ON PERIODS OF THE THIRD KIND FOR RANK 2 DRINFELD MODULES

Transcription

1 ON PERIODS OF THE THIRD KIND FOR RANK 2 DRINFELD MODULES CHIEH-YU CHANG Abstract. In analogy with the periods of abelian integrals of differentials of the third kind for an elliptic curve defined over a number field we introduce a notion of periods of the third kind for a rank 2 Drinfeld F q [t]-module ρ defined over an algebraic function field. In this paper we establish explicit formulae for these periods of the third kind for ρ. Combining with the main result in [Chang-Papanikolas 202] we show the algebraic independence of the periods of first second and third kinds for ρ.. Introduction In this paper we are concerned with some special values as transcendental invariants that occur naturally from algebro-geometric obects defined over algebraic function fields in characteristic p. Our obects here are Drinfeld modules. In analogy with the de Rham cohomology of elliptic curves a de Rham cohomology theory of Drinfeld modules was well-developed by Anderson-Deligne-Gekeler-Yu cf. [Gekeler 989 Yu 990] and the notion and transcendence of the periods of the first and second kinds for Drinfeld modules are well-understood now cf. [Yu 986 Yu 990]. The aim of this paper is to include periods of the third kind and use the recent advances [Chang-Papanikolas 20 Chang-Papanikolas 202] to establish an algebraic independence result for the three different kinds of periods for rank 2 Drinfeld modules... Motivation. Let E be an elliptic curve defined over Q given by y 2 = 4x 3 g 2 x g 3. Let Λ = Zω + Zω 2 be the period lattice of E comprising the periods of the abelian integral of the holomorphic differential form dx/y of the first kind. Let z ζz σz be the Weierstrass functions associated to E and let η : Λ C be the quasi-period map defined by ηω := ζz + ω ζz. The quasi-periods ηω for ω Λ arise from the abelian integral of the differential form of the second kind xdx/y having poles with zero residues. By the work of Siegel and Schneider in the 930s the nonzero periods and quasi-periods of E are known to be transcendental numbers. Given u C for which u Q we regard u u as an algebraic point in Ext E G m = E = E and then form the differential form of the third kind having poles with nonzero residues: δ := 2 y + u dx x u y. Date: May Mathematics Subect Classification. Primary J93; Secondary G09 J89. Key words and phrases. Algebraic independence Drinfeld modules periods. The author was partly supported by NCTS and NSC grant M MY3.

2 2 CHIEH-YU CHANG For ω Λ we set λω u := ωζu ηωu. If γ is any closed cycle on EC along which δ is holomorphic then the period of the integral of δ along the cycle takes the form.. λω u + 2mπ for some ω Λ and m Z. The values λω u + 2mπ as above occur in the second coordinate of period vectors of the exponential function z z 2 z z σz u σz σu expζuz z 2 of a commutative algebraic group which is an extension of E by the multiplicative group G m cf. [Hindry 988]. The transcendence of λω u + 2mπ was established by Laurent cf. [Laurent 980 Laurent 982] if it is nonzero and later on Wüstholz [Wüstholz 984] extended Laurent s result to arbitrary differentials of the third kind over Q. Given nonzero ω Λ and u... u n C with u i Q for i n the Q-linear independence of the following periods of three different kinds {ω ηω λω u... λω u n } was achieved by Wüstholz [Wüstholz 984] if ω u... u n are linearly independent over Q see also [Baker-Wüstholz 2007]. However the algebraic independence of the periods of three different kinds for E is still an open problem..2. The main results and outline. In the characteristic p function field setting our algebrogeometric obects are Drinfeld modules defined over algebraic function fields. It is well-known that Drinfeld F q [t]-modules of rank 2 are analogues of elliptic curves and the Carlitz module which we denote by C is an analogue of G m. We review the relevant background in 2.2. The de Rham cohomology theory of Drinfeld modules was developed by Anderson Deligne Gekeler and Yu in the late 980s cf. [Gekeler 989 Yu 990]. It follows that the notion of periods of the first kind and quasi-periods periods of the second kind of Drinfeld modules fits the situation of elliptic curves well. The transcendence of the nonzero periods of the first and second kinds in question was established by Yu [Yu 986 Yu 990]. Given a rank 2 Drinfeld F q [t]-module ρ defined over an algebraic function field to define its periods of the third kind we shall adopt the geometric point of view parallel to the classical case on elliptic curves. We consider the set Ext 0 ρ C which consists of the isomorphism classes of 2- dimensional t-modules that are extensions of ρ by C and whose Lie algebras are split cf Much as in the case of elliptic curves where Ext E G m = E the space Ext 0 ρ C has an F q[t]- module structure and is isomorphic to ρ cf. Woo [Woo 995] and Papanikolas-Ramachandran [Papanikolas-Ramachandran 2003]. So for any t-module φ corresponding to an algebraic point in Ext 0 ρ C we call the second coordinate of any period vector of the exponential function of φ a period of the third kind for ρ cf Unlike the classical situation where.. is obtained by integration here we use techniques of Frobenius difference equations to establish explicit formulae for the periods of third kind for ρ and it turns out that the formulae are completely analogous to... The result is stated as Theorem and its proof occupies 3.2. With the explicit formulae at hand using the main result in [Chang-Papanikolas 202] not only are we able to show the transcendence of any nonzero period of the third kind for ρ but also we can further show the algebraic independence of the periods of first second and third kinds for ρ see Theorem The context of this paper is presented in a self-contained way.

3 ON PERIODS OF THE THIRD KIND FOR RANK 2 DRINFELD MODULE 3 Acknowledgements. The author thanks W.D. Brownawell M.A. Papanikolas D.S. Thakur and J. Yu for many helpful discussions and suggestions. He particularly thanks W.D. Brownawell for reading carefully an earlier version of this paper. Finally the author thanks the referee for several useful suggestions which improve this paper. 2. Drinfeld modules and Periods of the third kind 2.. Notation and preliminaries. In this paper we adopt the following notation. F q = the finite field with q elements for q a power of a prime number p. θ t z = independent variables. A = F q [θ] the polynomial ring in θ over F q. k = F q θ the fraction field of A. k = F q /θ the completion of k with respect to the place at infinity. k = a fixed algebraic closure of k. k = the algebraic closure of k in k. C = the completion of k with respect to the canonical extension of. = the normalized absolute value for the completed field C so that θ = q. A = F q [t] the polynomial ring in t over F q. T = { f C [[t]] f converges on t } the Tate algebra over C. L = the fraction field of T. G a = the additive group. For n Z given a Laurent series f = i a i t i C t we define the n-fold twist of f by f n = i a qn i t i. For each n the twisting operation is an automorphism of C t and stabilizes several subrings e.g. k[[t]] k[t] T and L. For any matrix B with entries in C t we define B n by the rule B n i = B n i. Also we note cf. [Papanikloas 2008 Lem ] 2.. { f T f = f } = F q [t] { f L f = f } = F q t. As a matter of notation we define σ f := f for f C t and write τ for σ. In particular we have τ = x x q : C C. Finally we denote by C [τ] the twisted polynomial ring in τ over C with respect to the relation τc = c q τ for c C Drinfeld modules and quasi-periodic functions. A Drinfeld A-module of generic characteristic is an F q -linear ring homomorphism ρ : A C [τ] so that the coefficient of τ 0 in ρ t is θ and ρ t / C. Here we denote by ρ t the image of t under ρ. The degree of ρ t in τ is called the rank of ρ. Moreover the exponential function of ρ is defined to be the unique power series of the form exp ρ z = z + i= α iz qi with α i C satisfying the functional equation exp ρ θz = ρ t exp ρ z. Drinfeld showed that exp ρ is an entire function on C and its kernel Λ ρ := Ker exp ρ is a discrete A-module of rank r = rank ρ inside C cf. [Drinfeld 974]. This Λ ρ is called the period lattice of ρ and elements of Λ ρ are called periods of first kind of ρ. Finally we say that ρ is defined over k if the coefficients of ρ t lie in k. The ring Endρ := { x C ; xλ ρ Λ ρ } is called the multiplication ring of Λρ and it can be identified with the endomorphism ring of ρ cf. [Goss 996 Rosen 2002 Thakur 2004]. Given

4 4 CHIEH-YU CHANG two Drinfeld A-modules ρ and ν we say that ρ is isomorphic to ν if there exists ɛ C so that ν t = ɛ ρ t ɛ. Fix a rank 2 Drinfeld A-module ρ given by ρ t = θ + κτ + τ 2 with κ k = 0. Let Λ ρ = Aω + Aω 2 be the period lattice of ρ. When Endρ A we say that ρ has complex multiplication and the fraction field of Endρ is called the CM field of ρ. If Endρ = A then we say that ρ has no complex multiplication. The quasi-periodic function of ρ associated to τ denoted by F τ is the unique power series satisfying the conditions: F τ z 0 mod z q ; F τ θz θf τ z = exp ρ z q. The function F τ has the following properties: F τ is entire on C ; F τ z + ω = F τ z + F τ ω for ω Λ ρ ; F τ Λρ : Λ ρ C is A-linear. The values F τ ω for ω Λ ρ are called quasi-periods of ρ associated to τ or periods of second kind for ρ. For each ϕ = i a i τ i Mat d C [τ] with a i Mat d C we put ϕ := a 0. A t-module of dimension d is an F q -linear ring homomorphism φ : A Mat d C [τ] so that φ t θi d is a nilpotent matrix. We say that φ is defined over k if Imφ Mat d k[τ]. The exponential function of φ denoted by exp φ is defined to be the unique F q -linear entire map from C d to C d satisfying the conditions: exp φ z z mod deg q; exp φ φ t z = φ t exp φ z. The kernel of exp φ is called the period lattice of φ and we say that φ is uniformizable if exp φ is surective onto C d cf. [Anderson 986]. Now we turn to the periods of second kind for ρ. In fact for ω Λ ρ the vector ω F τ ω tr is a period vector of exp φ for the two dimensional t-module φ defined by φ t = ρ t 0 τ θ which is an extension of ρ by the additive group G a cf. [Yu 990 Brownawell-Papanikolas 2002]. In [Yu 986] and [Yu 990] Yu established fundamental results parallel to the work of Siegel and Schneider. That is nonzero periods of the first and second kinds for ρ are transcendental over k. Anderson proved an analogue of the Legendre relation: 2.2. ω F τ ω 2 ω 2 F τ ω = π/ q where π is a fixed generator of the Carlitz module C which is the rank one Drinfeld A-module defined by C t = θ + τ and q is a q -st root of. Throughout this paper we fix such a choice q as above for a given ρ Biderivations and extensions of Drinfeld modules. Fix a rank 2 Drinfeld A-module ρ given by ρ t := θ + κτ + τ 2 with κ k = 0. We also fix generators {ω ω 2 } of the period lattice Λ ρ over A. We are interested in the two dimensional t-modules φ which are extensions

5 ON PERIODS OF THE THIRD KIND FOR RANK 2 DRINFELD MODULE 5 of ρ by C and whose Lie algebras are split i.e. such a φ fits into a short exact sequence of A-modules 0 C φ ρ 0 and φ t = θi 2. Let Ext 0 ρ C be the set of isomorphism classes of the two dimensional t-modules φ as above; then it forms a group under Baer sum up to Yoneda equivalence. However Ext 0 ρ C can be described explicitly as follows see [Papanikolas-Ramachandran ]. Let Der 0 ρ C be the F q -vector space consisting of all F q -linear maps δ : A C [τ]τ satisfying δ ab = C a δ b + δ a ρ b for a b A. Any δ Der 0 ρ C is called a ρ C-biderivation and it is uniquely determined by the image δ t =: h C [τ]τ because of the above ρ C-biderivation rule. Hence any element h C [τ]τ defines an element in Der 0 ρ C given by t h. Thus we have a canonical isomorphism of F q -vector spaces h t h : C [τ]τ Der 0 ρ C. An element δ Der 0 ρ C is called inner if there exists U C [τ] so that δ a = δ U a := Uρ a C a U for all a A. The F q -vector space of all such inner ρ C-biderivations is denoted by Der inn ρ C. Given any δ Der 0 ρ C it defines a t-module φ Ext 0 ρ C in the following way ρ φ t := t 0 Mat 2 C [τ]. δ t C t Conversely one observes that every φ Ext 0 ρ C defines a unique δ Der 0ρ C. Note that if δ U is an inner derivation then the t-module φ associated to δ U is split. In this case the matrix 0 γ := provides the splitting: U γ ρ φ a γ = a 0 for all a A. 0 C a However every split extension in Ext 0 ρ C arises in this way. From the definition of the Yoneda equivalence for any two extensions in Ext 0 ρ C which are Yoneda equivalent their corresponding elements in Der 0 ρ C differ by an inner derivation. Since the Baer sum on Ext 0 ρ C corresponds to the usual addition on Der 0ρ C we see that Ext 0 ρ C is isomorphic to Der 0ρ C/Der inn ρ C as F q -vector spaces and hence the space Ext 0 ρ C is in biection with {ατ; α C }. Moreover Ext 0 ρ C has an F q[t]-module structure that is isomorphic to the rank 2 Drinfeld A-module ρ. For more details see [Woo 995 Papanikolas-Ramachandran 2003] Periods of the third kind for ρ. Given α k we let δ Der 0 ρ C be defined by δ t = ατ and let φ be its corresponding t-module given by ρ 2.4. φ t := t 0. δ t C t

6 6 CHIEH-YU CHANG Then its exponential function can be written as z exp exp φ = ρ z z 2 exp C z 2 + G δ z cf. [Papanikolas-Ramachandran 2003 p.422] where G δ is an F q -linear entire function on C satisfying the properties: G δ z 0 mod z q ; G δ aθz = C a G δ z + δ a exp ρ z a A. Thus φ is uniformizable and its period lattice is given by { } ω Λ φ = A-Span ω 2 0 λ λ 2 π ω for some λ λ 2 C. For any Λ φ we call λ a period of third kind for ρ associated to λ ατ. In analogy with.. we establish an explicit formula for such λ in the following theorem whose proof will occupy 3.2. Theorem Let ρ be a rank 2 Drinfeld A-module defined over k given by ρ t = θ + κτ + τ 2. Given α k let φ be the t-module defined in Let u C satisfy exp ρ u = α/ q. For any ω period vector Λ φ there exists f A so that λ λ = q uf τ ω ωf τ u + f π. Remark Given a Drinfeld A-module ρ of rank r > 2 defined over k we consider the space Ext 0 ρ C defined in same way as above but which is an r -dimensional t-module cf. [Papanikolas-Ramachandran 2003]. For each algebraic point in Ext 0 ρ C we can define the associated period of the third kind in the same way such as the rank 2 case and one can ask about its formula. Since Ext 0 ρ C is no longer one-dimensional the author does not know whether there has an expected formula like the result in the theorem above or analogous to.. for such periods of the third kind. 3. Algebraic independence of the periods of three different kinds We continue with the notation as above with a fixed rank 2 Drinfeld A-module ρ defined over k given by ρ t = θ + κτ + τ 2. We also fix generators {ω ω 2 } for the period lattice Λ ρ. 3.. Anderson generating functions. We let exp ρ z = z + i= α iz qi. Given any u C we consider the Anderson generating function: 3.. f u t := u exp ρ i=0 θ i+ t i = i=0 α i u qi θ qi t T cf. [Pellarin 2008 Chang-Papanikolas 20] and note that f u t is a meromorphic function on C. It has simple poles at θ θ q... with residues u α u q... respectively. Using ρ t exp ρ u = exp θ i+ ρ u we have θ i 3..2 κ f u + f u 2 = t θ f u + exp ρ u.

7 ON PERIODS OF THE THIRD KIND FOR RANK 2 DRINFELD MODULE 7 Since f u m t converges away from {θ qm θ qm+...} and Res t=θ f u t = u we have 3..3 κ f u θ + f u 2 θ = u + exp ρ u by specializing 3..2 at t = θ. Moreover using the difference equation F τ θz θf τ z = exp ρ z q one has 3..4 f u θ = F τ u cf. [Thakur ]. Furthermore we pick a suitable choice θ q of the q -st root of θ so that Ωθ = π where i= Ωt := θ q q t E θ qi cf. [Anderson et al. 2004] Cor Note that Ω satisfies the difference equation 3..5 Ω = t θω Proof of Theorem Recall that ρ t = θ + κτ + τ 2 for κ k = 0. First we assume that =. For = 2 we define f := g exp φ φ t i ω t i Mat 2 C [[t]] λ i=0 where λ is given in Then using the functional equation exp φ φ t z = φ t exp φ z one has φ t Hence for = 2 one has f g = θ f + κ f + f 2 α f + θg + g = t κ f + f 2 = t θ f α f + g = t θg. This leads to the following difference equation f f f f 2 0 = t θ κ 0 g g 2 0 α t θ Ω Put then we have Φ := κ 0 t θ 0 0 α 0 and Ψ := Ψ = ΦΨ. f g. f f 2 0 f f 2 0 g g 2 Ω Ω f Ω f Ω f Ω f 2 0 Ωg Ωg 2 For any vector v = v v 2 tr C 2 we put v := max{ v v 2 }. Then we have v + w max{ v w } for v w C 2. We claim that for each = 2 f converges on t < θ = q; g Res t=θ f = ω Res t=θ g = λ..

8 8 CHIEH-YU CHANG We write where a 0 = exp φ 0 0 For each l 2 by definition we have f l g l = =0 z z 2 = a i zqi i=0 z qi 2 and a i Mat 2 C for all i N. i=0 a i θ qi + ωqi l λ qi l t For any t < θ since exp φ is entire on C 2 there exists a positive integer N sufficiently large so that a i ωqi l i=0 θ qi + λ qi t max 0 i N a ωqi l i l λ qi l θ qi + t which converges to 0 as. Hence f l g l converges on t < θ for l = 2. For the second part of the above claim it suffices to prove that for each l 2 f l g l ω l /t θ λ l /t θ = f l g l =0 θ + ω l λ l t = a i =0 i= θ qi + ωqi l λ qi l t converges at t = θ. To prove it we use the entireness of exp φ again to deduce the following inequality: for N 0 a i i= θ qi + ωqi l λ qi l θ max i N a i ωqi l λ qi l θ θ q+ Note that the right hand side of the inequality converges to 0 as and so we show the desired claim. For each = 2 note that by we have f = f ω where f ω is the Anderson generating function of ω cf Since f θ = F τ ω for = 2 cf we have Ψθ = F τ ω / π F τ ω 2 / π 0 ω + κf τ ω / π ω 2 + κf τ ω 2 / π 0 λ + αf τ ω / π λ 2 + αf τ ω 2 / π As = we put ξ = q as given in Define ξ 0 0 B := ξκ ξ 0 and Φ := 0 0 then one has 3.2. B Φ = ΦB. 0 0 t θ κ 0 α/ξ 0..

9 ON PERIODS OF THE THIRD KIND FOR RANK 2 DRINFELD MODULE 9 Let f u be the Anderson generating function of u cf. 3.. we put G G 2 := t θ f u α/ξ f u ξω f 2 ξω f ξt θω f 2 ξt θω f and define Ψ := then using 3..2 and 3..5 one has ξω f 2 ξω f 0 ξt θω f 2 ξt θω f 0 G G 2 Ψ = Φ Ψ. Using 2.2. one can check that Ψ GL 3 T cf. [Chang-Papanikolas ]. Since det Ψ = det Ψ by Proposition 3..3 of [Anderson et al. 2004] both Ψ and Ψ are in GL 3 T Mat 3 E. ξ 2 Using 3.2. we have Ψ BΨ = Ψ BΨ. As Ψ BΨ GL 3 T we see that BΨ = Ψγ for some γ GL 3 A because the subring of T fixed by σ is A. specializing at t = θ it follows that γ is of the form 0 0 γ = 0 0 a b for some a b A whence λ = ξ uf τ ω ω F τ u + aθ π λ 2 = ξ uf τ ω 2 ω 2 F τ u + bθ π. According to 3..4 by Using the property that F τ Λρ is A-linear the desired result follows from For the general case that = we let ɛ k be a q 2 -st root of / satisfying ɛ q+ /ξ = / q. We define ν to be the rank 2 Drinfeld A-module given by ν t := ɛ ρ t ɛ whose leading coefficient in τ is. Then we have exp ν = ɛ exp ρ ɛ F τ = ɛ q F τ ɛ where F τ resp. F τ is the quasi-periodic function of ρ resp. ν associated to τ. We define ϕ to be the two dimensional t-module given by ɛ ϕ t := 0 ρ t 0 ɛ 0 ν = t 0 0 ατ C t 0 αɛ q τ C t then we have exp ϕ = ɛ 0 0 exp φ ɛ 0 0 Put ω i := ω i /ɛ for i = 2 and ū := u/ɛ. Let Λ ν be the period lattice of ν then we have Λ ν := A-Span { ω ω 2 } and exp ν ū = αɛ q /ξ. Since the leading coefficient of ν t is by and the period lattice Λ ϕ := Ker exp ϕ is the A-module generated by { ω λ ω 2 λ 2 0 π }.

10 0 CHIEH-YU CHANG where Now given any ω λ λ i = ξ ūf τ ω i ω i F τ ū for i = 2. ω/ɛ Ker exp φ by we see that Λ ϕ. Thus we derive λ that λ = ξɛ q uf τ ω ωf τ u + f π for some f A by using F τ = ɛ q F τ ɛ and the property that F τ Λρ : Λ ρ C is A-linear Algebraic independence results. Based on the recent developments of t-motivic transcendence theory cf. [Anderson et al Papanikloas 2008] Papanikolas and the author of the present paper proved the following result which we only state in the rank 2 version. Theorem Chang-Papanikolas [Chang-Papanikolas 202 Thm..2.3] Let ρ be a Drinfeld A-module of rank 2 defined over k. Let u... u m C satisfy exp ρ u i k for i =... m. If u... u m are linearly independent over Endρ then are algebraically independent over k. u... u m F τ u... F τ u m Using the theorem above we obtain the algebraic independence result concerning the periods of first second and third kinds for ρ. Theorem Let ρ be a Drinfeld A-module of rank 2 defined over k. Let u... u n C satisfy exp ρ u i k for i =... n. Given a nonzero period ω Λ ρ we set λω u i := ωf τ u i u i F τ ω for i =... n. If ω u... u n are linearly independent over Endρ then are algebraically independent over k. ω F τ ω λω u... λω u n Proof. Without loss of generality we let ω = aω + bω 2 for some a b A with b = 0. For any ω Λ ρ and u C we set λω u := ω F τ u uf τ ω. Note that by the fact that the restriction of F τ to Λ ρ is A-linear we have λω u i = aλω u i + bλω 2 u i for each i n. Moreover by 2.2. we have L := k = k = k ω ω 2 u... u n F τ ω F τ ω 2 F τ u... F τ u n ω ω 2 λω u... λω u n F τ ω F τ ω 2 λω 2 u... λω 2 u n ω ω λω u... λω u n F τ ω F τ ω λω u... λω u n. Let K ρ be the fraction field of Endρ and let r be the K ρ -dimension of the vector space over K ρ spanned by {ω ω 2 u... u n }. Then we have r n + since ω u... u n are linearly independent over K ρ. We first consider the case that ρ has complex multiplication. In this case ω is a K ρ -multiple of ω and F τ ω is a k-linear combination of { ω F τ ω} cf. [Brownawell-Papanikolas 2002

11 ON PERIODS OF THE THIRD KIND FOR RANK 2 DRINFELD MODULE 3.2 and 3.3] and hence by Theorem 3.3. we have tr. deg k L = 2n + 2. Therefore the following set { } ω λω u... λω u n F τ ω λω u... λω u n is a transcendence basis of L over k; in particular the elements ω F τ ω λω u... λω u n are algebraically independent over k. Now we consider the case that ρ has no complex multiplication. If r = n + 2 then by Theorem 3.3. L has full transcendence degree 2n + 4 over k whence the algebraic independence of {ω F τ ω λω u... λω u n } over k. Therefore the remaining case we consider is r = n +. Without loss of generality we suppose that ω ω u... u n are linearly independent over k. Let a ω + a 2 ω + c u c n u n = 0 for a a 2 c... c n A with a = 0 c n = 0 whence b ω + b 2 ω 2 + c u c n u n = 0 where b := a + a 2 a b 2 := a 2 b. Since F τ is F q -linear using the difference equation F τ θz = θf τ z + exp ρ z q one has F τ c u = c F τ u + β for some β k =... n. Let ρ t = θ + κτ + τ 2. Then using and the analogue of the Legendre relation 2.2. we obtain λω 2 c n u n = b π/ q c λω 2 u c n λω 2 u n + γ n ω 2 for some γ n k. Since λω 2 c n u n = c n λω 2 u n + β n ω 2 we have λω 2 u n = c n b π/ q c λω 2 u c n λω 2 u n + η n ω 2 On the other hand we also have λω u n = c n b 2 π/ q c λω u c n λω u n + η nω It follows that λω u n = aλω u n + bλω 2 u n = c n ba π/ q for some η n k. for some η n k. n c i λω u i + aη nω + bη n ω 2. i= To finish the proof we note that using 2.2. and F τ ω = af τ ω + bf τ ω 2 we have F τ ω = a + bω 2 F τ ω b π q ω. Since and u n k-span {ω ω 2 u... u n } F τ u n k-span { F τ ω F τ ω 2 F τ u... F τ u n }

12 2 CHIEH-YU CHANG we have that L = k ω ω 2 u... u n F τ ω F τ ω 2 F τ u... F τ u n = k = k ω ω 2 λω u... λω u n F τ ω F τ ω 2 λω 2 u... λω 2 u n = k ω ω 2 λω u... λω u n F τ ω π λω 2 u... λω 2 u n ω ω λω u... λω u n F τ ω π λω u... λω u n = k ω ω λω u... λω u n F τ ω λω u n λω u... λω u n where the third equality uses 2.2. the fourth equality uses and the last equality uses and the assumption ba = 0. It completes the proof since by Theorem 3.3. we have tr. deg k L = 2n +. References [Anderson 986] G. W. Anderson t-motives Duke Math. J [Anderson et al. 2004] G. W. Anderson W. D. Brownawell and M. A. Papanikolas Determination of the algebraic relations among special Γ-values in positive characteristic Ann. of Math [Brownawell 200] W. D. Brownawell Minimal group extensions and transcendence J. Number Theory [Brownawell-Papanikolas 2002] W. D. Brownawell and M. A. Papanikolas Linear independence of Gamma values in positive characteristic J. reine angew. Math [Baker-Wüstholz 2007] A. Baker and G. Wüstholz Logarithmic Forms and Diophatine Geometry Cambridge University Press [Chang-Papanikolas 20] C.-Y. Chang and M. A. Papanikolas Algebraic relations among periods and logarithms of rank 2 Drinfeld modules Amer. J. Math [Chang-Papanikolas 202] C.-Y. Chang and M. A. Papanikolas Algebraic independence of periods and logarithms of Drinfeld modules. With an appendix by Brian Conrad J. Amer. Math. Soc [Drinfeld 974] V. G. Drinfeld Elliptic modules Math. USSR-Sb [Gekeler 989] E.-U. Gekeler On the derham isomorphism for Drinfeld modules J. Reine Angew. Math. [Goss 996] D. Goss Basic Structures of Function Field Arithmetic Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge Vol. 35 Springer Berlin 996. [Hindry 988] M. Hindry Groupes algebriques commutatifs exemples explicites Séminaire d Arithmétique Saint-Etienne Exposé n 2 p.9 á 42. [Laurent 980] M. Laurent. Transcendance de périodes d intégrales elliptiques I J. Reine Angew. Math [Laurent 982] M. Laurent. Transcendance de périodes d intégrales elliptiques II J. Reine Angew. Math [Papanikloas 2008] M. A. Papanikolas. Tanakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms Invent. Math [Papanikolas-Ramachandran 2003] M. A. Papanikolas and N. Ramachandran A Weil-Barsotti formula for Drinfeld modules J. Number Theory

13 ON PERIODS OF THE THIRD KIND FOR RANK 2 DRINFELD MODULE 3 [Pellarin 2008] F. Pellarin Aspects de l indépendance algébrique en caractéristique non nulle Séminaire Bourbaki Vol. 2006/2007 Astérisque No Exp. no. 973 vii [Rosen 2002] M. Rosen Number Theory in Function Fields Grad. Texts in Math. 20 Springer New York [Thakur 2004] D. S. Thakur Function Field Arithmetic World Scientific Publishing River Edge NJ [Woo 995] S. S. Woo Extensions of Drinfeld modules of rank 2 by the Carlitz modules Bull. Korean Math Soc [Wüstholz 984] G. Wüstholz Transzendenzeigenschaften von Perioden elliptischer Integrale J. Reine Angew. Math [Yu 986] J. Yu Transcendence and Drinfeld modules Invent. Math [Yu 990] J. Yu On periods and quasi-periods of Drinfeld modules Compositio Math Department of Mathematics National Tsing Hua University and National Center for Theoretical Sciences Hsinchu City Taiwan R.O.C. address: cychang@math.cts.nthu.edu.tw