L_{χ_{2}} = L_{χ_{1}χ_{2}} for χ_{1}, χ_{2} ∈ Ĝ_{i}, we deduce that L_{χ} is a line bundle of degree zero for all χ ∈ Ĝ_{i}. Thus in fact (L_{χ}) ∈ Pic^{0}(C)(k^{c}) = J(k^{c}), where J denotes the Jacobian of C. Let H_{Q}(i) be the subgroup of J(k^{c}) generated by {(L_{χ})}_{χ∈Ĝ_{i}}. We shall refer to H_{Q}(i) as the associated subgroup of Q (at level i). H_{Q}(i) is cyclic of order dividing l^{i}, and it determines C_{Q}(i) up to isomorphism. It follows immediately from the definitions of C_{Q}(i) and H_{Q}(i) that H_{Q}(i − 1) = l.H_{Q}(i).

Let Q_{1}, …, Q_{r} be a basis of A(L) modulo torsion. For each Q_{j}, let λ^{s_{j}} denote the highest power of λ which exactly divides Q_{j} in A(L′). (So (η(i)_{∗}O_{Q}(i))) ∈ Cl(OG_{i}) is trivial for 0 ≤ i ≤ s_{j} and is non-trivial otherwise.) Write H_{j} = lim_{i^{⟵}} H_{Q_{j}}(i + s_{j}), where the inverse limit is taken with respect to the multiplication by l map. Theorem 1.2 implies that H_{j_{1}} ∩H_{j_{2}} = 0 if j_{1} ≠ j_{2}. Set H = H_{1} ×⋯×H_{r}. Then H is a subgroup of T_{l}(J), the l-adic Tate module of J.

Suppose that H = lim_{i^{⟵}} H_{i} is a subgroup of T_{l}(J) with each H_{i} cyclic of order dividing l^{i}. We shall say that H is realisable if H is a subgroup of H. This is equivalent to saying that for each i, there is a point Q^{(i)} ∈ A(L) such that H_{i} = H_{Q^{(i)}}(i).

Let σ ∈ Ω_{k} denote the Frobenius element of k. Write P_{A}(x) for the characteristic polynomial of σ acting on T_{l}(A). The following result characterises realisable subgroups. Theorem 5.9. Let H be a subgroup of T_{l}(J) as above. Then H is realisable if and only if H^{P_{A}(x)} = 0. Proof. Let O_{A} denote the structure sheaf of A/k^{c}. Exactly as in the discussion concerning the sheaf η(i)_{∗}O_{Q}(i), we have a direct sum decomposition

λ_{∗}^{i}O_{A} = ⨁_{χ∈Ĝ_{i}} L′_{χ}  (5.10)

where the L′_{χ} are line bundles of degree zero. For any point Q ∈ A(L), the group H_{Q}(i) is generated by {(Q̃^{∗}L′_{χ})}_{χ∈Ĝ_{i}}, where Q̃^{∗}L′_{χ} denotes the pullback of L′_{χ} along Q̃. As A and C are defined over k, we have

(Q̃^{∗}L′_{χ}^{σ}) = (Q̃^{∗}L′_{χ})^{σ}  ∀χ ∈ Ĝ_{i}.  (5.11)

It is a theorem of Tate (see e.g. the Main Theorem of [Ta] or Theorem 1 of Appendix 1 of [M]) that the natural homomorphism

ℤ_{l} ⊗_{ℤ} Hom_{k}(A, A) → Hom_{ℤ_{l}}(T_{l}(A), T_{l}(A))^{Ω_{k}}  (5.12)

is an isomorphism. This implies that P_{A}(σ) annihilates Pic^{0}(A)(k^{c}). It follows from (5.11) that H^{P_{A}(σ)} = 0, and so in particular that H^{P_{A}(σ)} = 0.

For the converse, recall that from standard theory we have

r = rank(A(L)) = rank_{ℤ}[Hom_{k}(J, A)].  (5.13)

Via the theorem of Tate quoted earlier,

rank_{ℤ}[Hom_{k}(J, A)] = rank_{ℤ_{l}}[Hom_{ℤ_{l}}(T_{l}(J), T_{l}(A))^{Ω_{k}}].  (5.14)

As σ acts semisimply on T_{l}(J) and T_{l}(A), it follows that

Ker(P_{A}(σ) : T_{l}(J) → T_{l}(J)) ≃ ℤ_{l}^{r}  (5.15)

(c.f. e.g. Theorem 1 of [Ta] or Theorem 2 of Appendix I of [M]). Write H′ = Ker(P_{A}(σ) : T_{l}(J) → T_{l}(J)). Then H ⊆ H′, H ≃ ℤ_{l}^{r}, and H_{j} is not contained in l.T_{l}(J) for j = 1, …, r. Hence H′ = H.

Recall that P_{A}(x) is the numerator of the zeta function of A/k. Thus, as in the tame number field case (see [T2] or [F]), and the cyclotomic function field case (see [C2]), we have yet another instance of the Galois module structure of rings of integers being determined by an L-function.

§5. References.

[A] A. Agboola, ‘Iwasawa theory of elliptic curves and Galois module structure of rings of integers’, to appear in Duke Math Journal.

[AT] A. Agboola, M. J. Taylor, ‘Class invariants of Mordell-Weil groups’, to appear.

[C1] R. J. Chapman, ‘Galois module structure in global function fields’, Ph.D. Thesis, Faculty of Technology, University of Manchester, 1988.

[C2] R. J. Chapman, ‘Classgroups of sheaves of locally free modules over global function fields’. In: The Arithmetic of Function Fields, eds D. Goss, D. R. Hayes and M. I. Rosen, Walter de Gruyter, Berlin, 1992.

[C3] R. J. Chapman, ‘Kummer theory and Galois module structure in global func-

tion fields’, Math. Z., 108, 375-388, (1991). [CN-S] Ph. Cassou-Noguès, A. Srivastav, ‘On Taylor’s conjecture for Kummer orders’,

Séminaire de Théorie des Nombres, Bordeaux, 2, 349-363, (1990). [CN-T] Ph. Cassou-Noguès, M. J. Taylor, Elliptic functions and rings of integers,

Birkhauser, Boston, (1987).

[CR] C. W. Curtis and I. Reiner, Methods of representation theory Vol. 1, Wiley,

1981.

[F] A. Fröhlich, Galois module structure of algebraic integers, Springer Verlag, New York, 1983.

[H] R. Hartshorne, Algebraic Geometry, Springer Verlag, New York, 1977.

[L] S. Lang, ‘Algebraic groups over finite fields’, American J. of Math. 78, 555-563, (1956).

[M] D. Mumford, Abelian varieties, Oxford University Press, Oxford, 1974.

[ST] A. Srivastav and M. J. Taylor, ‘Elliptic curves with complex multiplication and Galois module structure’, Inv. Math., 99, 165-184, (1990).

[Ta] J. Tate, ‘Endomorphisms of abelian varieties over finite fields’, Inv. Math., 2, 134-144, (1966).

[T1] M. J. Taylor, ‘Mordell-Weil groups and the Galois module structure of rings of integers’, Ill. J. Math., 32, 428-452, (1988).

[T2] M. J. Taylor, ‘On Fröhlich’s conjecture for rings of integers of tame extensions’, Invent. Math., 63, 41-79, (1981).

.

Abelian Varieties and Galois Module Structure In Global Function Fields A. Agboola

Department of Mathematics University of California Berkeley, CA 94720 U. S. A. Email: agboola@math.berkeley.edu Fax: (510)-642-8204 Running Title: Abelian Varieties and Galois Module Structure August 19, 2001

Abstract The purpose of this paper is to study the Galois module structure of certain

principal homogeneous spaces constructed via points of infinite order on a CM abelian variety defined over a global function field. We prove that, in general, such principal homogeneous spaces have non-trivial Galois structure. Under suitable hypotheses, we show how this fact may be described in terms of the zeta function of the abelian variety.