Isomorphism classes of Drinfeld modules over finite fields

We study isogeny classes of Drinfeld $A$-modules over finite fields $k$ with commutative endomorphism algebra $D$, in order to describe the isomorphism classes in a fixed isogeny class. We study when the minimal order $A[\pi]$ of $D$ occurs as an endomorphism ring by proving when it is locally maximal at $\pi$, and show that this happens if and only if the isogeny class is ordinary or $k$ is the prime field. We then describe how the monoid of fractional ideals of the endomorphism ring $\mathcal{E}$ of a Drinfeld module $\phi$ up to $D$-linear equivalence acts on the isomorphism classes in the isogeny class of $\phi$, in the spirit of Hayes. We show that the action is free when restricted to kernel ideals, of which we give three equivalent definitions, and determine when the action is transitive. In particular, the action is free and transitive on the isomorphism classes in an isogeny class which is either ordinary or defined over the prime field, yielding a complete and explicit description in these cases.


Introduction
Let F q be a finite field with q elements.Let A = F q [T ] be the ring of polynomials in indeterminate T with coefficients in F q , and let F = F q (T ) be the fraction field of A. Given a nonzero prime ideal p of A, we denote F p = A/p.Let k ∼ = F q n be a finite extension of F p .We consider k as an A-field via γ : A A/p ֒ k.Let τ be the Frobenius automorphism of k relative to F q , that is, the map α α q .Let k{τ } be the noncommutative ring of polynomials in τ with coefficients in k and commutation rule τ α = α q τ , α ∈ k.A Drinfeld module of rank r ≥ 1 over k is a ring homomorphism φ : A k{τ }, a φ a , such that (Note that φ is uniquely determined by φ T .)An isogeny u : φ ψ between two Drinfeld modules over k is a nonzero element u ∈ k{τ } such that uφ a = ψ a u for all a ∈ A (or equivalently such that uφ T = ψ T u).
The endomorphism ring E := End k (φ) of φ consists of the zero map and all isogenies φ φ; it is the centralizer of φ(A) in k{τ }.It is known that E is a free finitely generated A-module with r ≤ rank A E ≤ r 2 .We introduce a special element, π = τ n , the so-called Frobenius of k.Note that π lies in the center of k{τ }, and hence belongs to E.
Isogenies define an equivalence relation on the set of isomorphism classes of Drinfeld modules over k.The isogeny class of φ is determined by the minimal polynomial of π over F = φ(F ), cf.[16,Theorem 4.3.2].Since the properties of these polynomials are well understood, it is known how to classify Drinfeld modules over finite fields up to isogeny.
In this article, we investigate the isomorphism classes within a fixed isogeny class.This is an important and difficult question in the theory of Drinfeld modules, which can be approached from different viewpoints, cf.[14,10].Our approach is inspired by the work of Waterhouse [17] in the case of abelian varieties over finite fields and is partly aimed at producing efficient algorithms for explicitly computing a representative of each isomorphism class.We refer to Section 6 for a more in-depth comparison of our results to known results for abelian varieties.
When E is commutative, the endomorphism ring of a Drinfeld module isogenous to φ is an A-order in F (π) containing A[π].We start by investigating the natural question of when A[π] itself is an endomorphism ring of a Drinfeld module isogenous to φ.We prove the following: Theorem A. Let φ be a Drinfeld module over k such that End k (φ) is commutative.Then A[π] is the endomorphism ring of a Drinfeld module isogenous to φ if and only if either φ is ordinary or k = F p .
Next, we study isogenies from φ to other Drinfeld modules using the ideals of E. Let I E be a nonzero ideal.Since k{τ } has a right divison algorithm, we have k{τ }I = k{τ }u I for some u I ∈ k{τ }.This element defines an isogeny u I : φ ψ, where ψ is the Drinfeld module determined by ψ T = u I φ T u −1  I and is also denoted by I * φ.The map I I * φ induces a map S from the linear equivalences classes of ideals of E to the isomorphism classes of Drinfeld modules isogenous to φ.Generally, S is neither injective nor surjective.
It was observed by Waterhouse [17] in the setting of abelian varieties that S is injective when restricted to ideals of a special type, called kernel ideals.Kernel ideals were introduced in the context of Drinfeld modules by Yu [18].In Sections 3 and 4, we revisit Yu's definition, give two other equivalent definitions, and prove several general facts about kernel ideals.We also give an explicit example (Example 3.10) of a rank 3 Drinfeld module φ and an ideal I End k (φ) which is not kernel; as far as we know, this is the first such explicit example in the published literature.
In general, we have that End where (Equality holds when I is a kernel ideal; cf.Lemma 4.2.)Note that O I is an overorder of E, so S can be surjective only when E is the smallest order among the endomorphism rings of Drinfeld modules isogenous to φ.When E is a Gorenstein ring, we prove that any isogeny φ ψ such that End k (ψ) ∼ = O I for some (necessarily kernel) ideal I E arises from the map S via I I * φ = ψ.In other words, when E is Gorenstein, the image of S is the set of isomorphism classes in the isogeny class of φ whose endomorphism rings are overorders of E. Since A[π] is a Gorenstein ring, we arrive at the following: Theorem B. Assume that either k = F p or the isogeny class that we consider is ordinary, so that there is a Drinfeld module φ with End k (φ) = A[π].Then the map I I * φ from the linear equivalences classes of ideals of A[π] to the isomorphism classes of Drinfeld modules isogenous to φ is a bijection.
In [15], an algorithm is presented for computing the ideal class monoid of an order in a number field.In an ongoing project, we are working on adapting that algorithm to orders in function fields.Since for a given ideal I E computing I * φ is fairly straightforward, Theorem B combined with this algorithm will provide an efficient method for computing explicit representatives of isomorphism classes of all Drinfeld modules isogenous to φ such that End k (φ) ∼ = A[π].Let us mention that Assong [2] has recently described a brute-force algorithm to list isomorphism classes, based on a theoretical classification in terms of j-invariants and "fine isomorphy invariants", and implemented this for certain examples of isogeny classes of Drinfeld modules of rank 3. Our methods involve fractional ideals in endomorphism rings rather than invariants and explicit expressions for the coefficients of the Drinfeld module.
The outline of the paper is as follows.Section 2 contains our analysis of local maximality of A[π] at π, including a key result (Theorem 2.6) for the proof of Theorem A. Section 3 gives the definitions of kernel ideals and proves their equivalence, and Section 4 gives properties of kernel ideals and proves that every ideal is a kernel ideal when E is Gorenstein (Proposition 4.5).Section 5 contain our main results: we find which endomorphism rings can occur in a fixed isogeny class (Proposition 5.1), study the injectivity and surjectivity of the map I I * φ (Theorem 5.3), and prove when A[π] occurs as an endomorphism ring (Corollary 5.2), to obtain Theorem B (cf. Corollary 5.4).Finally, Section 6 contains a comparison between the results obtained in this paper and results from the literature ( [17,6,5]) on abelian varieties over finite fields.

Local maximality at π of the Frobenius order
As in the introduction, let A = F q [T ] and F = F q (T ), and let k = F q n be a finite A-field, i.e., a field equipped with a homomorphism γ : A k. We denote t = γ(T ).Let p A be the kernel of A. Then p is a maximal ideal such that F p := A/p = F q d is a subfield of F q n .We call d the degree of p; note that d divides n.By slight abuse of notation, we denote the monic generator of p in A by the same symbol.
Let φ : A k{τ } be a Drinfeld module of rank r, and let π = τ n .The results about the endomorphism algebra of φ that we use in this section are well-known and can be found, for example, in [9,14,18], but for the convenience of a single reference and consistency of notation, we will refer to [16].
Let K := F q (π) be the fraction field of F q [π] ⊆ k{τ } and define Then k(τ ) is a central division algebra over K of dimension n 2 , split at all places of K except at (π) and (1/π), where its invariants are 1/n and −1/n, respectively; see [16,Proposition 4.1.1].Extend φ to an embedding φ : F k(τ ).Then where Cent R (S) = {x ∈ R | xs = sx for all s ∈ S} denotes the centralizer of a subset S of a ring R. To simplify the notation, we will denote φ(A) by A and φ(F ) by F , with φ being fixed.Let Let B be the integral closure of A in F .There is a unique place p in F over the place (π) of K; see [16,Theorem 4.1.5].Let F p := F ⊗ K F q ((π)) be the completion of F at p, and let B p be the ring of integers of F p .
Definition 2.1.Given an A-order R in B containing π, let We say that R is locally maximal at Remark 2.2.Suppose E is commutative.Then E can be considered as an A ′ -order in F .It is observed in [18, p. 164] and [1, p. 514] that E is locally maximal at π. Therefore, for A[π] to be an endomorphism ring of a Drinfeld module isogenous to φ it is necessary for A[π] to be locally maximal at π.We investigate this condition in this section; later we will show that it is also sufficient; cf.Proposition 5.1.
Let m(x) be the minimal polynomial of π over F , so that F ∼ = F [x]/(m(x)).Note that π ∈ E is integral over A, so the polynomial m(x) is monic with coefficients in A and A[π] ∼ = A[x]/(m(x)).To analyze the local maximality of A[π], it will be convenient to change the perspective and express A[π] as a quotient of A ′ [x].To do so, consider T and π as two independent indeterminates over F q .Then consider Lemma 2.3.We have: (1) m(T ) is irreducible in K[T ] and has degree ] be the polynomial obtained by reducing the coefficients of m(T ) modulo π.Then, up to an F × q -multiple, m(T ) is equal to p [ F :K]/d .Proof.By [16,Theorem 4.2.7], the degree in T of m(0) is strictly larger than the degrees of the other coefficients of m(x).Hence, the leading term of m(T ) is the leading term of m(0) ∈ A, so its leading coefficient is in F × q .Moreover, by [16, Theorem 4.2.2 and Theorem 4.2.7], up to an F × q -multiple, m(0) is equal to p Lemma 2.4.The following hold: (1) The ideal M of A[π] p generated by π and p is maximal. (2) Proof.We have (Note that F ∼ = K[T ]/( m(T )) and, because there is a unique place in F over π, [16, proof of Theorem 2.8.5] implies that m(T ) remains irreducible over F q ((π)).) The element π of A[π] p is not a unit.Now This shows that p is also not invertible in A[π] p and where F p (resp.K π ) denotes the completion of F (resp.K) at p (resp.(π)), and where e and f denote the ramification index and the residue degree of the corresponding extension, respectively.

Proposition 2.5. A[π] is locally maximal at π if and only if one of the following holds:
• f F = 1 and e F = 1; • f F = 1 and e K = 1.
Proof.Let ord p be the normalized valuation on F corresponding to the place p. Then e K = ord p (π), e F = ord p (p).
Suppose that A[π] p = B p .Then, using the notation of Lemma 2.4, we have M = p and F p := B p / p = A[π] p /M = F p .Since π and p generate M, at least one of them must have ord p equal to 1. Hence, either e K = 1 or e F = 1.Next, f F , by definition, is the degree of the extension F p /F p .Hence f K = 1.
Conversely, suppose that one of the given conditions holds.Then the residue field of B p is F p and either p or π is a uniformizer of B p .By the structure theorem of local fields of positive characteristic, we have B p = F p p or B p = F p π .On the other hand, by Lemma 2.4, Theorem 2.6.Let H be the height of φ (see [16,Lemma 3.2.11]for the definition). Then with equality if and only if A[π] is locally maximal at π.
Proof.We have the following equalities: On the other hand, by [16,Proposition 4.1.10], Hence On the other hand, Thus, the inequality of the theorem is equivalent to Since e K , e F , f F are positive integers, the above inequality always holds, with equality if and only if f F = 1 and either e K = 1 or e F = 1.Now the theorem follows from Proposition 2.5.
Remark 2.7.The advantage of having the inequality of Theorem 2.6, rather than the statement of Proposition 2.5, is that instead of computing each of e K , e F , f F individually it combines these numbers into quantities that are easier to compute.
which implies that equality in Theorem 2.6 holds.
Therefore, the inequality of Theorem 2.6 becomes so it is an equality.Since n/d is a positive integer, an equality holds if and only if either Example 2.11.Let p = T , r = 2, and n = 3.Let φ T = τ 2 , so φ is supersingular.In this case, the characteristic polynomial of the Frobenius is Hence A[π] is locally maximal at π if and only if either For example, when q = 3, p = T 2 + T + 2, φ T = t + τ 4 , we calculate that φ p = (2t + 1)τ 2 + τ 8 , which tells us that H = 2.We also calculate the minimal polynomial for T over K, which is given by m Example 2.13.Suppose q = 3, n = 8, and p = T 2 + T + 2. Let Then, H = 2 and m(x so equality in Theorem 2.6 holds.In this case, A[π] is maximal at π.
The next four examples show that the quantities in Proposition 2.5 are essentially independent of each other.
Notice that e K /e F = 3/2 and e K f F = 3 imply that e K = 3, e F = 2, f F = 1, and Example 2.17 (Not locally maximal despite e F = e K = 1).Assume d is odd, q is odd, and n/d = [k : F p ] = 2. Then there is a supersingular Drinfeld module of rank 2 over k whose minimal polynomial is [16,Example 4.3.6].In this case, p remains inert in F , so e F = 1 and f F = 2. Since n/Hd = e K /e F and H = 2, we see that e K = 1.

Kernel ideals: Definitions
We keep the notation of the previous section but from now on we assume that E = End k (φ) is commutative.
Let I E be a nonzero ideal.Let k{τ } I be the left ideal of k{τ } generated by the elements of I. Then k{τ } I is generated by a single element u I ∈ k{τ } since k{τ } has right division algorithm.Thus, k{τ } I = k{τ } u I .It follows that , then ψ is a Drinfeld module over k of rank r and u I : φ ψ is an isogeny.We denote ψ = I * φ.Proof.Let J := k{τ } I ∩ D and J ′ := Ann E (φ[I]).Suppose u ∈ J. Then u ∈ E and u = wu I for some w ∈ k{τ }.But wu I annihilates ker(u I ) = φ[I], so u ∈ J ′ .This implies that J ⊆ J ′ .Conversely, if u ∈ J ′ , then by Lemma 2.1.1 in [14] we have u = wu I for some w ∈ k{τ }.Hence u ∈ k{τ } u I ∩ E = J, so J ′ ⊆ J.
Let φ and ψ be two Drinfeld module over k of rank r.Let l be a prime not equal to p = char A (k).Let u : φ ψ be an isogeny.Then u induces a surjective homomorphism φ k u − ψ k of A-modules with finite kernel, where the notation φ k means that the A-module structure on k is induced from φ : A k{τ } and likewise for ψ.From this, we get the short exact sequence A l (F l /A l , ker(u) l ) − 0, where ker(u) l denotes the l-primary part of ker(u) (this is an étale group scheme).Note that T l (φ) := Hom A l (F l /A l , φ k) is the l-adic Tate module of φ and that whose cokernel is isomorphic to ker(u) l .On the other hand, on Taking the A l -duals of (2), we obtain Hence to the isogeny u there corresponds a canonical sublattice of H l (φ) whose cokernel is isomorphic to ker(u) l .Now given a nonzero ideal I E, we would like to describe the sublattice of H l (φ) corresponding to u I .Before doing so we recall an elementary result about the duals of intersections of lattices.
Let R be a PID with field of fractions K. Let V = K n .A lattice in V is the R-span of a basis of V , i.e., a lattice is a free R-submodule Λ ⊆ V of rank n such that ΛK = V .Fix a basis {e 1 , . . ., e n } of V and define a symmetric K-bilinear pairing •, • : V × V K by defining e i , e j = δ ij (= Kronecker symbol) and extending it bilinearly to V × V .We identify V * := Hom K (V, K) with the linear functionals on V and take e * i (v) = e i , v as a basis of V * .For a lattice Λ in V , the dual lattice Λ * ⊆ V * is the lattice defined by If we identify V * with V by mapping e * i e i for all 1 ≤ i ≤ n, then is a lattice, and so is Lemma 3.5.We have Proof.The proof is omitted since it is fairly straightforward.Now returning to u I , let α, β ∈ I be nonzero elements.The overlattice of T l (φ) corresponding to ker(α) ∩ ker(β) is α −1 T l (φ) ∩ β −1 T l (φ).The sublattice of H l (φ) corresponding to ker(α) l is αH l (φ), so (α −1 T l (φ)) * = αH l (φ).From the previous lemma, we conclude that the sublattice of H l (φ) corresponding to ker(α) ∩ ker(β) is αH l (φ) + βH l (φ).Thus, the dual of u −1 I T l (I * φ) is IH l (φ) and we have proved: Lemma 3.6.The sublattice of H l (φ) corresponding to ker(u I ) l is IH l (φ).
Let O k be the ring of integers of the unramified extension F k of F p with residue field k.Let H p (φ) be the Dieudonné module of φ as defined in [14, Sec.2.5].Recall that H p (φ) is a free O k -module of rank r equipped with a τ deg(p) -linear map f φ,p : H p (φ) H p (φ) such that where the product is over all primes of A, including p.According to [14, Lemma 2.6.2],there is a bijection between the kernels of isogenies u : φ ψ and sublattices M = l A M l ⊆ H(φ) such that M l = H l (φ) for all but finitely many primes l and The quotient l =p (H l (φ)/M l ) defines a unique finite étale k-subscheme G p ⊆ G a,k in φ(A)-modules.Similarly, the quotient O k -module H p (φ)/M p endowed with the τ deg(p)linear map induced by f φ,p defines a unique k-subscheme G p ⊆ G a,k in φ(A)-modules.
The quotient of φ by G p × G p is the isogeny corresponding to M.
Proof.We already proved this for l = p.On the other hand, H p (φ) is the contravariant Dieudonné module, so u I (H p (I * φ)) is the submodule generated by all αH p (φ), α ∈ I. Hence u I (H p (I * φ)) = IH p (φ).
Definition 3.8.Let I be a nonzero ideal of E. We say that I is a kernel ideal if for any ideal J E the inclusion JH(φ) ⊆ IH(φ) implies J ⊆ I.
Proof.Note that by the previous discussion, JH(φ) ⊆ IH(φ) if and only if Suppose I is a kernel ideal in the sense of Definition 3.3 and Hence I is a kernel ideal in the sense of Definition 3.8.Conversely, suppose that I is a kernel ideal in the sense of Definition 3.8.Denote J = Ann E φ[I].We have I ⊆ J, and we need to show that this is an equality.For any α ∈ J, ker(α) contains φ[I], so φ[I] ⊆ φ[J].This implies JH(φ) ⊆ IH(φ).Hence J ⊆ I.
The next example shows that in general not every ideal of E is a kernel ideal.
We algorithmically compute, cf.[8], that an A-basis for E is given by e 1 , e 2 , e 3 , where We also compute that In k{τ }, we have These polynomials satisfy the equation where We also have Hence, (T + 1) 2 ∈ k{τ } w ⊆ k{τ } I.But I ∩ A = (T + 1) 3 A, so (T + 1) 2 ∈ I.This proves that I is not a kernel ideal.

Kernel ideals: Properties
We keep the notation and assumptions of the previous section.In particular, φ is a Drinfeld module over k such that E := End k (φ) is commutative, and The next lemma is the analogue of [17,Theorem 3.11].
for some u ∈ D.
).On the other hand, because u ∈ O I , we have Proof.First, consider a nonzero principal ideal αE.We have k{τ } αE = k{τ } α.Suppose u = gα ∈ k{τ } α and u ∈ D. Then g = uα −1 ∈ D and g ∈ k{τ }, so g ∈ E. Therefore, u ∈ αE.This implies that αE ⊆ k{τ } α ∩ D ⊆ αE, so (k{τ } (αE)) ∩ D = αE, i.e., αE is a kernel ideal.Now let I E be an arbitrary nonzero ideal.Since E is maximal, there is an ideal J E such that IJ = αE is principal.We have where the last equality follows from the earlier considered case of principal ideals.Now I ′ := k{τ } I ∩ D is an ideal of E, and we have I ′ J ⊆ IJ.Multiplying both sides by J −1 ⊆ D, we get I ′ ⊆ I. Since I ⊆ I ′ , we have I ′ = I, so I is a kernel ideal.Definition 4.4.We say that E is Gorenstein if E l := E ⊗ A A l is a Gorenstein ring for all primes l A, i.e., Hom A l (E l , A l ) is a free A l -module of rank 1; cf.[4].
Note that the maximal A-order in D is Gorenstein, so the next proposition implies Lemma 4.3.Proposition 4.5.If E is Gorenstein then every nonzero ideal of E is a kernel ideal.
Proof.Let I and J be nonzero ideals of ) is also a free E l -module of rank 1; cf.[8,Def. 4.8].Hence, the inclusion JH l (φ) ⊆ IH l (φ) implies that J l ⊆ I l , where J l := J ⊗ A A l and I l := I ⊗ A A l .
At p we consider the decomposition where the sum is over the places of F = D lying over p and D ν is the completion of D at ν.There is a natural isomorphism (cf.[14, Theorem 2.5.6]) where End(H p (φ)) denotes the ring of endomorphisms of H p (φ) compatible with the action of the Frobenius f φ,p .By [14, Corollary 2.5.8], the splitting (7) induces a compatible splitting Here E p is the completion of E in B p, and E ′ p = ⊕ j E j is a direct sum of finitely many local rings corresponding to places ν = p lying over p, and T p (φ) = lim − φ[p n ]( k) denotes the p-adic Tate module of φ.By [18, Corollary, p. 164] we have that E p = B p is maximal, hence a DVR, which implies that H c p (φ) is a free E p-module.Further, since E ′ p is Gorenstein by assumption, one can apply the argument in the proof of [8, Theorem 4.9] to (9) to conclude that H ét p (φ) is a free E ′ p -module.Combining these statements yields that JH p (φ) ⊆ IH p (φ) also implies that J p ⊆ I p .
Finally, consider I l as an A l -submodule of D ⊗ F F l for any place l including p. Then Hence I is a kernel ideal by Definition 3.8.

Endomorphism rings and ideal actions
We keep the notation and assumptions of the previous section.In particular, φ is a Drinfeld module over k of rank r such that E = End k (φ) is commutative.
Given an A-order R in F = D = E ⊗ A F and a prime l✁A, we denote R l = R⊗ A A l .Also, given a prime ν of B, we denote by B ν the completion of B at ν and by R ν the completion of R in B ν .
Proposition 5.1.Let R be an A-order in D containing π. Then there is a Drinfeld module ψ in the isogeny class of φ such that End k (ψ) = R if and only if R is locally maximal at π.
Proof.This is proved in [3,Theorem 1.5].We present a slightly different argument.
If R is the endomorphism ring of a Drinfeld module isogenous to φ, then R contains π and is locally maximal at π [18, Corollary, p. 164].
Conversely, assume R is locally maximal at π.It is enough to show that there is a Drinfeld module ψ in the isogeny class of φ such that End k (ψ) l = R l for all the primes l of A.
Pick any Drinfeld module φ 0 in the isogeny class.For any l = p, it follows from our assumptions that the rational Tate module V l (φ 0 ) is free of rank 1 over D l .It therefore contains lattices L with any order O L = {x ∈ D : xL ⊆ L} (cf.( 5)), and, identifying V l (φ 0 ) ≃ D l , we see that such a lattice is Galois invariant if and only if its order contains π.
For any prime l = p, we view both End k (φ 0 )) and R l as lattices in V l (φ 0 ).Hence, both End k (φ 0 ) and R are maximal at all but finitely many primes l.In particular, there exist only finitely many primes, l 1 , . . ., l n say, at which End k (φ 0 ) l = R l .
The lattice R l 1 has order {x ∈ D : xR l 1 ⊆ R l 1 } = R l 1 , and so does its dual R * l 1 ≃ R l 1 .As in (3), let H l 1 (φ 0 ) denote the dual of T l 1 (φ 0 ) and consider the intersection H l 1 (φ 0 ) ∩ R * l 1 .This is an order contained in R * l 1 and we consider the index χ := χ(R * l 1 /(H l 1 (φ 0 ) ∩ R * l 1 )), which is a product of non-zero A-ideals.We have (7).In this case, we have that the rational Dieudonné module p , where each summand (H p (φ 0 ) ⊗ F p ) ν is free over D ν , and therefore contains lattices with any order.Comparing End(φ 0 ) ν and R ν at each ν = p over p as lattices in D ν , and adjusting the former if necessary via an analogous procedure to that in the previous paragraph, yields a sublattice ⊕ ν =p L ν of H ét p (φ 0 ).At p, we set L p = H c p (φ 0 ).
By the dictionary between sublattices of H(φ 0 ) = l A H l (φ 0 ) and isogenies, the quotient of l =p H l (φ 0 ) × H p (φ 0 ) by l =p L l × ν|p L ν yields a finite A-invariant subgroup G; cf.[14,Section 2.6].The quotient φ 0 /G in turn yields a Drinfeld module ψ isogenous to φ 0 , for which End k (ψ) l = R l at all places l = p of A, and End k (ψ) ν = R ν at all primes ν|p of F with ν = p.Finally, by [18,Corollary,p. 164], End k (ψ) is locally maximal at π, so we also have End k (ψ) p = R p .(1) The map I I * φ defines an action of the monoid of fractional ideals of E up to linear equivalence on the set of isomorphism classes of Drinfeld modules in the isogeny class of φ whose endomorphism ring is the order of an E-ideal (and hence an overorder of E).
(2) Upon restricting to kernel ideals, the action is free.
(3) If E is Gorenstein, then the action is also transitive on the set of all Drinfeld modules whose endomorphism ring is the order of an E-ideal.In other words, if E is Gorenstein, then every submodule M of H(φ) is of the form IH(φ) for some nonzero ideal I E. By the dictionary between lattices and isogenies given above, the kernel G gives rise to a sublattice of N l ⊆ H l (φ) such that H l (φ)/N l ≃ G l for each l = p and a sublattice N p ⊆ H p (φ) satisfying (4)  Note that at all places, we may scale the ideal generators to lie in the local endomorphism ring.We conclude that p of E p .These local ideals I p and I l for all l = p, i.e., local integral lattices, are the localizations of a global lattice (again since I l = E l for all but finitely many l), which is closed under the action of E since it is so everywhere locally by construction.Hence, it is a global ideal I, as we had to show.(2) It follows from Theorem 5.3 that the number of isomorphism classes in the isogeny class is bounded below by the sum of the class numbers of the overorders of E and that equality holds if E is minimal and Bass, so that every overorder is Gorenstein.For rank 2 Drinfeld modules, this result can also be found in [10, §6] where the class numbers are given as products involving Dirichlet characters.In higher rank, analogous expressions for the class numbers of the orders in D could be given.(3) The problem of describing endomorphism rings of Drinfeld modules has been considered by several authors, see e.g.[1,9,18].Explicit algorithms to compute endomorphism rings were developed in [7] for Drinfeld modules of rank 2 over F p and in [8] for any Drinfeld module with commutative endomorphism algebra; in [12], the existence of an effective general algorithm is shown.

Comparison with abelian varieties over finite fields
There are striking resemblances of the theory of Drinfeld modules over finite fields with the theory of abelian varieties over finite fields.Isogeny classes of such abelian varieties are also determined by the minimal or characteristic polynomial of their Frobenius endomorphism π, and it is an important open problem to describe the isomorphism classes within a fixed isogeny class.Indeed, precisely when the varieties are ordinary or defined over the prime field F p , there exist categorical equivalences between isomorphism classes of abelian varieties over F q and certain Z[π, π]-ideals, where π = q/π is the dual of the Frobenius, also called the Verschiebung.
First, consider an isogeny class of simple ordinary abelian varieties over F q determined by a Frobenius endomorphism π.It is known that any ordinary variety A/F q admits a (Serre-Tate) canonical lifting Ã to the Witt vectors W = W (F q ), which may be embedded into C.In [6], Deligne shows that the functor A H 1 ( Ã ⊗ W C) induces an equivalence of categories between isomorphism classes in the isogeny class determined by π and free Z-modules of rank 2 dim(A) equipped with an endomorphism F acting as π and an endomorphism V such that F V = q playing the role of Verschiebung; these modules are often called Deligne modules.On the other hand, complex abelian varieties A C are determined by lattices via the equivalence A C A C (C) ∼ = C g /Λ induced from complex uniformization, and when A C has CM through a CM-type Φ, we may write Λ = Φ(I) for some fractional End(A C )-ideal I.
In this way, we may associate a fractional ideal I to each ordinary abelian variety A/F q , since each variety over F q has CM and therefore so does its canonical lifting Ã. Linearly equivalent fractional ideals yield homothetic lattices and hence isomorphic abelian varieties, and homomorphisms between abelian varieties are described by quotient ideals.Put differently, fractional ideals up to linear equivalence act on the isomorphism classes in the ordinary isogeny class.
By comparison, it should follow with a similar proof that ordinary Drinfeld modules over k admit a canonical lifting to C ∞ of A-characteristic zero.On the one hand, Drinfeld modules over C ∞ admit a analytic uniformization by a lattice Λ ⊆ C ∞ (where homothetic lattices describe isomorphic Drinfeld modules), which yields a bijection between lattices in C ∞ and Drinfeld modules over C ∞ .On the other hand, the ideal action φ I * φ may be defined for arbitrary Drinfeld modules over any A-field (i.e., of any characteristic).Indeed, as alluded to above, the Picard group of fractional ideals of an order O up to linear equivalence acts simply transitively on the isomorphism classes of Drinfeld modules over C ∞ with CM by O. Ideals of E may be embedded in C ∞ as lattices, and every lattice Λ ⊆ C ∞ yields an ideal χ(Λ/E)Λ E. One can

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Thus, deg T m(T ) = [ F : K].Since F is obtained by adjoining a root of m(T ), the equality deg T m(T ) = [ F : K] implies that m(T ) is irreducible over K. Finally, note that m(T ) = m(0), which completes the proof of the lemma.

Corollary 2 . 10 .
Assume End k (φ) is commutative.Then A[π] is locally maximal at π if and only if either φ is ordinary or k = F p .Proof.By [16, Theorem 4.1.5],End k (φ) is commutative if and only if [ F : F ] = r.Now, as in the previous proof, [ F : K] = n.The inequality of Theorem 2

and α 1
, . . ., α m ∈ I.This implies that φ[I] ⊆ ker(u I ).Each ker(α), α ∈ I, is an E-module scheme, so φ[I] is an E-module scheme.The annihilator Ann E (φ[I]) of this module scheme is an ideal of E. It follows immediately from the definition that I ⊆ Ann E (φ[I]).Definition 3.3.We say that I is a kernel ideal if I = Ann E (φ[I]).This definition is the analogue of the definition of this concept in the setting of abelian varieties; see[17, p. 533].Lemma 3.4.We have Ann E (φ[I]) = k{τ } I ∩ D, so Definitions 3.1 and 3.3 are equivalent.

)Lemma 4 . 2 .
Now assume that I * φ ∼ = J * φ, or equivalently cu I = u J u. Thenk{τ } cu I = k{τ } u I = k{τ } I and k{τ } u J u = k{τ } Ju.Note that k{τ } Ju ∩ D = (k{τ } J ∩ D)u, so if I and J are kernel ideals, then Ju = (k{τ } J ∩ D)u = k{τ } Ju ∩ D = k{τ } I ∩ E = I.Let (5) O I := {g ∈ D | Ig ⊆ I} be the (right) order of I in D. The next lemma is the analogue of [17, Proposition 3.9].Let I be a nonzero ideal in E and write k{τ } I = k{τ } u I with u I ∈ k{τ }. (1) We have u I O I u −1 I ⊆ End k (I * φ).(2) If I is a kernel ideal, then u I O I u −1 I = End k (I * φ).Proof.(1) Let u ∈ O I .By definition, u ∈ D, so it commutes with φ T in k(τ ).Therefore, a kernel ideal.Then k{τ } I ∩ D = I and (k{τ } I(u −1 I wu I )) ∩ D = (k{τ } I ∩ D)(u −1 I wu I ) = I(u −1 I wu I ), where the first equality follows from the fact that u −1 I wu I ∈ D. We see that I(u −1 I wu I ) ⊆ I, so u −1 I wu I ∈ O I .This proves that End k (I * φ) ⊆ u I O I u −1 I , which combined with the reverse inclusion proved earlier implies that End k (I * φ) = u I O I u −1 I .The next lemma is the analogue of [17, Theorem 3.15].Lemma 4.3.Assume E is the maximal A-order in D. Then every nonzero ideal of E is a kernel ideal.

Corollary 5 . 2 .Theorem 5 . 3 .
The ring A[π] is the endomorphism ring of a Drinfeld module isogenous to φ if and only if either φ is ordinary or k = F p .Proof.By Proposition 5.1, A[π] is the endomorphism ring of a Drinfeld module in the isogeny class of φ if and only if A[π] is locally maximal at π. On the other hand, Corollary 2.10 states that A[π] is locally maximal at π if and only if either φ is ordinary or k = F p .We saw in Section 3 that, given a Drinfeld module φ over k and an ideal I E = End k (φ), we can construct an isogenous Drinfeld module ψ = I * φ, which is determined by ψ T = u I φ T u −1 I and which satisfies End k (ψ) ⊇ u I O I u −1 I ≃ O I ⊇ E by Lemma 4.2.(2).Consider the isogeny class of a Drinfeld module φ over k with commutative endomorphism algebra.

Proof. ( 1 )
By Lemma 4.1.(1),we may consider the fractional ideals of E up to linear equivalence.The trivial ideal I = E, considered as a k{τ }-ideal, is generated by the trivial element, so E * φ = φ for any φ.For two ideals I, J it follows from the definition and commutativity that (I • J) * φ = I * (J * φ).As remarked above, for any ideal I, the Drinfeld modules φ and I * φ are isogenous via the generator u I of k{τ }I.(2) This follows from Lemma 4.1.(2).(3) The proof is inspired by [17, Proofs of Theorem 4.5 and Theorem 5.1].Suppose that φ and ψ are isogenous and that R := End k (ψ) is the order of an E-ideal, i.e., R ≃ O I for some ideal I E. We may write ψ = φ/G where the finite subgroup scheme G is the kernel of the isogeny.We want to show that ψ ∼ = I * φ.Since I is a kernel ideal by Proposition 4.5, by Proposition 3.7 this amounts to showing that the sublattice corresponding to the isogeny φ ψ with kernel G is IH(φ), up to linear equivalence.(Note also that the Drinfeld module I * φ indeed has endomorphism ring u I O I u −1 I ≃ O I by Lemma 4.2.(2).)

Corollary 5 . 4 . 1 I∼
Suppose that E = A[π] (so that either φ is ordinary or k = F p , by Lemma 5.2).Then the action I I * φ of the monoid of fractional ideals of A[π] is free and transitive on the isomorphism classes in the isogeny class of φ.Proof.Since we consider the fractional ideals up to linear equivalence, we may without loss of generality consider only integralA[π]-ideals.Since A[π] is Gorenstein (cf.[8,Proposition 4.10]), every A[π]-ideal is a kernel ideal by Proposition 4.5.The statement now follows from Theorem 5.3 since every endomorphism ring is an overorder of A[π]; note that all such overorders occur as endomorphism rings by Proposition 5.1.Remark 5.5.(1) The ideal action already appears in [11, Section 3] in a slightly different setting: fix an A-order O and consider the Picard group Pic(O), i.e., the quotient group of invertible O-ideals modulo principal ideals.Hayes shows that Pic(O) acts on the isomorphism classes of Drinfeld modules whose endomorphism ring contains O. Invertible O-ideals are proper and therefore have order O; so this statement is consistent with the statement End k (I * φ) ⊇ u I O I u −= O I which we prove in Lemma 4.2.
Hence A ֒ B p extends to an embedding A p ֒ B p .Since A[π] p is complete, the image of A p lies in A[π] p .This proves (3).
such that H p (φ)/N p ≃ G p .The lattice N p in H p (φ) is both a free left O k -module and a right E p -module; by the splittings of E p = E p ⊕ E ′ = N p ⊕ N ′ p splits as well, where N p is a sublattice of H c p (φ) and an O k ⊗ E p-module, and N ′ p is a sublattice of H ét p (φ) and an O k ⊗ E ′ p -module.As remarked in the proof of Proposition 4.5, the Gorenstein property implies that H l (φ) is free over E l of rank 1 for all l = p, and H ét p (φ) is free over E ′ p .Hence, any sublattice of H l (φ) is of the form I l • H l (φ) for some local ideal I l E l , and any sublattice of H ét p (φ) is of the form I ′ p • H ét p (φ) for some ideal I ′ p E ′ p .Since G is finite, we know that I l = E l for all but finitely many l; note also that there are only finitely many ν = p over p that contribute to I ′ p .Recall that E p is maximal by [18, Corollary, p. 164], hence a PID, so again any sublattice of H c p (φ) = (H p (φ)) p is of the form I p • H c p (φ) for a local principal ideal I p E p. p