SPECIAL VALUES of DRINFELD MODULAR FORMS and ALGEBRAIC INDEPENDENCE 1. Introduction 1.1. Motivation. One of the Main Themes in T

SPECIAL VALUES OF DRINFELD MODULAR FORMS AND ALGEBRAIC INDEPENDENCE

CHIEH-YU CHANG

Abstract. Let A be a polynomial ring in one variable over a finite field and k be its fraction field. Let f be a Drinfeld modular form of nonzero weight for a congruence subgroup of GL2(A) so that the coefficients of the q∞-expansion of f are algebraic over k. We consider n CM points α1, . . . , αn on the Drinfeld upper half plane for which the quadratic fields k(α1), . . . , k(αn) are pairwise distinct. Suppose that f is non-vanishing at these n points. Then we prove that f(α1), . . . , f(αn) are algebraically independent over k.

1. Introduction 1.1. Motivation. One of the main themes in transcendence theory is to study the values of a given special transcendental function at algebraic points. The classical work of Hermite and Lindemann in the 19th century established that the values of the exponential function at nonzero algebraic numbers are transcendental numbers. This fundamental theorem implies the transcendence of π and e. We are interested in the special transcendental functions which are modular forms of nonzero weight with algebraic Fourier coefficients for a congruence subgroup Γ ⊆ SL2(Z). We call such modular forms arithmetic. Let H be the complex upper half plane. We consider the moduli space SL2(Z)\H which classifies the isomorphism classes of elliptic curves over C. We say that α ∈ H is a CM point if Q(α) is quadratic over Q. Note that in this case the lattice Zα + Z is a period lattice of an elliptic curve that is isomorphic to an elliptic curve defined over Q with CM field Q(α). We are interested in the special values of arithmetic modular forms at CM points. For two values x, y ∈ C with y 6= 0, we denote by x ∼ y if x/y ∈ Q. Following Shimura, given an arithmetic modular form f of weight ` ∈ N and a CM point α ∈ H, there × exists ωα ∈ C which is unique up to algebraic multiple so that ω (1) f(α) ∼ ( √α )`. 2π −1

Note that ωα is a period of an elliptic curve defined over Q with CM field Q(α) and hence by a theorem of Schneider [Sch59] f(α) is transcendental over Q if f(α) is nonzero. Since imaginary quadratic fields parametrize isogeny classes of CM elliptic curves over C, it leads to the following conjecture concerning the algebraic independence of arithmetic modular forms at CM points. Conjecture 1.1.1. Let f be an arithmetic modular form of weight ` > 0 for a congruence subgroup Γ ⊆ SL2(Z). Let α1, . . . , αn ∈ H be CM points. Assume that the imaginary quadratic fields Q(α1),..., Q(αn) are pairwise distinct and that f is non-vanishing

Date: February 24, 2010. MSC: primary 11J93; secondary 11F52, 11G09. Key words: Algebraic independence; Drinfeld modular forms; CM points; periods; t-motives. The author was supported by an NCTS postdoctoral fellowship. 1 2 CHIEH-YU CHANG at these n CM points. Then the n special values f(α1), . . . , f(αn) are algebraically independent over Q. Concerning the conjecture above, using formula (1) and W¨ustholz’subgroup theorem [W89] the best known result involves only linear independence over Q due to Brownawell [Br01] in the case ` = 1. The purpose of this paper is to prove the analogue of Conjecture 1.1.1 in the setting of Drinfeld modular forms.

1.2. The main result. Let A := Fq[θ] be the polynomial ring in the variable θ over the finite field Fq of q elements, where q is a power of a prime number p. Let k := Fq(θ) be the fraction field of A. Let | · |∞ be the absolute value on k with respect to the place 1 ∞ so that |θ|∞ = q. Let k∞ := Fq(( θ )) be the completion of k with respect to | · |∞ and let C∞ be the completion of a fixed algebraic closure of k∞ with respect to the ¯ absolute value extending | · |∞. Throughout this paper k is a fixed algebraic closure of ¯ k inside C∞. It is well known that A, k, k∞, C∞ and k are good analogues of Z, Q, R, C and Q, respectively. 1 1 Consider the Drinfeld upper half plane H := P (C∞) \ P (k∞) = C∞ \ k∞. Then GL2(k∞) acts on H by fractional linear transformations: az + b a b γz := for z ∈ H, γ = ∈ GL (k ). cz + d c d 2 ∞

A congruence subgroup Γ of GL2(A) is a subgroup of GL2(A) that contains the kernel Γ(N) of the map GL2(A) → GL2(A/N) for some nonzero ideal N of A. Its cusps are defined to be the orbits Γ \ P1(k). A Drinfeld modular form for a congruence subgroup Γ of GL2(A) of weight ` ≥ 0 and of type m ∈ Z/(q − 1)Z is a holomorphic (i.e. rigid analytic) function f : H → C∞ satisfying: a b • f(γz) = (det γ)−m(cz + d)`f(z) for all γ = ∈ Γ; c d • f is holomorphic at the cusps of Γ. We first explain the second condition above for the cusp ∞. Let N = (n) be the ideal of A which is maximal among the ideals I of A with the property that Γ ⊇ 1 x ; x ∈ I . Let exp (z) be the Carlitz exponential function andπ ˜ be a fixed 0 1 C fundamental period of the Carlitz module defined in §2.1. Define q∞(z) = q∞(z; Γ) := πz˜ −1 expC ( n ) . Then we use q∞(z) as a uniformizing parameter at ∞. We say that f is holomorphic at ∞ if for |z|i := infx∈k∞ {|z − x|∞} sufficiently large f has an expansion P i of the form f(z) = i≥0 aiq∞(z) , which is called the q∞-expansion of f. (Note that it is called theπ ˜-normalized expansion of f in [Go80a, Go80b]). As classically for congruence subgroups of SL2(Z), to check holomorphy of f at any other cusp one maps that cusp to ∞ via an element of SL2(A). Given a Drinfeld modular form f for a congruence subgroup of GL2(A), we call f “arithmetic ”if all coefficients of the q∞-expansion of f are algebraic over k. A point α ∈ H is called a CM point if k(α) is quadratic over k. The main result of this paper is stated as follows. Theorem 1.2.1. Let f be an arithmetic Drinfeld modular form of nonzero weight for a congruence subgroup of GL2(A). Let α1, . . . , αn ∈ H be CM points so that the quadratic fields k(α1), . . . , k(αn) are pairwise distinct. Assume that f is non-vanishing at these n CM points. Then the n special values f(α1), . . . , f(αn) are algebraically independent over k. DRINFELD MODULAR FORMS AND ALGEBRAIC INDEPENDENCE 3

Remark 1.2.2. Let f be given in the theorem above. Suppose we are given two CM points β1, β2 ∈ H so that k(β1) = k(β2). If f is non-vanishing at β1 and β2, then we ¯ have f(β1)/f(β2) ∈ k, which is explained by combining Theorem 2.2.1 and Proposition 2.1.1. Note that in the situation of the theorem above we do not require that the composite ﬁeld of the pairwise distinct CM ﬁelds in question has degree 2n over k; actually, its degree might be smaller. 1.3. Outline of the paper. It is well-known that Drinfeld modules of rank 2 are analogues of elliptic curves. In §2, we review the theory of Drinfeld and establish an analogue of formula (1) for the arithmetic Drinfeld modular forms in question. This formula enables us to transfer the statement in Theorem 1.2.1 to an algebraic independence question in the transcendence theory for Drinfeld modules of rank 2 with complex multiplication. Our tool for proving the algebraic independence result is a fundamental theorem of Papanikolas [P08], which can be regarded as an analogue of Grothendieck’s conjecture on periods of abelian varieties over Q. The basic structures which come up in this theory are the t-motives which are dual notion of t-modules introduced by Anderson [A86] and their Galois groups. In §3, we review Papanikolas’ theory and use these t-motivic techniques to give a proof of Theorem 1.2.1.

2. Special values of Drinfeld modular forms 2.1. Review of Drinfeld modules. Let t be an independent variable of θ and let τ : C∞ → C∞ be the Frobenius qth power operator. Let C∞[τ] be the twisted polynomial q ring in τ over C∞ subject to the relation τc = c τ for c ∈ C∞. A Drinfeld Fq[t]-module (of generic characteristic) is an Fq-linear ring homomorphism ρ : Fq[t] → C∞[τ] so that r ρt = θ + a1τ + ··· + arτ and the image of ρ is not contained in C∞. The degree of ρt ¯ in τ is called the rank of ρ. We say that ρ is deﬁned over k if all coeﬃcients of ρt lie in k¯. Fix a rank r Drinfeld Fq[t]-module ρ. Its exponential function, denoted by expρ(z), P∞ qi is the unique power series of the form z + i=1 αiz satisfying the functional equation

expρ(θz) = ρt(expρ(z)).

By a rank r A-lattice Λ ⊆ C∞ we mean that Λ is a free A-module of rank r and it is discrete under the topology induced from | · |∞. Drinfeld showed that expρ(z) is entire on C∞ and the kernel of expρ, denoted by Λρ, is a rank r A-lattice inside C∞. Elements in Λρ are called periods of ρ. Conversely, given a rank r A-lattice Λ ⊆ C∞, we consider the entire function Y z e (z) := z (1 − ). Λ λ 06=λ∈Λ Λ For any a ∈ Fq[t], let ρa ∈ C∞[τ] be deﬁned by the following equality Λ eΛ(a(θ)z) = ρa (eΛ(z)). Λ Λ Then the map ρ : Fq[t] → C∞[τ] given by (a 7→ ρa ) deﬁnes a rank r Drinfeld Fq[t]- module with exponential function eΛ(z). For more details, see [Go96, R02, T04]. Λ Λ0 We say that two Drinfeld Fq[t]-modules ρ , ρ of the same rank are isomorphic if × Λ0 −1 Λ 0 −1 there exists ∈ C∞ so that ρt = ρt . Note that in this setting Λ = Λ, and we say that the two A-lattices Λ and Λ0 are homothetic. Conversely, given two homothetic 0 Λ Λ0 A-lattices Λ, Λ , their corresponding Drinfeld Fq[t]-modules ρ and ρ are isomorphic. 4 CHIEH-YU CHANG

The simplest example of a Drinfeld Fq[t]-module is the Carlitz module C, which is the rank one Drinfeld Fq[t]-module given by Ct = θ + τ. Its exponential function is given by ∞ i X zq exp (z) := z + . C (θqi − θ)(θqi − θq) ··· (θqi − θqi−1 ) i=1 It can be also written as Y z exp (z) = z (1 − ), C aπ˜ 06=a∈A whereπ ˜ is a generator of ΛC , given by √ ∞ q−1 Y i π˜ = θ −θ (1 − θ1−q )−1 i=1 √ for a ﬁxed (q − 1)st root q−1 −θ of −θ. Throughout this paper, we ﬁx such aπ ˜. Given two Drinfeld Fq[t]-modules ρ1 and ρ2, we say that ρ1 and ρ2 are isogenous if there exists φ ∈ C∞[τ] so that φ(ρ1)a = (ρ2)aφ for all a ∈ Fq[t]; otherwise, we say that ρ1 and ρ2 are non-isogenous. In the case ρ1 = ρ2 = ρ, we set

End(ρ) := {φ ∈ C∞[τ]; φρa = ρaφ for all a ∈ Fq[t]} and note that End(ρ) can be identiﬁed with the multiplication ring of Λρ:

{x ∈ C∞; xΛρ ⊆ Λρ} . Suppose that ρ is of rank 2. We say that ρ has complex multiplication if End(ρ) % A; otherwise, we say that ρ has no complex multiplication. When ρ has complex multiplication, the fraction ﬁeld Kρ of End(ρ) is called the CM ﬁeld of ρ and it is quadratic over k. In analogy with a classical result for CM elliptic curves, we have the basic properties concerning rank 2 Drinfeld Fq[t]-modules with complex multiplication.

Proposition 2.1.1. Let ρ1 and ρ2 be two rank 2 Drinfeld Fq[t]-modules with complex multiplication. Then we have:

(i) ρ1 and ρ2 are isogenous if and only if their CM fields are the same. ¯ (ii) Suppose that ρ1 and ρ2 are defined over k with the same CM field. Then for ¯ any nonzero period λi of ρi, i = 1, 2, we have λ1/λ2 ∈ k. 2.2. Special values of Drinfeld modular forms at algebraic points. Before proving Theorem 1.2.1, we shall focus on the case of rank 2 Drinfeld modules and make a connection with Drinfeld modular forms. Λ Λ 2 Given a rank 2 Drinfeld Fq[t]-module ρ , we write ρt = θ + g(Λ)τ + ∆(Λ)τ . The q+1 Λ g(Λ) j-invariant of ρ is defined by j(Λ) := ∆(Λ) . For a given z ∈ H, we set Λz := Az + A and define g(z) := g(Λz), ∆(z) := ∆(Λz) and j(z) := j(Λz). When we regard g(z), ∆(z) as functions on H, then g (resp. ∆) is a Drinfeld modular form of weight q − 1 (resp. 2 q − 1) and type 0 for GL2(A). Moreover, when we take the normalization gnew := 1−q 1−q2 π˜ g, ∆new :=π ˜ ∆, then gnew and ∆new are arithmetic (cf. [Go80b, Ge88]). Note that the function j(z) is called a Drinfeld modular function for GL2(A). Λ Λ0 Λ Λ0 Given two Drinfeld Fq[t]-modules φ , φ of rank two, we see that φ and φ are 0 isomorphic if and only if j(Λ) = j(Λ ). Let GL2(A) act on the rank 2 A-lattice Λz := Az + A with z ∈ H by

γ · Λz := Λγz, for γ ∈ GL2(A). DRINFELD MODULAR FORMS AND ALGEBRAIC INDEPENDENCE 5

0 Then we see that any two lattices Λz and Λz0 with z, z ∈ H are homothetic if and only 0 if z = γz for some γ ∈ GL2(A). As any rank 2 A-lattice Λ is homothetic to a lattice of the form Λz for some z ∈ H, we have the identiﬁcations set of isomorphism classes of rank ∼ GL2(A)\H = = C∞ two Drinfeld Fq[t]-modules over C∞ z 7→ Λz 7→ j(z).

Therefore, the holomorphic function j : H → C∞ identifies the rigid analytic space 1 GL2(A) \ H with the affine line A both set-theoretically and analytically. Whence the ¯ ¯ function field of the affine line GL2(A) \ H over k is identified with k(j). In what follows, for any two elements x, y ∈ C∞ with y 6= 0, we denote by x ∼ y if x/y ∈ k¯. The following formula is the key bridge connecting the special values of Drinfeld modular forms and periods of Drinfeld modules of rank 2. Theorem 2.2.1. Let f be an arithmetic Drinfeld modular form of nonzero weight ` for ¯ a congruence subgroup Γ of GL2(A). For any α ∈ H with j(α) ∈ k, we have ` f(α) ∼ (ω/π˜) , where ω is a nonzero period of the rank 2 Drinfeld Fq[t]-module ρ given by ρt = θ + q+1pj(α)τ + τ 2 for some (q + 1)st root q+1pj(α) of j(α).

Λα Proof. Let Λα = Aα + A and let ρ be its corresponding rank 2 Drinfeld Fq[t]-module 2 Λα 2 × q −1 given by ρt = θ+g(α)τ +∆(α)τ . We choose an element α ∈ C∞ so that ∆(α)α = −1 Λα 1. Define ρ to be the rank 2 Drinfeld Fq[t]-module given by ρt = α ρt α. Then we q+1p 2 q+1p have ρt = θ + j(α)τ + τ for some (q + 1)st root j(α) of j(α). Put ω1 := α/α and ω2 := 1/α, then the period lattice of ρ is given by Λρ = Aω1 + Aω2. It follows 2 2 q −1 ω2 q −1 that ∆(α) = ω2 and hence ∆new(α) = ( π˜ ) . Suppose that Γ ⊇ Γ(N) for some nonzero ideal N of A. Let X(N) be the smooth projective model of the affine algebraic curve Y (N) := Γ(N)\H. Then X(N) is defined ¯ ¯ over k (cf. [Ge84]) and we denote by k(X(N)) (resp. C∞(X(N))) the function field of ¯ q2−1 ` ¯ ¯ X(N) over k (resp. C∞). We claim that f /∆new is algebraic over k(j). As j(α) ∈ k, 2 q −1 ` ¯ ω2 ` this claim implies f (α)/∆new(α) ∈ k and hence we have that f(α) ∼ ( π˜ ) . To prove the claim above, we let u, v ∈ k be two elements, not both in A, with the monic generator n of the ideal N as a common denominator. Define X 1 g(z) Eu,v(z) := , and fu,v(z) := q−1 . az + b Eu,v(z) (a, b) ∈ k × k (a, b) ≡ (u, v) mod A × A We list the following results due to Gekeler [Ge86]: ¯ ¯ • k(X(N)) is identified with the field over k(j) generated by fu,v for all (u, v) ∈ −1 2 DN := (n A/A) \{(0, 0)}. πz˜ −1 ¯ • Each fu,v has a Laurent series expansion in expC ( n ) with coefficients in k. Note that since Γ(N) is of finite index in GL2(A), the map X(N) X(1) is finite and ¯ ¯ ¯ hence k(X(N)) is a finite extension of k(X(1)). Therefore, k({j, fu,v;(u, v) ∈ DN }) is an algebraic extension of k¯(j). ¯ Let KN be the field over k generated by all modular functions for Γ(N) for which πz˜ −1 the coefficients of their expansions in expC ( n ) are algebraic over k. Here a modular function for Γ(N) is a meromorphic function on H that is invariant under the action of Γ(N) and has a Laurent expansion at each cusp of Γ(N) with only finitely many ¯ polar terms. Note that k({j, fu,v;(u, v) ∈ DN }) ⊆ KN and C∞({j, fu,v;(u, v) ∈ DN }) 6 CHIEH-YU CHANG is identified with C∞(X(N)). One checks directly that KN and C∞ are linearly disjoint ¯ ¯ ¯ over k. It follows that KN = k({j, fu,v;(u, v) ∈ DN }), which is algebraic over k(j). Let N = (n) be the ideal chosen for Γ (see §1.2 for the definition) and recall that πz˜ 0 q∞(z) = q∞(z; Γ) := 1/ expC ( n ). Note that n|n. Put n := n/n, then we have πz˜ πz˜ 1/ expC ( n ) = 1/Cn0 (expC ( n )). It follows that all coefficients of the expansion of f in πz˜ −1 ¯ expC ( n ) are in k since by assumption all coefficients of the expansion of f in q∞(z) ¯ are in k. As ∆new is an arithmetic Drinfeld modular form for GL2(A) , all coefficients of πz˜ −1 ¯ q2−1 the expansion of ∆new in expC ( n ) are also in k. It follows that f /∆new ∈ KN , whence the claim. Note that if α ∈ H is a CM point, then we have j(α) ∈ k¯ (cf. [Ge83, Satz 4.3]). Let α ∈ H be a CM point and ρ be the rank 2 Drinfeld Fq[t]-modules defined in Theorem 2.2.1. Then the CM field of the rank 2 Drinfeld module ρ is equal to k(α), since ρ is isomorphic to φΛα . As f(α) is related to the ratio (ω/π˜)` for some period ω of ρ, by Proposition 2.1.1 we see that Theorem 1.2.1 is a consequence of the following theorem, which will be proved in the next section: ¯ Theorem 2.2.2. Let ρ1, . . . , ρn be rank 2 Drinfeld Fq[t]-modules defined over k with complex multiplication. Suppose that ρi and ρj are non-isogenous for all i 6= j. Then for any nonzero period ωi of ρi for i = 1, . . . , n, the n + 1 quantities

π,˜ ω1, . . . , ωn are algebraically independent over k¯.

3. t-Motivic Galois groups and algebraic independence

Let C∞((t)) be the field of Laurent series in the variable t over C∞. Let T ⊆ C∞((t)) be the ring of power series that are convergent on the closed unit disc. (T is called the Tate algebra over C∞). Let L be the fraction field of T. P i For n ∈ Z, given a Laurent series f = i ait ∈ C∞((t)) we define the n-fold twist n −n (n) P q i of f by the rule σ f := f := i ai t . For each n, the twisting operation is an ¯ automorphism of the Laurent series field C∞((t)) stabilizing several subrings, e.g., k[[t]], ¯ (n) k[t] and T. More generally, for any matrix B with entries in C∞((t)) we define B (n) (n) σ by the rule B ij := Bij . For any subfield F of C∞((t)) we denote by F the set σ ¯ σ consisting of the elements in F fixed by σ. Then we have L = k(t) = Fq(t) (cf. [P08, Lemma 3.3.2]). P∞ i A power series f = i=0 ait ∈ C∞[[t]] that satisfies

pi lim |ai|∞ = 0 and [k∞(a0, a1, a2, ··· ): k∞] < ∞ i→∞ is called an entire power series. As a function of t, such a power series f converges on all of C∞ and, when restricted to k∞, f takes values in k∞. The ring of entire power series is denoted by E. 3.1. Review of Papanikolas’ theory. In this subsection we follow [P08] for back- ground and terminology of t-motives. Let k¯(t)[σ, σ−1] be the noncommutative ring of Laurent polynomials in σ with coefficients in k¯(t), subject to the relation σf = f (−1)σ, ∀ f ∈ k¯(t). Let k¯[t, σ] be the noncommutative subring of k¯(t)[σ, σ−1] generated by t and σ over k¯. An Anderson t-motive is a left k¯[t, σ]-module M which is free and finitely generated both as a left k¯[t]-module and a left k¯[σ]-module and which satisfies (t − θ)N M ⊆ DRINFELD MODULAR FORMS AND ALGEBRAIC INDEPENDENCE 7

σM for integers N sufficiently large. Given an Anderson t-motive M of rank r over ¯ ¯ k[t], let m ∈ Matr×1(M) comprise a k[t]-basis of M. Then multiplication by σ on ¯ M is represented by σm = Φm for some Φ ∈ Matr(k[t]). Note that the condition (t − θ)N M ⊆ σM for N 0 implies det Φ = c(t − θ)s for some c ∈ k¯× and s ≥ 0. We say that a left k¯(t)[σ, σ−1]-module P is a pre-t-motive if it is finite dimensional over k¯(t). Let P be the category of pre-t-motives. Morphisms in P are left k¯(t)[σ, σ−1]- module homomorphisms. Given a pre-t-motive P which is of dimension r over k¯(t), let ¯ p ∈ Matr×1(P ) comprise a k(t)-basis of P . Then multiplication by σ on P is given by ¯ σp = Φp for some Φ ∈ GLr(k(t)). Furthermore, we say that P is rigid analytically (−1) trivial if there exists Ψ ∈ GLr(L) so that Ψ = ΦΨ. Such Ψ is called a rigid analytic trivialization for Φ and it is unique up to right multiplication by a matrix in GLr(Fq(t)). −1 B Note that in this situation, the entries of Ψ p comprise an Fq(t)-basis of P , where † P := L ⊗k¯(t) P , on which σ acts diagonally, B † P :=the Fq(t)-submodule of P fixed by σ. Given an Anderson t-motive M, we obtain a pre-t-motive M by setting M := ¯ (−1) ¯ k(t) ⊗k¯[t] M with the σ-action given by σ(f ⊗ m) := f ⊗ σm for f ∈ k(t), m ∈ M. Rigid analytically trivial pre-t-motives that can be constructed from Anderson t-motives using direct sums, subquotients, tensor products, duals and internal Hom’s, are called t-motives. These t-motives form a neutral Tannakian category T over Fq(t) with fiber functor M 7→ M B. For any t-motive M, let TM be the strictly full Tannakian subcategory of T generated by M, then by Tannakian duality TM is equivalent to the category of finite-dimensional representations over Fq(t) of an affine algebraic group scheme ΓM defined over Fq(t). We call ΓM the (t-motivic) Galois group of M and naturally we have a faithful repre- B sentation ϕM :ΓM ,→ GL(M ), which is called the tautological representation of M. Such ΓM can be constructed explicitly as follows. ¯ Let Φ ∈ GLr(k(t)) represent multiplication by σ on M and let Ψ ∈ GLr(L) be a rigid analytic trivialization for Φ. Now set Ψ1,Ψ2 ∈ GLr(L ⊗k¯(t) L) to be the matrices such −1 that (Ψ1)ij = Ψij ⊗ 1 and (Ψ2)ij = 1 ⊗ Ψij, and let Ψe := Ψ1 Ψ2 ∈ GLr(L ⊗k¯(t) L). We define an Fq(t)-algebra homomorphism µΨ : Fq(t)[X, 1/ det X] → L ⊗k¯(t) L by setting µΨ(Xij) = Ψe ij, where X = (Xij) is an r × r matrix of independent variables. Define

(2) ΓΨ := Spec ImµΨ.

Thus Γ is the smallest closed subscheme of GL such that Ψ ∈ Γ ( ⊗¯ ). Ψ r/Fq (t) e Ψ L k(t) L We collect the results developed by Papanikolas. ¯ Theorem 3.1.1 (Papanikolas [P08]). Let M be a t-motive and let Φ ∈ GLr(k(t)) represent multiplication by σ on M and let Ψ ∈ GLr(L) be a rigid analytic trivialization for Φ. Then ΓΨ has the following properties. (a) Γ is a closed (t)-subgroup scheme of GL . Ψ Fq r/Fq (t) (b) ΓΨ is absolutely irreducible and smooth over Fq(t). ¯ (c) dim ΓΨ = tr.degk¯(t)k(t)({Ψij; 1 ≤ i, j ≤ r}). (d) ΓΨ is isomorphic to ΓM over Fq(t). ¯ ¯ s ¯× (e) If Φ ∈ Matr(k[t]) ∩ GLr(k(t)) and detΦ = c(t − θ) for some c ∈ k , then there (−1) exists Ψ ∈ Matr(E) ∩ GLr(T) so that Ψ = ΦΨ and one has ¯ dim ΓΨ = tr. degk¯ k ({Ψij(θ); 1 ≤ i, j ≤ r}) .

Remark 3.1.2. Theorem 3.1.1 implies that ΓΨ can be regarded as a linear algebraic group over Fq(t), and we always identify ΓM with ΓΨ in this paper. Moreover, the 8 CHIEH-YU CHANG tautological representation of M is described as follows. For any Fq(t)-algebra R, the map ϕ(R) :Γ (R) → GL(R ⊗ M B) is given by M M Fq (t) γ 7→ 1 ⊗ Ψ−1m 7→ (γ−1 ⊗ 1) · (1 ⊗ Ψ−1m) .

Remark 3.1.3. Given two t-motives M1,M2, we consider the t-motive as direct sum

M := M1 ⊕ M2. By (2) and Theorem 3.1.1(d) we have ΓM ⊆ ΓM1 × ΓM2 . Since Mi is a sub-t-motive of M, by Tannakian theory we have a surjective Fq(t)-morphism

πi :ΓM ΓMi for i = 1, 2. Let TM (resp. TMi ) be the strictly full Tannakian subcategory of T generated by M (resp. Mi). Note that the surjective map πi comes from the restriction of the ﬁber functor of TM to TMi . Using the explicit description of ϕM in the remark above, we see that πi is the restriction to ΓM of the canonical projection ΓM1 × ΓM2 ΓMi . 3.2. t-Motivic Galois groups of CM Drinfeld modules of rank 2. From now 2 ¯ on, we ﬁx a rank 2 Drinfeld Fq[t]-module ρ given by ρt = θ + gτ + ∆τ with g, ∆ ∈ k, ∆ 6= 0. Let Λρ = Aω1 + Aω2. The quasi-periodic function of ρ (associated to τ) is the unique entire function Fτ : C∞ → C∞ satisfying the two conditions: q • Fτ (z) ≡ 0 (mod z ); q • Fτ (θz) = θFτ (z) + expρ(z) .

The Fτ also has the property that Fτ |Λρ :Λρ → C∞ is A-linear (for more details, see [Ge89]). The values Fτ (ω) for ω ∈ Λρ, are called quasi-periods of ρ. In analogy with the classical work of Schneider, Yu [Yu86] proved that nonzero periods and quasi-periods of ρ are transcendental over k. The matrix ω1 −Fτ (ω1) ω2 −Fτ (ω2) is called the period matrix of ρ. In analogy with the Legendre relation for elliptic curves, Anderson proved √ q−1 (3) ω1Fτ (ω2) − ω2Fτ (ω1) =π/ξ ˜ ∆ (cf. [T04, Thm. 6.4.6]), √ q−1 × 1/q ¯× where ∆ is a (q − 1)st root of ∆ and ξ ∈ Fq satisfies ξ = −ξ. For any ∈ k , it is not hard to show that the field over k¯ generated by the entries of the period matrix of ρ is equal to the field over k¯ generated by the entries of the period matrix of φ given by −1 φt := ρt. Hence for the purpose of investigating transcendence properties, we need 2 only consider a fixed rank 2 Drinfeld Fq[t]-module given in the form ρt = θ + κτ + τ with κ ∈ k¯. Let 0 1 (4) Φ := ∈ Mat (k¯[t]) ∩ GL (k¯(t)) ρ (t − θ) −κ(−1) 2 2 ¯ define a pre-t-motive Mρ, i.e., with respect to a fixed k(t)-basis m ∈ Mat2×1(Mρ) of the ¯ two dimensional k(t)-vector space Mρ, multiplication by σ on Mρ is given by Φρ. Note ∼ that we have EndT (Mρ) = Kρ, where Kρ is the fraction field of End(ρ) (cf. [CP08]). In fact, Mρ is a t-motive and we review the construction of a rigid analytic trivialization for Φρ as follows (cf. [CP08, §2.5]). P∞ qi Let expρ(z) := z + i=1 αiz be the exponential function of ρ. Given u ∈ C∞, we consider the Anderson generating function ∞ ∞ qi X u X αiu (5) f (t) := exp ti = ∈ u ρ θi+1 θqi − t T i=0 i=0 DRINFELD MODULAR FORMS AND ALGEBRAIC INDEPENDENCE 9

q and note that fu(t) is a meromorphic function on C∞. It has simple poles at θ, θ ,... q u u with residues −u, −α1u ,... respectively. Using ρt(expρ( θi+1 )) = expρ( θi ), we have

(1) (2) (6) κfu (t) + fu = (t − θ)fu(t) + expρ(u).

(m) qm qm+1 Since fu (t) converges away from {θ , θ ,...} and Rest=θ fu(t) = −u, we have

(1) (2) (7) κfu (θ) + fu (θ) = −u + expρ(u) by specializing (6) at t = θ. 1 q−1 −1 Pick a suitable choice (−θ) of the (q − 1)-st root of −θ so that Ω(θ) = π˜ , where ∞ −q Y t Ω(t) := (−θ) q−1 1 − ∈ , θqi E i=1

(see [ABP04, Cor. 5.1.4]). Now put fi := fωi (t) for i = 1, 2, and deﬁne

(1) (1) ! −f2 f1 (8) Ψρ := ξΩ (1) (2) (1) (2) . κf2 + f2 −κf1 − f1

Then Ψρ ∈ Mat2(E) and det Ψρ = ξΩ, whence Ψρ ∈ GL2(T). Furthermore, by (6) we (−1) have Ψρ = ΦρΨρ. By specializing Ψρ at t = θ, we obtain −1 ω1 −Fτ (ω1) (9) Ψρ (θ) = . ω2 −Fτ (ω2)

Therefore by Theorem 3.1.1 we have the following equality ¯ dim ΓΨρ = tr. degk¯ k(ω1, ω2,Fτ (ω1),Fτ (ω2)). When ρ has complex multiplication, Thiery [Thi92] showed that ¯ tr. degk¯ k(ω1, ω2,Fτ (ω1),Fτ (ω2)) = 2,

whence dim ΓΨρ = 2. Actually one can prove dim ΓΨρ = 2 directly without using Thiery’s result, but we do not discuss the details here. However, we have an explicit description of the motivic Galois group of Mρ:

Lemma 3.2.1. Let ρ, Mρ and Ψρ be deﬁned as above. Suppose ρ has complex multiplication and set Kρ := EndT (Mρ). Then for any Fq(t)-algebra R, one has the isomorphism on R-valued points of group schemes:

Γ (R) ∼ (K ⊗ R)×. Ψρ = ρ Fq (t)

Proof. See the proof of [C09, Lemma 2.3.2].

∼ ¯ We note that any g ∈ EndT (Mρ) = Kρ is represented by a matrix G ∈ Mat2(k(t)) satisfying g(m) = Gm, and we have a natural embedding in the following way

−1 (10) (g 7→ Ψρ GΨρ) : EndT (Mρ) ,→ Mat2(Fq(t)) (cf. [CP08, §3.2]), which will be used in the proof of Theorem 2.2.2. 10 CHIEH-YU CHANG

3.3. Some properties of algebraic tori. In what follows, K is always a separable quadratic ﬁeld over Fq(t) together with a ﬁxed embedding K ,→ Mat2(Fq(t)). We denote by Res ( ) the Weil restriction of scalars for , and note that its K/Fq (t) Gm/K Gm/K × R-valued points is given by R ⊗ K for any (t)-algebra R. So we have regarded Fq (t) Fq Res ( ) as an algebraic subgroup of GL via the given embedding K ,→ K/Fq (t) Gm/K 2/Fq (t) Mat2(Fq(t)). Consider the determinant map det : Res ( ) → . Note that the restric- K/Fq (t) Gm/K Gm tion of det to the (t)-rational points of Res ( ) coincides with the usual Fq K/Fq (t) Gm/K norm map N : K× → (t)×. The kernel of det is a one-dimensional non-split K/Fq (t) Fq torus over (t) and we denote it by Res ( )(1). It is also called the norm Fq K/Fq (t) Gm/K torus corresponding to K/Fq(t).

Proposition 3.3.1. Let K1 and K2 be two non-isomorphic separable quadratic ﬁelds over Fq(t). Let Gi be the norm torus corresponding to Ki/Fq(t), i = 1, 2. Then any Fq(t)-morphism f : G1 → G2 is constant. Proof. Note that since Res ( ) is split over K , G is split over K for i = Ki/Fq (t) Gm/Ki i i i ¯ 1, 2. Let Gi (resp. f) be the scalar extension of Gi (resp. f) to K1, i = 1, 2. By the assumption on K1 and K2, we see that G2 is still non-split over K1 and hence is anisotropic (cf. [S98, §13.2.1]). Note that G1 is isomorphic to Gm/K1 . By [S98, Prop. ¯ 13.2.2] we have that f is constant, whence the result.

Given n pairwise non-isomorphic separable quadratic ﬁelds K1,..., Kn over Fq(t) (with n ≥ 2), let T be the (n+1)-dimensional (t)-subtorus of ×n Res ( ) n Fq i=1 Ki/Fq (t) Gm/Ki deﬁned by (11) T := (γ , . . . , γ ) ∈ ×n Res ( ); det γ = det γ for all i 6= j . n 1 n i=1 Ki/Fq (t) Gm/Ki i j

(Note that Tn−1 is deﬁned in the same way corresponding to K1,..., Kn−1). We denote by π the canonical projection ×n Res ( ) → Res ( ) and n i=1 Ki/Fq (t) Gm/Ki Kn/Fq (t) Gm/Kn π the canonical projection ×n Res ( ) → ×n−1 Res ( ). [n] i=1 Ki/Fq (t) Gm/Ki i=1 Ki/Fq (t) Gm/Ki

Theorem 3.3.2. Let n ≥ 2 and let K1,..., Kn be n pairwise non-isomorphic separable quadratic ﬁelds over Fq(t). Let Tn,Tn−1, πn, π[n] be deﬁned above. Suppose that G is an Fq(t)-subtorus of Tn satisfying the two properties that the restriction to G of π[n] (resp. π ) is surjective onto T (resp. Res ( )). Then G = T . n n−1 Kn/Fq (t) Gm/Kn n

Proof. Suppose on the contrary that G $ Tn. Since π[n] is surjective onto Tn−1 and G is connected, dim G = n. For an algebraic torus H over Fq(t) and any integer s, we denote by s · Id the map (x 7→ xs): H → H. By [Ono61, Prop. 1.3.1] there exists ∨ an Fq(t)-isogeny (i.e. surjective Fq(t)-morphism with ﬁnite kernel) π[n] : Tn−1 G so ∨ that π[n] ◦ π[n] = degπ[n] · Id. It follows that we have the surjective Fq(t)-morphism ν := π ◦ π∨ : T Res ( ). n n [n] n−1 Kn/Fq (t) Gm/Kn Notice that ×n−1 Res ( )(1) is the maximal anisotropic (t)-subtorus of i=1 Ki/Fq (t) Gm/Ki Fq Tn−1. Moreover, it is of dimension n − 1. By the deﬁnitions of Tn, π[n], πn and the property π∨ ◦ π = degπ · Id, we see that the image of ×n−1 Res ( )(1) [n] [n] [n] i=1 Ki/Fq (t) Gm/Ki under ν is contained inside Res ( )(1) and hence is constant by Proposition n Kn/Fq (t) Gm/Kn 3.3.1. It follows that νn is not a surjection, whence a contradiction. 3.4. Proof of Theorem 1.2.1. In §2, we have seen that Theorem 1.2.1 is a consequence of Theorem 2.2.2. In §3.2, we have seen that without loss of generality, the rank 2 Drinfeld Fq[t]-modules ρi with complex multiplication given in Theorem 2.2.2 are of DRINFELD MODULAR FORMS AND ALGEBRAIC INDEPENDENCE 11

2 ¯ the form (ρi)t = θ + κiτ + τ with κi ∈ k, for i = 1, . . . , n. For each 1 ≤ i ≤ n, we let

Φρi (resp. Ψρi ) be given in (4) (resp. (8)), and let Mi be the t-motive defined by Φρi . Moreover, for each 1 ≤ i ≤ n we set Ki := EndT (Mi), which is naturally embedded into Mat2(Fq(t)) by (10). Note that Ki is isomorphic to Ki, which is the CM field of n ρi. Since we have assumed the Drinfeld modules {ρi}i=1 are pairwise non-isogenous, n the n quadratic fields {Ki}i=1 are pairwise non-isomorphic. n Now, we define M to be the direct sum of the t-motives M := ⊕i=1Mi, on which the n ¯ action of σ is represented by the block diagonal matrix Φ := ⊕i=1Φρi ∈ GL2n(k(t)) ∩ ¯ n Mat2n(k[t]). Note that the block diagonal matrix Ψ := ⊕i=1Ψρi ∈ GL2n(T)∩Mat2n(E) gives a rigid analytic trivialization for Φ. Since each ρi has complex multiplication for i = 1, . . . , n, by (9), (3) and Theorem 3.1.1 we have dim ΓΨ ≤ n + 1. Hence proving Theorem 2.2.2 is equivalent to proving dim ΓΨ = n + 1. We are going to prove dim ΓΨ = n + 1 by induction on n. We discuss the following two cases: (I) p is odd. In this case, we list the following results: n • {Ki}i=1 are pairwise non-isomorphic separable quadratic fields over Fq(t) together with the embedding Ki ,→ Mat2(Fq(t)) via (10). • By Lemma 3.2.1, we have Γ =∼ Res ( ). Ψρi Ki/Fq (t) Gm/Ki −1 • Since for each 1 ≤ i ≤ n, we have det Ψe ρi = Ω ⊗ Ω ∈ L ⊗k¯(t) L (cf. §3.1), by (2) we have Γ ⊆ T ⊆ ×n Γ , where T is defined in (11). Ψ n i=1 Ψρi n n−1 • Set M[n] := ⊕i=1 Mi. Then the induction hypothesis implies ΓM[n] = Tn−1. • Let π be the canonical projection ×n Γ → ×n−1Γ and π be the [n] i=1 Ψρi i=1 Ψρi n canonical projection ×n Γ → Γ . By Remark 3.1.3 we have a surjective i=1 Ψρi Ψρn

Fq(t)-morphism ΓM ΓM[n] (resp. ΓM ΓMn ), which coincides with the restriction to ΓM of π[n] (resp. πn). The properties listed above satisfy the conditions of Theorem 3.3.2 and hence we have ΓΨ = Tn, whence dim ΓΨ = n + 1. (II) p is even. In√ this case we first note that the only inseparable quadratic field over k is given by k( θ). In other words, there is only one isogeny class of CM points whose corresponding CM fields are inseparable over k. According to Theorem 2.2.1 and Proposition 2.1.1, without loss of generality we can assume√ that√ the CM fields of q 2 ρ1, . . . , ρn−1 are separable over k and ρn is given by (ρn)√t = θ+( θ + θ)τ +τ . (Note that the endomorphism ring of ρ is identified with [ θ]). We further note that Γ n √ Fq Mn is isomorphic to Gm × Ga over Fq( t) (see [CP08, Remark 3.3.3]). We continue with n the notation {Mi}i=1 ,M,M[n] defined in the case (I). In (I), the condition p 6= 2 is only used to ensure that each quadratic field in question is separable over Fq(t). Therefore, we have proved that dim ΓM[n] = n. Let F be the composite field of K1,..., Kn. Then ΓM[n] is split over F since we have Γ ⊆ ×n−1 Res ( ). M[n] i=1 Ki/Fq (t) Gm/Ki

By Remark 3.1.3, we have ΓM ⊆ ΓM[n] × ΓMn and hence ΓM is a commutative algebraic group over Fq(t). Let π[n] (resp. πn) be the projection map ΓM[n] × ΓMn →

ΓM[n] (resp. ΓM[n] × ΓMn → ΓMn ). Note that by Remark 3.1.3 the restriction to

ΓM of π[n] (resp. πn) is surjective onto ΓM[n] (resp. ΓMn ). Let ΓM , ΓM[n] , ΓMn and π¯[n], π¯n be the scalar extension of algebraic groups and morphisms to F respectively.

Let ΓMs (resp. ΓMu ) be the closed subgroup consisting all semisimple (resp. unipotent) elements of ΓM , then ΓM is isomorphic to ΓMs × ΓMu . Asπ ¯[n](ΓMs ) is contained in the subgroup of ΓM[n] consisting of semisimple elements, we see thatπ ¯[n](ΓMs ) = ΓM[n] , whence dim ΓMs ≥ n. On the other hand, sinceπ ¯n(ΓMu ) is contained in the unipotent 12 CHIEH-YU CHANG

subgroup of ΓMn andπ ¯n is surjective, we have that dim ΓMu ≥ 1. Combining these two inequalities, we obtain that dim ΓM = n + 1.

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Mathematics Division, National Center for Theoretical Sciences, National Tsing Hua University, Hsinchu City 30042, Taiwan R.O.C.

Department of Mathematics, National Central University, Chung-Li 32054, Taiwan R.O.C. E-mail address: [email protected]