Journal of Number Theory

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1 Journal of Number Theory Contents lists available at ScienceDirect Journal of Number Theory Anoteonarefinedversionof Anderson Brownawell Papanikolas criterion Chieh-Yu Chang a,b,, a National Center for Theoretical Sciences, Department of Mathematics, National Tsing Hua University, Hsinchu City 300, Taiwan, ROC b Department of Mathematics, National Central University, Chung-Li 32054, Taiwan, ROC article info abstract Article history: Received 2 March 2008 Availableonline3December2008 Communicated by Dinesh S. Thakur MSC: primary J93, G09 Keywords: ABP criterion Linear independence Algebraic independence We give a refinement of the linear independence criterion over function fields developed by Anderson, Brownawell and Papanikolas [Greg W. Anderson, W. Dale Brownawell, Matthew A. Papanikolas, Determination of the algebraic relations among special Γ -values in positive characteristic, Ann. of Math ]. As a consequence, a function field analogue of the Siegel Shidlovskii theorem is derived Elsevier Inc. All rights reserved.. Introduction Let F q be the finite field of q elements with characteristic p. LetA := F q [θ] be the polynomial ring in variable θ over F q, with fraction field k := F q θ. Define an absolute value at the infinite place of k so that θ = q. Letk := F q θ be the -adic completion of k, letk be a fixed algebraic closure of k,letc be the -adic completion of k,let k be the algebraic closure of k in C and let F q be the algebraic closure of F q in k. Let t be an independent variable of θ. LetT be the Tate algebra of power series in C [[t]] that are convergent on the closed unit disk in C, and let L C t be the fraction field of T. LetE be the subring of T consisting of power series that are everywhere convergent and whose coefficients * Address for correspondence: Mathematics Division, National Center for Theoretical Sciences, National Tsing Hua University, Hsinchu City 300, Taiwan, ROC. address: cychang@math.cts.nthu.edu.tw. The author was supported by NCTS postdoctoral fellowship X/$ see front matter 2008 Elsevier Inc. All rights reserved. doi:0.06/j.jnt

2 730 C.-Y. Chang / Journal of Number Theory lie in a finite extension of k. Finally for a Laurent series f = i a it i C t and an integer n Z, we set f n := i aqn i t i and extend the operation f f n entrywise to matrices whose entries are in C t. In 2004, Anderson, Brownawell and Papanikolas [2] developed a criterion for linear independence over function fields, the so-called ABP criterion, to deal with the special values of the geometric Γ -function over A. As a break through, they proved that all the algebraic relations among those special Γ -values are explained by the standard functional equations. Now, we state the ABP criterion as the following: Theorem. Anderson Brownawell Papanikolas. Fix a matrix Φ = Φt Mat l k[t] such that det Φ is a polynomial in t vanishing if at all only at t = θ. Fixacolumn vector ψ = ψt Mat l E satisfying the functional equation ψ = Φψ. Evaluating ψ at t = θ, thus obtaining a column vector ψθ Mat l k. For every row vector ρ Mat l k such that ρψθ = 0 there exists a row vector P = Pt Mat l k[t] such that Pθ = ρ, Pψ = 0. In other words, in the situation of Theorem., every k-linear relation among entries of the specialization ψθ is explained by a k[t]-linear relation among entries of ψ itself. The main theorem of this paper is the following: Theorem.2. Fix a matrix Φ = Φt Mat l k[t] such that det Φ is a polynomial in t satisfying det Φ0 0. Fixavectorψ =[ψ t,...,ψ l t] tr Mat l E satisfying the functional equation ψ = Φψ.Let k \F q satisfy det Φ i 0 for all i =, 2, 3,... Then we have: For every vector ρ Mat l k such that ρψ = 0 there exists a vector P = Pt Mat l k[t] such that P = ρ, Pψ = 0. 2 tr.deg kt ktψ t,...,ψ l t = tr.deg k kψ,..., ψ l. In the situation of above theorem, we note that for any ψ in Mat l T satisfying ψ = Φψ,by Proposition 3..3 of [2] the condition det Φ0 0impliesψ Mat l E. Theorem.2 is an extension of ABP criterion. Theorem.22 is a consequence of Theorem.2. It can be thought of as a function field analogue of the Siegel Shidlovskii theorem concerning E-functions satisfying linear differential equations: Theorem.3 Siegel Shidlovskii, 956. Let f,..., f n be a set of E-functions which satisfy the system of first-order equations d dz f.. = B f n f.., f n where B is an n n matrix with entries in Qz. Denote the common denominator of the entries of B by T z. Then, for any Q such that T 0, tr.deg Qz Q z, f z,..., f n z = tr.deg Q Q f,..., f n.

3 C.-Y. Chang / Journal of Number Theory As a refined version of the Siegel Shidlovskii theorem, Beukers [3] showed that for any Q with T 0, any Q-linear relation among the values f,..., f n is the specialization of a linear relation among f,..., f n over Qz. For more details, we refer readers to [3]. In analogy with classical Galois theory of differential equations, Papanikolas [9] developed a Galois theory of systems of Frobenius difference equations. More precisely, let Φ GL l kt and suppose that there exists Ψ GL l L such that Ψ = ΦΨ, then one has an affine algebraic group scheme Γ Ψ defined over F q t so that dim Γ Ψ = tr.deg kt ktψ,. where ktψ is the field generated by all entries of Ψ over kt. Such difference equation Ψ = ΦΨ defines a rigid analytically trivial pre-t-motive M Φ in the terminology of [9]. Papanikolas [9] proved that the category R of rigid analytically trivial pre-t-motives forms a neutral Tannakian category over F q t. Once we consider the strictly full Tannakian subcategory of R generated by M Φ,then by Tannakian duality it is equivalent to the category RepΓ MΦ, F q t of finite dimensional representations of Γ MΦ over F q t, where Γ MΦ is an affine algebraic group scheme over F q t. Furthermore, such Γ MΦ is shown to be isomorphic to Γ Ψ over F q t by Papanikolas. Let Φ Mat l k[t] and k \F q satisfy the conditions of Theorem.2 and suppose that there exists Ψ Mat l T GL l L so that Ψ = ΦΨ. Note that in this situation all entries of Ψ are entire by [2, Proposition 3..3]. Then it is not hard to see that combining Theorem.2 and. one has dim Γ Ψ = tr.deg k k Ψ,.2 where kψ is the field generated by all entries of Ψ over k. Observe that.2 is a generalization of Papanikolas transcendence degree theorem cf. [9, Theorem..7] which is a function field analogue of Grothendieck s conjecture on periods of abelian varieties. Papanikolas theorem has been used to deal with algebraic independence concerning Carlitz logarithms, Carlitz polylogarithms, gamma values, logarithms of Drinfeld modules, etc. cf. [4 6,8,9]. Here we note that the conditions of Theorem.2 are weaker than those of Papanikolas transcendence degree theorem, and hence we have more choices of difference equations and specializations to deal with algebraic independence of certain special values. For example, using a formula of Anderson and Thakur [] we construct a difference equation satisfying the conditions of Theorem.2 to show the algebraic independence of Carlitz zeta values with varying constant fields see [7]. The present paper is organized as follows. In Section 2, we follow [2] and [9] closely to prove Theorem.2. In Section 3, we give some examples to explain that the conditions of Theorem.2 make sense. 2. Proof of Theorem Notations Given a polynomial f k[t] let deg t f denote its degree in t as usual deg 0 = and, more generally, given a matrix F with entries in k[t] put deg t F := max ij deg t F ij. Given any algebraic number x k we set x :=max τ τ x, where τ ranges over the automorphisms of k/k, therebydefining the size of x. More generally given a polynomial f = i a it i k[t], wedefine f :=max i a i.yet more generally, given a matrix F with entries in k[t] we define F :=max ij F ij.thenwehave D + E max D, E, FG F G for all matrices D, E, F, G with entries in k[t] such that D + E and FG are defined.

4 732 C.-Y. Chang / Journal of Number Theory Reductions and further notations We of course assume that ρ 0 and we may assume without loss of generality that K, Φ Mat l K[t], ρ Mat l K for some field extensions k K 0 K k, where K 0 /k is a finite separable extension and K is the closure of K 0 in k under the extraction of qth roots. Let O be the integral closure of A in K. After making suitable replacements Φ a q Φ, ψ a q ψ, ρ bρ for suitably chosen a, b A, ab 0, we may assume without loss of generality that Φ Mat l O[t], ρ Mat l O. If l>, then we fix a matrix ϑ Mat l l O of maximal rank such that ρϑ = 0. Thus, the K-subspace of Mat l K annihilated by right multiplication by ϑ is the K-span of ρ. Let Θ Mat l O[t] be the transpose of the matrix of cofactors of Φ. Then, ΦΘ = ΘΦ = det Φ l. Here l denotes the identity matrix of size l Proof of Theorem.2 We follow [2] closely to give a detailed proof of Theorem.2 as follows The case l = For the case of l =, since we have assumed that ρ 0, we have that ψ = 0. In this case, our task is to show that ψ vanishes identically. For any nonnegative integer ν we have ψ q ν q = ψ q ν+ = Φ q ν+ ψ q ν+. Our assumption implies that ψ q ν = 0 ν = 0,, 2,... Since is transcendental over F q,, q, q 2, q 3,... are distinct. Thus, ψ vanishes identically since ψ vanishes infinitely many times in the disc t if or in the disc t if < Construction of the auxiliary function E For the case of l>, let N be a parameter taking values in the set of positive integers divisible by 2l. We claim that there exists h = ht Mat l O[t] depending on the parameter N such that i h =O as N, and with the following properties for each value of N: ii h 0. iii deg t h < 2l N. iv E q N+ν = 0forν = 0,...,N, where E := hψ E. We call E the auxiliary function.

5 C.-Y. Chang / Journal of Number Theory We note that the auxiliary function E satisfies the following identity: hθ 0 Θ N+ν ψ N+ν = hθ 0 Θ N+ν Φ N+ν Φ 0 ψ = det Φ N+ν det Φ 0 E. 2. Further, the following identity will imply condition iv of the above claim: hθ 0 Θ N+ν ϑ N+ν t= q N+ν = 0 for ν = 0,...,N. 2.2 Assuming 2.2, then by the definition of ϑ, we see that for each 0 ν N, hθ 0 Θ N+ν t= q N+ν is spanned by ρ N+ν.Sincethehypothesisρψ= 0isequivalentto we have ρ N+ν ψ N+ν q N+ν = 0, hθ 0 Θ N+ν ψ N+ν t= q N+ν = 0 for ν = 0,...,N, and hence by 2., 0 = det Φ N+ν det Φ 0 E t= q N+ν = [ det Φ ] N+ν [ det Φ N+ν ] 0 E q N+ν for ν = 0,...,N. Thus, by our assumption det Φ i 0, for i =, 2, 3,..., we have that E q N+ν = 0 for ν = 0,...,N. Now, our task is to find h Mat l O[t] satisfying i, ii, iii as above and the identity 2.2. Here we shall note that any nonzero x O has the property x. Now we let r := l N, s := l N, 2 and pick any u A so that u > and u O. Foreach0 ν N wemultiplyby u q N+ν 2l N+N+ν deg t Θ on the both sides of 2.2, then with respect to the evident choice of bases, the homogeneous system of O-linear equations that we need to solve is described by a matrix M Mat r s O depending on N such that

6 734 C.-Y. Chang / Journal of Number Theory or M u q N 2l N+2N deg t Θ Θ q q ϑ =O as N if >, M u q N 2l N+2N deg t Θ Θ q q ϑ =O as N if. The solution we need to find is described by a vector x Mat s O depending on N such that Then [2, Lemma 3.3.5] proves our claim A functional equation for E We claim that there exist polynomials x 0, Mx = 0, x =O as N. depending on the parameter N such that max l i=0 a i =O as N a 0,...,a l O[t] and with the following properties for each value of N : Not all the a i vanish identically. a 0 E + a E + +a l E l = 0. For the proof of above claim, we need the following identity: a 0 h 0 + a h Φ 0 + +a l h l Φ l Φ 0 = Since E ν = hψ ν = h ν Φ ν Φ 0 ψ for integer ν > 0, making a right multiplication by ψ on both sides of 2.3 we obtain the functional equation a 0 E + a E + +a l E l = 0. To solve a 0,...,a l satisfying the first two properties of above claim and the identity 2.3, we reduce to solve a system of homogeneous linear equations which is with respect to the evident choice of bases described by a matrix M Mat l l+ O[t] depending on N such that M =O as N andthesolutionwehavetofindisdescribedbyavectorx Mat l+ O[t] depending on N such that x 0, Mx = 0, x =O as N. Then [2, Lemma 3.3.6] proves our claim. After dividing out common factors of t, we may further assume that for each value of N: Not all the constant terms a i 0 vanish.

7 C.-Y. Chang / Journal of Number Theory Vanishing of E We claim that E vanishes identically for some N. Suppose that this is not the case. Let λ be the leading coefficient of the Maclaurin expansion of E and note that a 0 0λ q0 + +a l 0λ q l = 0. Hence by [2, Lemma 3.3.3] Liouville inequality we have Using the Schwarz Jensen formula cf. [2, 2.5], for all N we have or λ = O as N. 2.4 λ N q q sup Ex sup max ψi x x C x C i l h N 2l if >, x x N+μ λ sup Ex x C x sup max ψi x N 2l x C i l h if <, x where μ := μ N := q N + q N+ + +q 2N.Hencewehave λ = O 2l N as N if > 2.5 or N 2l μ λ = O as N if <. 2.6 Inthecaseof =, we pick any α k so that α >. Then using Schwarz Jensen formula again we have and hence λ α N sup Ex sup max ψi x x C x C i l h α N 2l x α x α λ = O α 2l N as N. 2.7 In either case, the bound of 2.5 or 2.6 or 2.7 for λ contradicts to 2.4 as N 0.

8 736 C.-Y. Chang / Journal of Number Theory The case E = 0 Now we fix a value of N such that the auxiliary function E vanishes identically. Since the entries of h are polynomials in t of degree < N and not all vanishing identically, there exists some 0 ν < N such that h N+ν = h q N+ν q N+ν 0. Define P = Pt := h N+ν Θ N+ν Θ Mat l O[t]. We claim that P 0. To prove this claim, we need only show that det Θ N+ν Θ t= 0. Suppose that this is not the case, then we have 0 = det Φ Φ N+ν Θ N+ν Θ t= = det Φ N+ν det Φ t= = [ det Φ N+ν] N+ν [ det Φ ]. This contradicts to our assumption. Thus, P 0. On the other hand, by 2.2 we have Pϑ = hθ 0 Θ N+ν ϑ N+ν t= q N+ν q N+ν = 0, and hence P K-span of ρ Mat l K. Finally, from 2., we see that Pψ = h N+ν Θ N+ν Θ ψ = [ hθ 0 Θ N+ν ψ N+ν] N+ν = [ det Φ N+ν det Φ 0 ] N+ν E = 0. Therefore up to a nonzero correction factor of K, the vector P is the vector we want, and the proof of Theorem.2 is completed.

9 C.-Y. Chang / Journal of Number Theory Proof of Theorem.22 We follow [9] closely to prove Theorem.22. Let Q := k[ψ,..., ψ l ] and S := kt[ψ,..., ψ l ], then as rings, Q = k[x,...,x l ]/a, S = kt[x,...,x l ]/b, for some ideals a and b. HereX,...,X l are l independent variables. For d, let k[x,...,x l ] d and a d be the elements of k[x,...,x l ] and a of total degree d, andletq d Q correspond their quotient. Similarly, we define b d and S d. Fix d, let N = l d+ /l and define ψ Mat N E to be the column vector whose entries are the concatenation of and each column vector ψ n Mat l n E for n =,...,d. Define Φ Mat N k[t] GL N kt to be the diagonal block matrix then we have Φ := [] Φ Φ 2 Φ d, We observe that ψ = Φψ. S d := kt-span in E of the entries of ψ; Q d := k-span in k of the entries of ψ. Using Theorem.2 it can be shown that for all d, dim kt S d = dim k Q d for detailed argument, see [9, Proposition 5..5]. Thus the homogenizations of Q and S have the same Hilbert series and hence 3. Some remarks tr.deg kt kt ψ t,...,ψ l t = tr.deg k k ψ,..., ψ l. Remark 3.. Here we give a counterexample for Theorem.2 if det Φ j = 0 for some positive integer j inthecaseof >. Define where q qi as i.put Φ := t j and Ω := q q i= t qi, 3. is a fixed choice of q th root of. Note that Ω is an entire power series since ψ := t j+ t 0 Ω, then we have ψ = Φψ. Hence Theorem.2 does not hold because ψ = 0 and ψ is transcendental over kt since ψ has infinitely many zeros.

10 738 C.-Y. Chang / Journal of Number Theory Remark 3.2. In this remark, we give examples which assert that if we take F q, then Theorem.2 does not hold. Let F q ν for some ν N and let Ω := Ω θ be defined as in 3.. Note that from the functional equation Ω = t θω we have Ω q ν = θ ν... θ θω, and hence Ω k.sinceω is transcendental over kt, Theorem.22 does not hold. Furthermore, we consider [ ] [ Ω ] = 0. Ω Define [ ] 0 Φ := 0 t θ [ ] and ψ :=, Ω then we have ψ = Φψ. Since Ω is transcendental over kt, it is impossible to find Mat 2 k[t] so that P = [ Ω ] and Pψ = 0. Hence Theorem.2 does not hold. P Remark 3.3. We are interested in Φ Mat l k[t] GL l kt so that there exists Ψ Mat l T GL l L satisfying Ψ = ΦΨ since in this situation we can use the Galois theory of systems of Frobenius difference equations, in particular the equality.. We claim that the condition det Φ0 0 is a necessary condition for the existence of such Ψ. Note that Ψ = ΦΨ implies det Ψ = det Φ det Ψ.IfdetΦ0 = 0, i.e., det Φ is divisible by t, then writing down det Ψ as a formal power series in t and solving its coefficients recursively from the functional equation det Ψ = det Φ det Ψ shows that det Ψ 0. Thus, we complete the proof of the claim. Acknowledgments The author thanks F. Beukers, W.D. Brownawell, L.-C. Hsia, M.A. Papanikolas, D.S. Thakur and J. Yu for many helpful discussions and comments concerning the contents of this paper. He further thanks NCTS for hospitality. References [] Greg W. Anderson, Dinesh S. Thakur, Tensor powers of the Carlitz module and zeta values, Ann. of Math [2] Greg W. Anderson, W. Dale Brownawell, Matthew A. Papanikolas, Determination of the algebraic relations among special Γ -values in positive characteristic, Ann. of Math [3] Frits Beukers, A refined version of the Siegel Shidlovskii theorem, Ann. of Math [4] Chieh-Yu Chang, Matthew A. Papanikolas, Algebraic relations among periods and logarithms of rank 2 Drinfeld modules, preprint. [5] Chieh-Yu Chang, Matthew A. Papanikolas, Dinesh S. Thakur, Jing Yu, Algebraic independence of arithmetic gamma values and Carlitz zeta values, preprint. [6] Chieh-Yu Chang, Matthew A. Papanikolas, Jing Yu, Determination of algebraic relations among special gamma values and zeta values in positive characteristic, preprint. [7] Chieh-Yu Chang, Matthew A. Papanikolas, Jing Yu, Frobenius difference equations and algebraic independence of zeta values in positive equal characteristic, preprint. [8] Chieh-Yu Chang, Jing Yu, Determination of algebraic relations among special zeta values in positive characteristic, Adv. Math [9] Matthew A. Papanikolas, Tannakian duality for Anderson Drinfeld motives and algebraic independence of Carlitz logarithms, Invent. Math