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 q1-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)nH which classifies the isomorphism classes of elliptic curves over C. We say that α 2 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 2 C with y 6= 0, we denote by x ∼ y if x=y 2 Q. Following Shimura, given an arithmetic modular form f of weight ` 2 N and a CM point α 2 H, there × exists !α 2 C which is unique up to algebraic multiple so that ! (1) f(α) ∼ ( pα )`: 2π −1 Note that !α 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 congru- ence subgroup Γ ⊆ SL2(Z). Let α1; : : : ; αn 2 H be CM points. Assume that the imagi- nary 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 in- dependent 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 j · j1 be the absolute value on k with respect to the place 1 1 so that jθj1 = q. Let k1 := Fq(( θ )) be the completion of k with respect to j · j1 and let C1 be the completion of a fixed algebraic closure of k1 with respect to the ¯ absolute value extending j · j1. Throughout this paper k is a fixed algebraic closure of ¯ k inside C1. It is well known that A; k; k1; C1 and k are good analogues of Z; Q; R; C and Q, respectively. 1 1 Consider the Drinfeld upper half plane H := P (C1) n P (k1) = C1 n k1. Then GL2(k1) acts on H by fractional linear transformations: az + b a b γz := for z 2 H; γ = 2 GL (k ): cz + d c d 2 1 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 n P1(k). A Drinfeld modular form for a congruence subgroup Γ of GL2(A) of weight ` ≥ 0 and of type m 2 Z=(q − 1)Z is a holomorphic (i.e. rigid analytic) function f : H ! C1 satisfying: a b • f(γz) = (det γ)−m(cz + d)`f(z) for all γ = 2 Γ; c d • f is holomorphic at the cusps of Γ. We first explain the second condition above for the cusp 1. 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 2 I . Let exp (z) be the Carlitz exponential function andπ ~ be a fixed 0 1 C fundamental period of the Carlitz module defined in x2.1. Define q1(z) = q1(z; Γ) := πz~ −1 expC ( n ) . Then we use q1(z) as a uniformizing parameter at 1. We say that f is holomorphic at 1 if for jzji := infx2k1 fjz − xj1g sufficiently large f has an expansion P i of the form f(z) = i≥0 aiq1(z) , which is called the q1-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 1 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 q1-expansion of f are algebraic over k. A point α 2 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 2 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 2 H so that k(β1) = k(β2). If f is non-vanishing at β1 and β2, then we ¯ have f(β1)=f(β2) 2 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 x2, 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 indepen- dence 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 x3, 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 : C1 ! C1 be the Frobenius qth power operator. Let C1[τ] be the twisted polynomial q ring in τ over C1 subject to the relation τc = c τ for c 2 C1. A Drinfeld Fq[t]-module (of generic characteristic) is an Fq-linear ring homomorphism ρ : Fq[t] ! C1[τ] so that r ρt = θ + a1τ + ··· + arτ and the image of ρ is not contained in C1. 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), P1 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 ⊆ C1 we mean that Λ is a free A-module of rank r and it is discrete under the topology induced from j · j1.