EXPLICIT CLASS FIELD THEORY FOR GLOBAL FUNCTION FIELDS. 1. Introduction C F

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1 EXPLICIT CLASS FIELD THEORY FOR GLOBAL FUNCTION FIELDS DAVID ZYWINA Astract. Let F e a gloal function field and let F a e its maximal aelian extension. Following an approach of D. Hayes, we shall construct a continuous homomorphism ρ: Gal(F a /F ) C F, where C F is the idele class group of F. Using class field theory, we shall show that our ρ is an isomorphism of topological groups whose inverse is the Artin map of F. As a consequence of the construction of ρ, we otain an explicit description of F a. Fix a place of F, and let A e the suring of F consisting of those elements which are regular away from. We construct ρ y comining the Galois action on the torsion points of a suitale Drinfeld A-module with an associated -adic representation studied y J.-K. Yu. 1. Introduction In the memory of David Hayes Let F e a gloal field, that is, a finite field extension of either Q or a function field F p (t). Fix an algeraic closure F of F, and let F sep e the separale closure of F in F. Class field theory gives a description of the maximal aelian extension F a of F in F sep. Let θ F : C F Gal(F a /F ) e the Artin map where C F is the idele class group of F ; see 1.5 for notation and [Tat67] for ackground on class field theory. The map θ F is a continuous homomorphism. By taking profinite completions, we otain an isomorphism of topological groups: θ F : ĈF Gal(F a /F ). So θ F gives a one-to-one correspondence etween the finite aelian extensions L of F in F sep and the finite index open sugroups of C F. For a finite aelian extension L/F, the corresponding open sugroup of C F is the kernel U of the homomorphism C F θ F Gal(F a /F ) Gal(L/F ) where the second homomorphism is the restriction map σ σ L (the group U is computale; it equals N L/F (C L ) where N L/F : C L C F is the norm map). However, given an open sugroup U with finite index in C F, the Artin map θ F does not explicitly produce the corresponding extension field L/F (in the sense, that it does not give a concrete set of generators for L over F ; though it does give enough information to uniquely characterize it). Explicit class field theory for F entails giving constructions for all the aelian extensions of F. We shall give a construction of F a, for a gloal function field F, that at the same time produces the inverse of θ F (without referring to θ F or class field theory itself) Context. David Hayes [Hay74] provided an explicit class field theory for rational function fields F = k(t). He uilt on and popularized the work of Carlitz from the 1930s [Car38], who had constructed a large class of aelian extensions using what is now called the Carlitz module (this is the prototypical Drinfeld module; we will recall the asic definitions concerning Drinfeld modules in 2). Drinfeld and Hayes have oth descried explicit class field theory for an aritrary gloal function field F, see [Dri74] and [Hay79]. Both proceed y first choosing a distinguished place 2000 Mathematics Suject Classification. Primary 11R37; Secondary 11G09. 1

2 of F, and their constructions give the maximal aelian extension K of F that splits completely at. Drinfeld defines a moduli space of rank 1 Drinfeld (elliptic) modules with level structure arising from the finite places of F whose spectrum turns out to e K. Hayes fixes a normalized Drinfeld module φ of rank 1 whose field of definition along with its torsion points can e used to construct K (this approach is more favourale for explicit computations). One approach to computing the full field F a is to chose a second place of F, since F a will then e the compositum of K and K. We wish to give a version of explicit class field theory that does not require this second unnatural choice. In Drinfeld s second paper on Drinfeld modules [Dri77], he achieves exactly this y considering another moduli space of rank 1 Drinfeld modules ut now with additional -adic structure. As remarked y Goss in [Gos96, 7.5], it would e very useful to have a modification of Drinfeld s construction that can e applied directly to φ to give the full aelian closure F a. We shall do exactly this! J.-K. Yu [Yu03] has studied the additional -adic structure introduced y Drinfeld and has teased out the implicit Galois representation occurring there. Yu s representation, which may also e defined for higher rank Drinfeld modules, can e viewed as an analogue of the Sato-Tate law, cf. [Yu03] and [Zyw11] Overview. The goal of this paper is to given an explicit construction of the inverse θ F. Moreover, we will construct an isomorphism of topological groups ρ: W a F C F, where WF a is the sugroup of Gal(F a /F ) that acts on the algeraic closure k of k in F sep as an integral power of Froenius map x x q (we endow WF a with the weakest topology for which Gal(F a /k) is an open sugroup and has its usual profinite topology). The inverse of our ρ will e the homomorphism α θ F (α) 1 (Theorem 3.5). For an open sugroup U of C F with finite index, define the homomorphism ρ U : WF a C F C F /U; it factors through Gal(L U /F ) where L U is the field corresponding to U via class field theory. Everything aout ρ is computale, and in particular one can find generators for L U. In 2, we shall give the required ackground on Drinfeld modules; in particular, we focus on normalized Drinfeld modules of rank 1. The representation ρ will then e defined in 3.3. The rest of the introduction serves as further motivation and will not e needed later. After a quick recap of explicit class field for Q, we shall descrie the aelian extensions of F = k(t) constructed y Carlitz which will lead to a characterization of ρ. We will treat the case F = k(t) in more depth in 5 and recover Hayes description of k(t) a as the compositum of three linearly disjoint fields The rational numers. We riefly recall explicit class field theory for Q. For each positive integer n, let µ n e the sugroup of n-torsion points in Q ; it is a free Z/nZ-module of rank 1. Let χ n : Gal(Q a /Q) (Z/nZ) e the representation for which σ(ζ) = ζ χn(σ) for all σ Gal(Q a /Q) and ζ µ n. By taking the inverse image over all n, ordered y divisiility, we otain a continuous and surjective representation χ: Gal(Q a /Q) Ẑ. The Kronecker-Weer theorem says that Q a is the cyclotomic extension of Q, and hence χ is an isomorphism of topological groups. The group Ẑ R + can e viewed as a sugroup of A Q, where R+ is the group of positive elements of R. The quotient map Ẑ R + C Q is an isomorphism of topological groups, and 2

3 y taking profinite completions we otain an isomorphism Ẑ = Ẑ R + ĈQ. Composing χ with this map, we have a isomorphism ρ: Gal(Q a /Q) ĈQ. One can show that the inverse of the Artin map θ Q : ĈQ ρ(σ) 1. Gal(Q a /Q) is simply the map σ 1.4. The rational function field. Let us riefly consider the rational function field F = k(t) where k is a finite field with q elements; it is the function field of the projective line P 1 k. Let denote the place of F that has uniformizer t 1. The suring of F consisting of functions that are regular at all places except possily is A = k[t]. Let End k (G a,f ) e the ring of k-linear endomorphisms of the additive group scheme G a over F. More concretely, End k (G a,f ) is the ring of polynomials i c ix qi F [X] with the usual addition and the multiplication operation eing composition. The Carlitz module is the homomorphism φ: A End k (G a,f ), a φ a of k-algeras for which φ t = tx + X q. Using the Carlitz module, we can give F sep an interesting new A-module structure; i.e., for a A and ξ F sep, we define a ξ := φ a (ξ). For a monic polynomial m A, let φ[m] e the m-torsion sugroup of F sep, i.e., the set of ξ F sep for which m ξ = 0 (equivalently, the roots of the separale polynomial φ m F [X]). The group φ[m] is free of rank 1 as an A/(m)-module, and we have a continuous surjective homomorphism χ m : Gal(F a /F ) (A/(m)) such that σ(ξ) = χ m (σ) ξ for all σ Gal(F a /F ) and ξ φ[m]. By taking the inverse image over all monic m A, ordered y divisiility, we otain a surjective continuous representation χ: Gal(F a /F ) Â. However, unlike the cyclotomic case, the map χ is not an isomorphism. The field m F (φ[m]) is a geometric extension of F that is tamely ramified at ; in particular, it does not contain extensions of F that are wildly ramified at or the constant extensions. Following J.K. Yu, and Drinfeld, we shall define a continuous homomorphism ρ : W a F F where F = k((t 1 )) is the completion of F at the place. We will put off its definition, and simply note that ρ can e characterized y the fact that it satisfies ρ (Fro p ) = p for each monic irreducile polynomial p of A. The image of ρ is thus contained in the open sugroup F + := t (1 + t 1 k [t 1 ]) of F. We can view  F + as an open sugroup of the ideles A F. We define the continuous homomorphism ρ: W a F χ ρ  F + C F where the first map takes σ WF a to (χ(σ), ρ (σ)) and the second map is the compositum of the inclusion map to A F with the quotient map A F C F. The main result of this paper, for F = k(t), says that the aove homomorphism ρ: WF a C F is an isomorphism of topological groups. Moreover, the inverse of the Artin map θ F : C F WF a is the homomorphism σ ρ(σ) 1. In particular, oserve that ρ does not depend on our initial choice of place! Taking profinite completions, we otain an isomorphism ρ: Gal(F a /F ) ĈF. After first developing the theory for a general gloal function field, we will return to the case F = k(t) in 5 where we will explicitly descrie the aelian extension of k(t) arising from ρ. 3

4 The constructions for a general gloal function field F are more involved; they more closely resemle the theory of complex multiplication for elliptic curves than the cyclotomic theory. We first choose a place of F. In place of the Carlitz module, we will consider a suitale rank 1 Drinfeld module φ. We cannot always take φ to e defined over F, ut we can choose a model defined over the maximal aelian extension H A of F that is unramified at all places and splits completely at (we will actually work with a slightly larger field H + A ) Notation. Throughout this paper, we consider a gloal function field F with a fixed place. Let A e the suring consisting of those functions that are regular away from. Denote y k the field of constants of F and let q e the cardinality of k. For each place λ of F, let F λ e the completion of F at λ. Let ord λ : F λ Z e the discrete valuation corresponding to λ and let O λ F λ e its valuation ring. The idele group A F of F is the sugroup of (α λ ) λ F λ, where the product is over all places of F, such that α λ elongs to O λ for all ut finitely many places λ. The group A F is locally compact when endowed with the weakest topology for which λ O λ, with the profinite topology, is an open sugroup. We emed F diagonally into A F ; it is a discrete sugroup. The idele class group of F is C F := A F /F which we endow with the quotient topology. Let m λ e the maximal ideal of O λ. Define the finite field F λ = O λ /m λ whose cardinality we will denote y N(λ). The degree of the place is d := [F : k]. Take a place p of F. We denote y Fro p an aritrary element of Gal(F sep /F ) such that for each finite Galois extension K F sep of F for which p is unramified, Fro p K is contained in the (arithmetic) Froenius conjugacy class of p in Gal(K/F ). Let L e an extension field of k. We fix an algeraic closure L of L and let L sep e the separale closure of L in L. We shall take k to e the algeraic closure of k in L sep. Let L perf e the perfect closure of L in L. We shall denote the asolute Galois group of L y Gal L := Gal(L sep /L). The Weil group W L of L is the sugroup of Gal L consisting of those automorphisms σ for which there exists an integer deg(σ) that satisfy σ(x) = x qdeg(σ) for all x k. The map deg: W L Z is a group homomorphism with kernel Gal(L sep /Lk). We endow W L with the weakest topology for which Gal(L sep /Lk) is an open sugroup with its usual profinite topology. Let WL a e the image of W L under the restriction map Gal L Gal(L a /L) where L a is the maximal aelian extension of L in L sep. Let L e an extension field of k. We define L[τ] e the ring of polynomials in τ with coefficients in L that oey the commutation rule τ a = a q τ for a L. In particular, note that L[τ] will e non-commutative if L k. We can identify L[τ] with the k-algera End k (G a,l ) consisting of the k-linear endomorphism of the additive group scheme G a,l ; identify τ with the endomorphism X X q. Suppose that L is perfect. Let L((τ 1 )) e the skew-field consisting of Laurent series in τ 1 ; it contains L[τ] as a suring (we need L to e perfect so that τ 1 a = a 1/q τ is always valid). Define the valuation ord τ 1 : L((τ 1 )) Z {+ } y ord τ 1( i a iτ i ) = inf{i : a i 0} and ord τ 1(0) = +. The valuation ring of ord τ 1 is L[τ 1 ], i.e., the ring of formal power series in τ 1. Again note that L((τ 1 )) and L[τ 1 ] are non-commutative if L k. For a topological group G, we will denote y Ĝ the profinite completion of G. We will always consider profinite groups, for example Galois groups, with their profinite topology. Acknowledgements. Thanks to David Goss, Bjorn Poonen and the referee. 4

5 2. Background For an in-depth introduction to Drinfeld modules, see [Dri74, DH87, Gos96]. The introduction [Hay92] of Hayes and Chapter VII of [Gos96] are particularly relevant to the material of this section Drinfeld modules. Let L e a field extension of k. A Drinfeld module over L is a homomorphism φ: A L[τ], x φ x of k-algeras whose image contains a non-constant polynomial. Let : L[τ] L e the ring homomorphism that takes a twisted polynomial to its constant term. We say that φ has generic characteristic if the homomorphism φ: A L is injective; using this map, we then otain an emedding F L that we will view as an inclusion. The ring L[τ] is contained in the skew field L perf ((τ 1 )). The map φ is injective, so it extends uniquely to a homomorphism φ: F L perf ((τ 1 )). The function v : F Z {+ } defined y v(x) = ord τ 1(φ x ) is a discrete valuation that satisfies v(x) 0 for all non-zero x A; the valuation v is non-trivial since the image of φ contain a non-constant element of L[τ]. Therefore, v is equivalent to ord and hence there exists a positive r Q that satisfies (2.1) ord τ 1(φ x ) = rd ord (x) for all x F. The numer r is called the rank of φ and it is always an integer. Since L perf ((τ 1 )) is complete with respect to ord τ 1, the map φ extends uniquely to a homomorphism φ: F L perf ((τ 1 )) that satisfies (2.1) for all x F. This extension of φ was first constructed in [Dri77] and will e the key to the -adic part of the construction of the inverse of the Artin map in 3.2. It will also lead to a more straightforward definition of the leading coefficient map µ φ of [Hay92, 6], see 2.2. Restricting our extended map φ to F gives a homomorphism F L perf [τ 1 ]. After composing φ F with the homomorphism L perf [τ 1 ] L perf which takes a power series in τ 1 to its constant term, we otain a homomorphism F L perf of k-algeras whose image must lie in L. In particular, the Drinfeld module φ gives L the structure of an F -algera. Fix two Drinfeld modules φ, φ : A L[τ]. An isogeny from φ to φ over L is a non-zero f L[τ] for which fφ a = φ af for all a A. An isomorphism from φ to φ over L is an f L[τ] = L that is an isogeny from φ to φ Normalized Drinfeld modules. Fix a Drinfeld module φ: A L[τ] of rank r and also denote y φ its extension F L perf ((τ 1 )). For each x F, we define µ φ (x) (L perf ) to e the first non-zero coefficient of the Laurent series φ x L perf ((τ 1 )). By (2.1), the first term of φ x is µ φ (x)τ rd ord (x). For a non-zero x A, one can also define µ φ (x) as the leading coefficient of φ x as a polynomial in τ. For x, y F, the value µ φ (xy) is equal to the coefficient of µ φ (x)τ rd ord (x) µ φ (y)τ rd ord (y), and hence (2.2) µ φ (xy) = µ φ (x)µ φ (y) 1/qrd ord (x). With respect to our emedding F L arising from φ, we have µ φ (x) = x for all x F. We say that φ is normalized if µ φ (x) elongs to F for all x F (equivalently, for all non-zero x A). If φ is normalized, then y (2.2) the map µ φ : F F is a group homomorphism that equals the identity map when restricted to F ; this is an example of a sign function. Definition 2.1. A sign function for F is a group homomorphism ε: F F that is the identity map when restricted to F. We say that φ is ε-normalized if it is normalized and µ φ : F F is equal to ε composed with some k-automorphism of F. 5

6 A sign function ε is trivial on 1 + m, so it determined y the value ε(π) for a fixed uniformizer π of F. Lemma 2.2. [Hay92, 12] Let ε e a sign function for F and let φ : A L[τ] e a Drinfeld module. Then φ is isomorphic over L to an ε-normalized Drinfeld module φ: A L[τ] The action of an ideal on a Drinfeld module. Fix a Drinfeld module φ: A L[τ] and take a non-zero ideal a of A. Let I a,φ e the left ideal in L[τ] generated y the set {φ a : a a}. All left ideals of L[τ] are principal, so I a,φ = L[τ] φ a for a unique monic polynomial φ a L[τ]. Using that a is an ideal, we find that I a,φ φ x I a,φ for all x A. Thus for each x A, there is a unique polynomial (a φ) x in L[τ] that satisfies (2.3) φ a φ x = (a φ) x φ a. The map a φ: A L[τ], x (a φ) x is also a Drinfeld module, and hence (2.3) shows that φ a is an isogeny from φ to a φ. Lemma 2.3. [Hay92, 4] (i) Let a and e non-zero ideals of A. Then φ a = ( φ) a φ and a ( φ) = (a) φ. (ii) Let a = wa e a non-zero principal ideal of A. Then φ a = µ φ (w) 1 φ w and (a φ) x = µ φ (w) 1 φ x µ φ (w) for all x A. Lemma 2.4. Let σ : L L e an emedding of fields. Let σ(φ): A L [τ] e the Drinfeld module for which σ(φ) x = σ(φ x ), where σ acts on the coefficients of L[τ]. For each non-zero ideal a of A, we have σ(a φ) = a σ(φ) and σ(φ a ) = σ(φ) a. Proof. The left ideal of L [τ] generated y σ(i a,φ ) is I a,σ(φ), and hence σ(φ a ) = σ(φ) a. Applying σ to (2.3), we have σ(φ) a σ(φ) x = σ(φ a φ x ) = σ((a φ) x φ a ) = σ(a φ) x σ(φ) a, for all x A. This shows that σ(a φ) = a σ(φ) Hayes modules. Fix a sign function ε for F. Let C e the completion of an algeraic closure F of F with respect to the -adic norm; it is oth complete and algeraically closed. Definition 2.5. A Hayes module for ε is a Drinfeld module φ: A C [τ] of rank 1 that is ε- normalized and for which φ: A C is the inclusion map. Denote y X ε the set of Hayes modules for ε. We know that Hayes modules exist ecause Drinfeld A-modules over C of rank 1 can e constructed analytically [Dri74, 3] and then we can apply Lemma Take any Hayes module φ X ε. Using that φ a C [τ] is monic along with (2.3), we see that the Drinfeld module a φ also elongs to X ε. By Lemma 2.3(i), we find that the group I of fractional ideals of A acts on X ε. Let P + e the sugroup of principal fractional ideals generated y those x F that satisfy ε(x) = 1. The group P + acts trivially on X ε y Lemma 2.3(ii), and hence induces an action of the finite group Pic + (A) := I/P + on X ε. 1 In our construction of the inverse of the Artin map, this is the only part that we have not made explicit. It is analogous to how one analytically constructs an elliptic curve with complex multiplication y the ring of integers O K of a quadratic imaginary field K [The quotient C/O K gives such an elliptic curve over C. We can then compute the j-invariant to high enough precision to identify what algeraic integer it is (it elongs to the Hilert class field of K and we know its degree over Q).]. The current version of Magma [BCP97] has a function AnalyticDrinfeldModule that can compute rank 1 Drinfeld modules that are defined over the maximal aelian extension H A of F that is unramified at all places and splits completely at. 6

7 Proposition 2.6. [Hay92, 13] The set X ε is a principal homogeneous space for Pic + (A) under the action. We now consider the arithmetic of the set X ε. Take any φ X ε and choose a non-constant y A. Let H + A e the sufield of C generated y F and the coefficients of φ y as a polynomial in τ. We call H + A the normalizing field for the triple (F,, ε). Lemma 2.7. [Hay92, 14] The extension H + A /F is finite and normal, and depends only on the triple (F,, ε). So for every φ X ε, we have φ(a) H + A [τ]. From now on, we shall view φ as a Drinfeld module φ: A H + A [τ]. By [Hay92, Proposition 11.4], we actually have φ(a) B[τ] where B is the integral closure of A in H + A. Since φ is normalized, we find that φ has good reduction at every place of H + A not lying over (for each non-zero prime ideal P of B, we can compose φ with a reduction modulo P map to otain a Drinfeld module of rank 1 over B/P). There is thus a natural action of the Galois group Gal(H + A /F ) on X ε. With a fixed φ X ε, Proposition 2.6 implies that there is a unique function ψ : Gal(H + A /F ) Pic+ (A) such that σ(φ) = ψ(σ) φ. Lemmas 2.3 and 2.4 imply that ψ is a group homomorphism that does not depend on the initial choice of φ. A consequence of following important proposition is that ψ is surjective, and hence an isomorphism. Proposition 2.8. The extension H + A /F is unramified away from. For each non-zero prime ideal p of A, the class ψ(fro p ) of Pic + (A) is the one containing p. 3. Construction of the inverse of the Artin map Fix a place of F. Throughout this section, we also fix a sign function ε for F and a Hayes module φ X ε (as descried in the previous section). Let F + e the open sugroup of F consisting of those x F for which ε(x) = 1. So as not to clutter the construction, all the lemmas of 3 will e proved in λ-adic representations. Fix a place λ of F ; we shall also denote y λ the corresponding maximal ideal of A. Take any automorphism σ Gal F. Since the map ψ of 2.4 is surjective, we can choose a non-zero ideal a of A for which σ(φ) = a φ. For each positive integer e, let φ[λ e ] e the set of F that satisfy φ x () = 0 for all x λ e ; equivalently, φ[λ e ] is the set of F such that φ λ e() = 0 (recall that we can identify each element of L[τ] with a unique polynomial i 0 c ix qi L[X]). We have φ[λ e ] F sep since the polynomials φ x are separale for all x λ e. Using the A-module structure coming from φ, we find that φ[λ e ] is an A/λ e -module of rank 1. The λ-adic Tate module of φ is defined to e T λ (φ) = Hom A (F λ /O λ, φ[λ ]) where φ[λ ] = e 1 φ[λ e ]. The Tate module T λ (φ) is a free O λ -module of rank 1, and hence V λ (φ) := F λ Oλ T λ (φ) is a one-dimensional vector space over F λ. For each e, the map φ[λ e ] σ(φ)[λ e ], ξ σ(ξ) is an isomorphism of A/λ e -modules. Comining over all e, we otain an isomorphism V λ (σ): V λ (φ) V λ (σ(φ)) of F λ -vector spaces. The isogeny φ a from φ to a φ induces a homomorphism φ[λ e ] (a φ)[λ e ] of A/λ e -module for each e. Comining together, we otain an isomorphism V λ (φ a ): V λ (φ) V λ (a φ) of F λ -vector spaces. Using our assumption σ(φ) = a φ, the map V λ (φ a ) 1 V λ (σ) elongs to Aut Fλ (V λ (φ)) = F λ ; we denote this element of F λ y ρa λ (σ). Lemma

8 (i) Take σ, γ Gal F and fix ideals a and of A such that σ(φ) = a φ and γ(φ) = φ. Then (σγ)(φ) = (a) φ and ρ a λ (σγ) = ρa λ (σ)ρ λ (γ). (ii) Take σ Gal F and fix ideals a and of A such that σ(φ) = a φ = φ. Then ρ a λ (σ)ρ λ (σ) 1 is the unique generator w F of the fractional ideal a 1 that satisfies ε(w) = 1. (iii) Take σ Gal F and fix an ideal a such that σ(φ) = a φ. Identifying λ with a non-zero prime ideal of A, let f 0 e the largest power of λ dividing a. Then ord λ (ρ a λ (σ)) = f. By Lemma 3.1(i), the map ρ λ : Gal H + A O λ, σ ρa λ (σ) is a homomorphism. It is continuous and it is unramified at all places not lying over λ or y the Drinfeld module analogue of the Néron-Ogg-Shafarevich criterion, cf. [Gos96, Theorem ] (as mentioned after Lemma 2.7, φ has good reduction at all places of H + A not lying over ) adic representation. By 2.2, our Drinfeld module φ: A H + A [τ] extends uniquely to a homomorphism φ: F (H + A )perf ((τ 1 )) that satisfies (2.1) with r = 1 for all x F. Recall that we defined a homomorphism deg : W F Z for which σ(x) equals x qdeg(σ) = τ deg(σ) xτ deg(σ) for all x k. Given a series u F ((τ 1 )) with coefficients in F sep and an automorphism σ Gal F, we define σ(u) to e the series otained y applying σ to the coefficients of u. Lemma 3.2. (i) There exists a series u F [τ 1 ] such that u 1 φ(f )u k((τ 1 )). Any such u has coefficients in F sep. (ii) Fix any u as in (i). Take any σ W F and fix a non-zero ideal a of A for which σ(φ) = a φ. Then φ 1 a σ(u)τ deg(σ) u 1 F ((τ 1 )) elongs to φ(f + ) and is independent of the choice of u. Take σ W F and fix a non-zero ideal a of A such that σ(φ) = a φ. Choose a series u F [τ 1 ] as in Lemma 3.2(i). Using that φ: F (H + A )perf ((τ 1 )) is injective and Lemma 3.2(ii), we define ρ a (σ) to e the unique element of F + for which φ ( ρ a (σ) ) = φ 1 a σ(u)τ deg(σ) u 1. We now state some results aout ρ a (σ); they are analogous to those concerning ρ a λ (σ) in the previous section. Lemma 3.3. (i) Take σ, γ W F and fix ideals a and of A such that σ(φ) = a φ and γ(φ) = φ. Then (σγ)(φ) = (a) φ, and we have ρ a (σγ) = ρ a (σ)ρ (γ). (ii) Take σ W F and fix ideals a and of A such that σ(φ) = a φ = φ. Then ρ a (σ)ρ (σ) 1 is the unique generator w F of the fractional ideal a 1 that satisfies ε(w) = 1. Lemma 3.4. The map ρ : W H + A F +, σ ρ A (σ) is a continuous homomorphism that is unramified at all places of H + A 8 not lying over.

9 3.3. The inverse of the Artin map. For each σ W F, fix a non-zero ideal a of A such that σ(φ) = a φ. By Lemma 3.1(iii), (ρ a λ (σ)) λ is an idele of F. We define ρ(σ) to e the element of the idele class group C F that is represented y (ρ a λ (σ)) λ. By Lemmas 3.1(ii) and 3.3(ii), we find that ρ(σ) is independent of the choice of a. Lemmas 3.1(i) and 3.3(i) imply that the map ρ: W F C F is a group homomorphism. The restriction of ρ to the finite index open sugroup W H + agrees with W H + A Q λ ρ λ F + λ O λ C F where the second homomorphism is otained y composing the natural inclusion into A F with the quotient map A F C F. Since the representations ρ λ are continuous, we deduce that λ ρ λ, and hence ρ, is continuous. So we may view ρ as a continuous homomorphism WF a C F. By taking profinite completions, we otain a continuous homomorphism ρ: Gal(F a /F ) ĈF. Recall that the Artin map θ F : C F Gal(F a /F ) of class field theory gives an isomorphism θ F : ĈF Gal(F a /F ) of topological groups. Our main result is then the following: Theorem 3.5. The map ρ: Gal(F a /F ) ĈF is an isomorphism of topological groups. The inverse of the isomorphism Gal(F a /F ) ĈF, σ ρ(σ) 1 is the Artin map θ F. Before proving the theorem, we mention the following arithmetic input. Lemma 3.6. Fix a place λ of F. Let p λ, e a place of F which we identify with the corresponding non-zero prime ideal of A. Then ρ p λ (Fro p) = 1. Proof of Theorem 3.5. Take an open sugroup U of C F with finite index. Let L U e the fixed field in F sep of the kernel of the homomorphism WF a ρ C F C F /U; this gives an injective group homomorphism ρ U : Gal(L U /F ) C F /U. Let S U e the set of places p of F for which p = or for which there exists an idele α A F whose class in C F does not lie in U and satisfying α λ = 1 for λ p and α p O p. The set S U is finite since U is open in C F. Take any place p / S U. Choose a uniformizer π p of F p and let α(p) e the idele of F that is π p at the place p and 1 at all other places. Define the idele β := (ρ p λ (Fro p)) λ α(p) A F. Lemma 3.6 says that β λ = 1 for all λ p while Lemma 3.1(iii) tells us that ord p (β p ) = 0. By our choice of S U, the image of β in C F must lie in U. Therefore, ρ U (Fro p ) is the coset of C F /U represented y α(p) 1. In particular, note that L U /F is unramified at all p / S U. The group C F /U is generated y the elements α(p) with p / S U, and hence ρ U is surjective. Therefore ρ U : Gal(L U /F ) C F /U is an isomorphism of groups. Define the isomorphism θ U : C F /U Gal(L U /F ), α (ρ 1 U (α)) 1. For each p / S U, it takes the coset containing α(p) to the Froenius automorphism corresponding to p. Composing the quotient map C F C F /U with θ U, we find that the resulting homomorphism C F Gal(L U /F ) equals the map α θ F (α) LU where θ F : C F Gal(F a /F ) is the Artin map of F. Recall that class field theory gives a one-to-one correspondence etween the finite aelian extensions L of F and the open sugroups U with finite index in C F. Let L F sep e an aritrary finite aelian extension of F. Class field theory says that L corresponds to the kernel U of the map C F Gal(L/F ), α θ F (α) L. By comparing with the computation aove, we deduce that L = L U. Since L was an aritrary finite aelian extension, we deduce that F a = U L U. 9 A

10 Taking the inverse limit of the isomorphisms ρ U : Gal(L U /F ) C F /U as U varies, we find that the corresponding homomorphism ρ: Gal(F a /F ) ĈF is an isomorphism (the injectivity is precisely the statement that F a = U L U). The inverse of the isomorphism Gal(F a /F ) Ĉ F, σ ρ(σ) 1 is otained y comining the homomorphisms θ U : C F /U Gal(L U /F ); from the calculation aove, this equals θ F. Corollary 3.7. The homomorphism ρ: WF a C F is an isomorphism of topological groups. The inverse of the isomorphism WF a C F, σ ρ(σ) 1 is the Artin map θ F. Proof. This follows directly from the theorem. Oserve that the natural maps WF a Ŵ F a = Gal(F a /F ) and C F ĈF from the group to their profinite completion are oth injective since F is a gloal function field. Remark 3.8. (i) The isomorphism ρ: WF a C F depends only on F (and not on our choices of, ε, and φ X ε ). (ii) Our proof only requires class field theory to prove that ρ is injective, i.e., to show that we have constructed all finite aelian extensions of F. 4. Proofs of the lemmas 4.1. Proof of Lemma 3.1. (i) By Lemma 2.3(i) and 2.4, we have (σγ)φ = (a) φ and σ(φ ) = σ(φ). By Lemma 2.3(i) and our choice of a, we find that φ a = (a φ) φ a = σ(φ) φ a = σ(φ )φ a. We have σ(φ (σ 1 (ξ))) = σ(φ) (ξ) for all ξ σ(φ)[λ e ], so V λ (σ) V λ (φ ) V λ (σ) 1 and V λ (σ(φ )) are the same automorphism of V λ (σ(φ)). Therefore, V λ (φ a ) 1 V λ (σγ) = V λ (φ a ) 1 V λ (σ(φ )) 1 V λ (σγ) = V λ (φ a ) 1 (V λ (σ) V λ (φ ) V λ (σ) 1 ) 1 V λ (σγ) = V λ (φ a ) 1 V λ (σ) V λ (φ ) 1 V λ (σ) 1 V λ (σγ) = (V λ (φ a ) 1 V λ (σ)) (V λ (φ ) 1 V λ (γ)). Thus ρ a λ (σγ) = ρa λ (σ)ρ λ (γ). (ii) Since a φ = φ, Lemma 2.6 implies that the fractional ideal a 1 is the identity class in Pic + (A). There are thus non-zero w 1, w 2 A such that (w 1 )a = (w 2 ) and ε(w 1 ) = ε(w 2 ) = 1. In particular, w := w 1 /w 2 is the unique generator of a 1 satisfying ε(w) = 1. By Lemma 2.3(ii), we have (w 1 ) φ = φ and φ (w1 ) = φ w1. Therefore, y Lemma 2.3(i) we have φ a(w1 ) = ((w 1 ) φ) a φ (w1 ) = φ a φ w1. Similarly, φ (w2 ) = φ φ w2, and hence φ a φ w1 = φ φ w2. Thus ρ a λ (σ)ρ λ (σ) 1 = V λ (φ a ) 1 V λ (φ ) = V λ (φ w1 ) V λ (φ w2 ) 1. The automorphisms V λ (φ w1 ) and V λ (φ w2 ) oth elong to Aut Fλ (V λ (φ)) = F λ and correspond to w 1 and w 2, respectively. We conclude that ρ a λ (σ)ρ λ (σ) 1 = w 1 w2 1 = w. (iii) We view λ as oth a place of F and a non-zero prime ideal of A. For each e f, the kernel of φ[λ e ] φ[λ e ], ξ σ 1 (φ a (ξ)) has cardinality φ[λ f ] = N(λ) f. So ρ a λ (σ) 1 = V λ (σ 1 ) V λ (φ a ), which is an element of Aut Fλ (V λ (φ)) = F λ, gives an O λ-module homomorphism T λ (φ) T λ (φ) whose cokernel has cardinality N(λ) f. Therefore, ord λ (ρ a λ (σ) 1 ) = f. In particular, we have ord λ (ρ a λ (σ)) = f. 10

11 4.2. Proof of Lemma 3.2. Fix a non-constant y A that satisfies ε(y) = 1 and define h := d ord (y) 1. Since φ is ε-normalized and ε(y) = 1, we have φ y = τ h + h 1 j=0 jτ j for unique j H + A. We set u = i=0 a iτ i with a i F to e determined where a 0 0. Expanding out the series φ y u and uτ h, we find that φ y u = uτ h holds if and only if (4.1) a qh i a i = 0 j h 1, i+j h 0 j a qj i+j h holds for all i 0. We can use the equations (4.1) to recursively solve for a 0 0, a 1, a 2,.... The a i elong to F sep since (4.1) is a separale polynomial in a i and the j elong to H + A F sep. Let k h e the degree h extension of k in k. The elements of the ring F ((τ 1 )) that commute with τ h are k h ((τ 1 )). Since τ h elongs to the commutative ring u 1 φ(f )u, we find that u 1 φ(f )u is a suset of k h ((τ 1 )). Thus u F ((τ 1 )) has coefficients in F sep and satisfies u 1 φ(f )u k((τ 1 )). Recall that φ induces an emedding F L; this gives inclusions F k L. Fix a uniformizer π of F. There is a unique homomorphism ι: F k((τ 1 )) that satisfies the following conditions: ι(x) = x for all x F, ι(π) = τ d, ord τ 1(ι(x)) = d ord (x) for all x F. We have ι(f ) = F ((τ d )). Let C e the centralizer of ι(f ) in F ((τ 1 )). Using that F and τ d are in ι(f ), we find that C = F ((τ d )) = ι(f ). Take any v F [τ 1 ] that satisfies v 1 φ(f )v k((τ 1 )). By [Yu03, Lemma 2.3], there exist w 1 and w 2 k [τ 1 ] such that w 1 1 (u 1 φ x u)w 1 = ι(x) = w 1 2 (v 1 φ x v)w 2 for all x F. So for all x F, we have (uw 1 )ι(x)(uw 1 ) 1 = φ x = (vw 2 )ι(x)(vw 2 ) 1 and hence (w 1 2 v 1 uw 1 )ι(x)(w 1 2 v 1 uw 1 ) 1 = ι(x). Thus w 1 2 v 1 uw 1 elongs to C k((τ 1 )), and so v = uw for some w k [τ 1 ]. The coefficients of v thus lie in F sep since the coefficients of u lie in F sep and w has coefficients in the perfect field k F sep. This completes the proof of (i). Since w has coefficients in k, we have σ(w) = τ deg(σ) wτ deg(σ) and hence σ(v)τ deg(σ) v 1 = σ(uw)τ deg(σ) (uw) 1 = σ(u)σ(w)τ deg(σ) w 1 u 1 = σ(u)(τ deg(σ) wτ deg(σ) )τ deg(σ) w 1 u 1 = σ(u)τ deg(σ) u 1. This proves that φ 1 a σ(u)τ deg(σ) u 1 is independent of the initial choice of u. We now show that φ 1 a σ(u)τ deg(σ) u 1 elongs to φ(f ). We have seen that φ(f ) is conjugate to ι(f ) in F ((τ 1 )) and that ι(f ) is its own centralizer in F ((τ 1 )). Therefore, the centralizer of φ(f ) in F ((τ 1 )) is φ(f ). So it suffices to prove that φ 1 a σ(u)τ deg(σ) u 1 commutes with φ x for all x A. Take any x A. We have σ(u 1 φ x u) = τ deg σ (u 1 φ x u)τ deg σ since u 1 φ x u has coefficients in k. By our choice of a, we have σ(φ) x = (a φ) x = φ a φ x φ 1 a. Therefore, τ deg σ (u 1 φ x u)τ deg σ = σ(u 1 φ x u) = σ(u) 1 σ(φ) x σ(u) = σ(u) 1 φ a φ x φ 1 a σ(u) and one concludes that φ 1 a σ(u)τ deg σ u 1 commutes with φ x. 11

12 The only thing that remains to e proved is that φ 1 a σ(u)τ deg σ u 1 elongs to φ(f ). + Since φ is ε-normalized, this is equivalent to showing that the first non-zero coefficient of the Laurent series φ 1 a σ(u)τ deg σ u 1 in τ 1 is 1. Since φ a is a monic polynomial of τ, we need only show that the first non-zero coefficient of σ(u)τ deg σ u 1 F ((τ 1 )) is 1, i.e., σ(a 0 )a qdeg σ 0 = 1. This is true since a 0 k ; indeed, a 0 is non-zero and satisfies a qh 0 a 0 = 0 y (4.1). Remark 4.1. Take any place p of F and any valuation v : F sep Q {+ } extending ord p. We have v( j ) 0 for 0 j h (in 2.4, we noted that the coefficients of φ y are integral over A). Using (4.1) repeatedly, we find that v(a i ) 0 for all i 0 (the roots of (4.1), as a polynomial in a i, differ y a value in k h ). For each i 1, the extension F (a i )/F (a i 1 ) is an Artin-Schreier extension. Since the right-hand side of (4.1) is integral at each place not lying over, we deduce that F (a i )/F (a i 1 ) is unramified at all places not lying over. Let L F sep e the extension of F generated y k and the set {a i } i 0, i.e., the extension of F k generated y the coefficients of u. We find that L is unramified at all places of F away from Proof of Lemma 3.3. Fix a series u F [τ 1 ] as in Lemma 3.2(i). (i) We have φ(ρ a (σγ)) = φ 1 a σγ(u)τ deg(σγ) u 1 and φ(ρ (γ)) = φ 1 γ(u)τ deg(γ) u 1. So it suffices to show that φ(ρ a (σ)) equals φ(ρ a (σγ)ρ (γ) 1 ) = φ 1 a σγ(u)τ deg(σγ) u 1 uτ deg(γ) γ(u) 1 φ = φ 1 a σ(γ(u))τ deg(σ) γ(u) 1 φ = φ 1 a σ(φ )φ a φ 1 a σ(φ 1 γ(u))τ deg(σ) (φ 1 γ(u)) 1. We showed in 4.1 that φ a = σ(φ )φ a, so we need only prove that φ 1 a σ(φ 1 γ(u))τ deg(σ) (φ 1 γ(u)) 1 and φ(ρ a (σ)) agree. By Lemma 3.2(ii), it thus suffices to show that (φ 1 γ(u)) 1 φ(f )φ 1 γ(u) k((τ 1 )). Take any x F. Since u 1 φ x u elongs to k((τ 1 )), so does γ(u 1 φ x u). By our choice of, we have γ(u 1 φ x u) = γ(u) 1 γ(φ) x γ(u) = γ(u) 1 ( φ) x γ(u) = γ(u) 1 φ φ x φ 1 γ(u), and hence (φ 1 γ(u)) 1 φ x (φ 1 γ(u)) is an element of k((τ 1 )). (ii) First note that φ(ρ a (σ)ρ (σ) 1 ) = φ 1 a σ(u)τ deg σ u 1 uτ deg σ σ(u) 1 φ = φ 1 a φ. In 4.1, we showed that φ 1 a φ = φ w1 φ 1 w 2 where w 1 and w 2 elong to A and w = w 1 /w 2 is the unique generator of a 1 satisfying ε(w) = 1. Therefore, ρ a (σ)ρ (σ) 1 = w as desired Proof of Lemma 3.4. The map ρ is a homomorphism y Lemma 3.3(i). Fix a series u = i 0 c iτ i as in Lemma 3.2(i). For σ W H +, we have ord (ρ (σ)) = d ord τ 1(φ(ρ (σ))) = A d deg(σ). So to prove that ρ is continuous, we need only show that Gal(F sep /H + ρ A k) O is continuous. It suffices to show that for each e 1, the homomorphism β e : Gal(F sep /H + ρ A k) O (O /m e ) has open kernel. For each σ Gal(F sep /H + A k), we have φ(ρ (σ)) = σ(u)u 1. One can check that β e (σ) = 1, equivalently ord (ρ (σ) 1) e, if and only if ord τ 1(σ(u)u 1 1) = ord τ 1(σ(u) u) is at least ed. Thus the kernel of β e is Gal(F sep /L e ) where L e is the finite extension of H + A k generated y the set {c i } 0 i<ed 12

13 It remains to prove that ρ is unramified at all places of H + A not lying over. Let L e the sufield of F sep fixed y ker(ρ ); it is the extension of H + A k generated y the set {c i} i 0. The field L does not depend on the choice of u, since ρ does not. In Remark 4.1, we saw that the extension L of F k generated y the coefficients of a particular u was unramified at all places of F away from. Therefore, L is unramified at all places of F away from, since L and H + A oth have this property Proof of Lemma 3.6. First consider the case λ. For each e 1, we have Fro p (ξ) = φ p (ξ) for all ξ φ[λ e ] F sep ; this was oserved y Hayes [Hay92, p.28], for a proof see [Gos96, 7.5]. Thus the map V λ (φ p ): V λ (φ) V λ (φ) equals V λ (Fro p ), and hence ρ p λ (Fro p) = V λ (φ p ) 1 V λ (Fro p ) = 1. We may now assume that λ =. Let F p e an algeraic closure of F p. The field F p has a unique place extending the p-adic place of F p and let O p F p e the corresponding valuation ring. The residue field of O p is an algeraic closure of F p that we denote y F p, and let r p : O p F p e the reduction homomorphism. Choose an emedding F F p. The restriction map Gal(F p /F p ) Gal(F /F ) is an inclusion that is well defined up to conjugation. So after conjugating, we may assume that Fro p lies in Gal(F p /F p ); it thus acts on O p and we have r p (Fro p (ξ)) = r p (ξ) N(p) for all ξ O p. We will also denote y r p the map O p ((τ 1 )) F p ((τ 1 )) where one reduces the coefficients of the series. In 2.4, we noted that the coefficients of φ x are integral over A for each x A. This implies that φ(a) O p [τ]. Define the homomorphism φ: A φ O p [τ] rp F p [τ]. The map φ is a Drinfeld module over F p of rank 1 since φ is normalized. Since φ is normalized, we find that φ x O p ((τ 1 )) for all non-zero x A. Therefore, φ(f ) O p ((τ 1 )). Using that O p ((τ 1 )) is complete with respect to ord τ 1 and that (2.1) holds with r = 1, we deduce that φ(f ) O p ((τ 1 )). We claim that r p induces an isomorphism φ(f ) φ(f ). Since φ is a Drinfeld module, and hence injective, the map r p induces an isomorphism φ(a) φ(a) which then extends to an isomorphism φ(f ) φ(f ) of their quotient fields. The map r p : φ(f ) φ(f ) preserves the valuation ord τ 1 so y the uniqueness of completions, r p also gives an isomorphism φ(f ) φ(f ). Thus to prove that ρ p (Fro p ) equals 1, it suffices to show that r p (φ(ρ p (Fro p ))) = 1. Lemma 3.2(i), along with Remark 4.1, shows that there is a series u O p [τ 1 ] with coefficients in F sep such that u 1 φ(f )u k((τ 1 )). With such a series u, we have φ(ρ (Fro p )) = φ 1 p Fro p (u)τ d u 1 where d = deg(fro p ). The polynomial φ p F [τ] has coefficients in O p and r p (φ p ) = τ d ; thus φ 1 p elongs to O p ((τ 1 )) and r p (φ 1 p ) = τ d. We have r p (Fro p (u)) = τ d r p (u)τ d. Therefore, r p (φ(ρ (Fro p ))) = r p (φ 1 p )r p (Fro p (u))τ d r p (u) 1 = τ d τ d r p (u)τ d τ d r p (u) 1 = The rational function field We return to the rational function field F = k(t) where k is a finite field with q elements. Using our constructions, we shall recover the description of F a given y Hayes in [Hay74]; he expressed F a as the compositum of three linearly disjoint fields over F. In particular, we will explain how two of these fields arise naturally from our representation ρ. We define A = k[t]; it is the suring of F consisting of functions that are regular away from a unique place of F. We have F = k, F = k((t 1 )) and ord : F Z is the valuation for 13

14 which ord (t 1 ) = 1. Let ε: F k e the unique sign function of F that satisfies ε(t 1 ) = 1. Those x F for which ε(x) = 1 form the sugroup F + := t (1 + t 1 k [t 1 ]) = t (1 + m ). Recall that the Carlitz module is the homomorphism φ: A F [τ], a φ a of k-algeras for which φ t = t + τ. In the notation of 2.4, φ is a Hayes module for ε. The coefficients of φ t lie in F, so the normalizing field H + A for (F,, ε) equals F. We saw that Gal(H+ A /F ) acts transitively on X ε, so X ε = {φ}. For each place λ of F (including!), we have defined a continuous homomorphism ρ λ : WF a F λ, σ ρa λ (σ). The representation ρ λ is characterized y the property that ρ λ (Fro p ) = p holds for each monic irreducile polynomial p A = k[t] not corresponding to λ (comine Lemmas 3.1(ii), 3.3(ii) and 3.6 to show that ρ A λ (Fro p)ρ (p) λ (Fro p) 1 = ρ A λ (Fro p) is the unique generator w of pa that satisfies ε(w) = 1). For λ, the representation ρ λ has image in O λ, so it extends to a continuous representation Gal(F a /F ) O λ. The image of ρ lies in F, + ut is unounded (so it does not extend to a Galois representation). Comining the representations ρ λ together, we otain a continuous homomorphism (5.1) ρ λ : WF a F + O λ. λ By composing λ ρ λ with the quotient map A F C F, we otain a continuous homomorphism ρ: WF a C F. Corollary 3.7 says that ρ is an isomorphism of topological groups (and that the inverse of σ ρ(σ) 1 is the Artin map θ F ). The quotient map F + λ O λ C F is actually an isomorphism, so the map (5.1) is also an isomorphism. Taking profinite completions, we otain an isomorphism (5.2) Gal(F a /F ) F + O λ = t (1 + m ) O λ. λ Using this isomorphism, we can now descrie three linearly disjoint aelian extensions of F whose compositum is F a Torsion points. The representation χ := λ ρ: Gal(F a /F ) λ O λ = Â arises from the Galois action on the torsion points of φ as descried in 1.4. The fixed field in F a of ker(χ) is the field K := m F (φ[m]) where the union is over all monic polynomials m of A, and we have an isomorphism Gal(K /F ) Â. The field K, which was first given e Carlitz, is a geometric extension of F that is tamely ramified at Extension of constants. Define the homomorphism deg: WF a Z y σ ord (ρ (σ)); this agrees with our usual definition of deg(σ) [it is easy to show that ord τ 1(φ(ρ (σ))) equals ord τ 1(τ deg(σ) ) = deg(σ), and then use (2.1) with r = d = 1]. The map deg thus factors through W (k(t)/k(t)) Z, where W (k(t)/k(t)) is the group of σ Gal(k(t)/k(t)) that act on k as an integral power of q. Of course, k(t)/k(t) is an aelian extension with Gal(k(t)/k(t)) Ẑ. λ Wildly ramified extension. Define the homomorphism W a F 1 + m, σ ρ (σ)/t deg(σ) ; it is well defined since ord (ρ (σ)) = deg(σ) and since the image of ρ is contained in F + = t (1 + m ). Since 1 + m is compact, this gives rise to a Galois representation β : Gal(F a /F ) 1 + m. 14 λ

15 Let L e the fixed field of ker(β) in F a. The field L /F is an aelian extension of F that is unramified away from and is wildly ramified at ; it is also a geometric extension of F. We will now give an explicit description of this field. Define a 0 = 1 and for i 1, we recursively choose a i F sep that satisfy the equation (5.3) a q i a i = ta i 1. This gives rise to a chain of field extensions, F F (a 1 ) F (a 2 ) F (a 3 ). We claim that L = i F (a i). The construction of ρ starts y finding an appropriate series u F [τ 1 ] with coefficients in F sep. Define u := i 0 a iτ i with the a i defined as aove. The recursive equations that the a i satisfy φ t u = uτ (just multiply them out and check!). As shown in 4.2, this implies that u 1 φ(f )u k((τ 1 )). For σ WF a, ρ (σ) is then the unique element of F + for which φ(ρ (σ)) = σ(u)τ deg(σ) u 1 where σ(u) is otained y letting σ act on the coefficients of u. In particular, φ(β(σ)) = σ(u)u 1 for all σ Gal(L /F ) since L /F is a geometric extension. We find that β(σ) = 1 if and only if σ(u) = u, and thus L is the extension of F generated y the set {a i } i 1. Using the isomorphism (5.2), we find that F a is the compositum of the fields K, k(t) and L, and that they are linearly disjoint over F. This is exactly the description given y Hayes in [Hay74, 5]; the advantage of our representation ρ is that the fields k(t) and L arise naturally. References [BCP97] Wie Bosma, John Cannon, and Catherine Playoust, The Magma algera system. I. The user language, J. Symolic Comput. 24 (1997), no. 3-4, Computational algera and numer theory (London, 1993). 1 [Car38] Leonard Carlitz, A class of polynomials, Trans. Amer. Math. Soc. 43 (1938), no. 2, [DH87] Pierre Deligne and Dale Husemoller, Survey of Drinfel d modules, Current trends in arithmetical algeraic geometry (Arcata, Calif., 1985), 1987, pp [Dri74] V. G. Drinfel d, Elliptic modules, Mat. S. (N.S.) 94(136) (1974), , , 2, 2.4 [Dri77], Elliptic modules. II, Mat. S. (N.S.) 102(144) (1977), no. 2, , 325. English translation: [Gos96] Math. USSR-S. 31 (1977), , 2.1 David Goss, Basic structures of function field arithmetic, Ergenisse der Mathematik und ihrer Grenzgeiete (3), vol. 35, Springer-Verlag, Berlin, , 2, 3.1, 4.5 [Hay74] David R. Hayes, Explicit class field theory for rational function fields, Trans. Amer. Math. Soc. 189 (1974), , 5, [Hay79], Explicit class field theory in gloal function fields, Studies in algera and numer theory, 1979, pp [Hay92], A rief introduction to Drinfeld modules, The arithmetic of function fields (Columus, OH, 1991), 1992, pp , 2.1, 2.2, 2.3, 2.6, 2.7, 2.4, 4.5 [Tat67] J. T. Tate, Gloal class field theory, Algeraic Numer Theory (Proc. Instructional Conf., Brighton, 1965), 1967, pp [Yu03] Jiu-Kang Yu, A Sato-Tate law for Drinfeld modules, Compositio Math. 138 (2003), no. 2, , 4.2 [Zyw11] David Zywina, The Sato-Tate law for Drinfeld modules (2011). arxiv: Department of Mathematics and Statistics, Queen s University, Kingston, ON K7L 3N6, Canada address: zywina@mast.queensu.ca URL: 15