Table Of ContentLinear independence of values of G-functions
7
1 S. Fischler and T. Rivoal
0
2
February 1, 2017
n
a
J
1
3 Abstract
T] Givenanynon-polynomialG-functionF(z) = ∞k=0Akzk ofradiusofconvergence
N R, we consider the G-functions Fn[s](z) = ∞k=0 (PkA+kn)szk for any integers s ≥ 0 and
. n 1. For any fixed algebraic numberα such that 0 < α < R and any numberfield
h ≥ P | |
t K containing α and the Ak’s, we define Φα,S as the K-vector space generated by the
a
[s]
m valuesFn (α),n 1and0 s S. WeprovethatuK,F log(S) dimK(Φα,S) vFS
≥ ≤ ≤ ≤ ≤
for any S, with effective constants u > 0 and v > 0, and that the family
[ K,F F
F[s](α) contains infinitely many irrational numbers. This theorem ap-
1 n 1 n vF,s 0
v plies in pa≤rti≤cular≥when F is an hypergeometric series with rational parameters or a
(cid:0) (cid:1)
1
multiple polylogarithm, and it encompasses a previous result by the second author
5
and Marcovecchio in the case of polylogarithms. The proof relies on an explicit con-
0
9 struction of Pad´e-type approximants. It makes use of results of Andr´e, Chudnovsky
0
andKatzon G-operators, of anewlinear independencecriterion `alaNesterenko over
.
1 number fields, of singularity analysis as well as of the saddle point method.
0
7
1
1 Introduction
:
v
i
X
The class of G-functions was defined by Siegel [33] to generalize the Diophantine properties
r
a of the logarithmic function, by opposition to the exponential function which he generalized
with the class of E-functions. A series F(z) = ∞k=0Akzk ∈ Q[[z]] is a G-function if the
following three conditions are met (we fix an embedding of Q into C):
P
1. There exists C > 0 such that for any σ Gal(Q/Q) and any k 0, σ(A ) Ck+1.
k
∈ ≥ | | ≤
2. Define D as the smallest positive integer such that D A is an algebraic integer for
n n k
any k n. There exists D > 0 such that for any n 0, D Dn+1.
n
≤ ≥ ≤
3. F(z) is a solution of a linear differential equation with coefficients in Q(z).
The first property implies that the radius of convergence of F is positive. In the second
property, the existence of D is enough for the purpose of this paper, but we mention that
a famous conjecture of Bombieri implies that D always divides cn+1db for some integers
n an
a,b 0,c 1, where d := lcm 1,2,...,n = en+o(n) (see [20]). The third property shows
n
≥ ≥ { }
that there is a number field containing all the coefficients A . In the case where they are
k
all rational numbers, the three conditions become A Ck+1, D A Z for k n and
k n k
| | ≤ ∈ ≤
1
D Dn+1, and F(z) is in fact a solution of a linear differential equation with coefficients
n
≤
in Q(z).
G-functions can be either algebraic over Q(z), like
∞ 1 ∞ 2k 2 ∞ 4k 1+√1 6z
zk = , k zk = , zk = − ,
1 z k +1 1+√1 4z 2k √2 12z
k=0 − k=0 (cid:0) (cid:1) − k=0(cid:18) (cid:19) p −
X X X
∞ 3k 2cos 1 arcsin(3√3z) ∞ (30k)!k!
zk = 3 2 , zk, (1.1)
2k √4 27z (15k)!(10k)!(6k)!
k=0(cid:18) (cid:19) (cid:0) − (cid:1) k=0
X X
or transcendental over Q(z), like
∞ zk+1 ∞ 2k 1+√1 4z
= log(1 z), k zk+1 = 1 √1 4z +log − ,
k +1 − − (k +1)2 − − 2
(cid:0) (cid:1)
Xk=0 Xk=0 (cid:16) (cid:17)
∞ 2k ∞ z2k+2 z 2
k z2k+1 = 2arcsin(2z), = 2arcsin . (1.2)
2k +1 (k +1)2 2k+2 2
Xk=0 (cid:0) (cid:1) Xk=0 k+1 (cid:16) (cid:17)
Transcendental G-functions also include the polylogarit(cid:0)hms(cid:1)Lis(z) = ∞k=1 zkks for s ≥ 1.
All the above examples are special cases of the generalized hypergeometric series with
P
rational parameters, which is a G-function:
a ,a ,...,a ∞ (a ) (a ) (a )
F 1 2 p+1;z = 1 k 2 k··· p+1 kzk, (1.3)
p+1 p b ,b ,...,b (1) (b ) (b )
(cid:20) 1 2 p (cid:21) k=0 k 1 k··· p k
X
where (α) = 1 and (α) = α(α + 1) (α + k 1) for k 1; we assume that b
0 k j
··· − ≥ − 6∈
N = 0,1,2,... for any j. Not all G-functions are hypergeometric, for instance the
{ }
algebraic function 1 = ∞ k k k+j zk or the transcendental functions
√1 6z+z2 k=0 j=0 j j
∞k=0 kj=0 kj 2 k+jj−2 zk, 12 logP(1−z)(cid:0)2P= ∞k(cid:0)=1(cid:1)((cid:0)k1 (cid:1)jk(cid:1)=−11 1j)zk, andmoregenerallymultiple
polylogarithms zn1 with s ,s ,...,s Z.
P (cid:0)P (cid:0) (cid:1) (cid:0)n1>·(cid:1)··>(cid:1)nk≥1 ns11ns22···nskk P1 2 P k ∈
In this papePr, we are interested in the Diophantine properties of the values of G-
functions at algebraic points. We first recall that there is no definitive theorem about
the irrationality or transcendance of values of G-functions, like the Siegel-Shidlovsky The-
orem for values of E-functions: transcendental G-functions may take rational values or
algebraic values at some non-zero algebraic points, see [6, 10, 36] for examples related to
Gauss F hypergeometric function. Moreover, very few values of classical G-functions
2 1
are known to be irrational: apart from logarithms of algebraic numbers (proved to be
transcendental by other methods, namely the Hermite-Lindemann theorem), we may cite
Ap´ery’s Theorem [5] that ζ(3) = Li (1) / Q, and the Chudnovsky-Andr´e Theorem [3] on
3
∈
the algebraic independence over Q of the values F [1, 1;1;α] and F [ 1, 1;1;α] for any
2 1 2 2 2 1 −2 2
α Q, 0 < α < 1 (1).
∈ | |
1
This result was first proved by G. Chudnovsky in the 70’s by an indirect method not related to
G-functions, and it was reproved by Andr´e in the 90’s by a method designed for certain G-functions
(simultaneous adelic uniformization), but which has been applied so far only to these 2F1 functions.
2
Up to now, known results on values of G-functions can be divided into two families.
The first one gathers theorems on F(α), where α Q C is sufficiently close to 0 in terms
∈ ⊂
of F (and, often, of other parameters including the degree and height of α). One of the
most general results of this family is the following.
Theorem 1 (Chudnovsky [13, 14]). Let Y(z) = t(F (z),...,F (z)) be a vector of G-
1 S
functions solution of a differential system Y (z) = A(z)Y(z), where A(z) M (Q(z)).
′ S
∈
Assume that 1,F (z), ..., F (z) are Q(z)-algebraically independent. Then for any integer
1 S
d 1, there exists C = C(Y,d) > 0 such that, for any algebraic number α = 0 of degree d
≥ 6
4S
with α < exp( Clog(H(α))4S+1), there does not exist a polynomial relation of degree d
| | −
and coefficients in Q(α) between the values 1,F (α),...,F (α).
1 S
Here, H(α) is the naive height of α, i.e. the maximum of the modulus of the integer
coefficients of the (normalized) minimal polynomial of α over Q. See [1] for a general
strategy recently obtained to prove algebraic independence of G-functions. Chudnovsky’s
theorem refines the works of Bombieri [12] and Galochkin [22]. Andr´e [2] generalized
Chudnovsky’s theorem to the case of an inhomogenous system Y (z) = A(z)Y(z)+B(z).
′
Thus, if we consider the case where α = a/b Q and d = 1, the values 1,F (α),...,F (α)
1 S
∈
are Q-linearly independent provided b (c a )c2 > 0, for some constants c > 0 and
1 1
≥ | |
c > 1 depending on the vector Y. The best value known so far for c is quadratic in
2 2
S; see [21, 38] for related results. When (1,F (z),...,F (z)) = (1,Li (z),...,Li (z)), we
1 S 1 S
refer to [23, 28] for the best linear independence results, where c is “only” linear in S.
2
The second family consists in more recent results where α is a fixed algebraic point in
the disk of convergence: lower bounds are obtained for the dimension of the vector space
generated over a given number field by F(α), where F ranges through a suitable set of
G-functions. In general, this lower bound is not large enough to imply that all these values
F(α) are irrational. In this family, we quote the theorem that infinitely many odd zeta
values ζ(2n + 1) = Li (1), n 1, are irrational (see [7, 30]). Let us also quote the
2n+1
≥
following result, first proved in [31] when α is real.
Theorem 2 (Marcovecchio [25]). Let α Q, 0 < α < 1. The dimension of the Q(α)-
∈ | |
vector space spanned by 1,Li (α),...,Li (α) is larger than 1+o(1) log(S) as S + .
1 S [Q(α):Q]log(2e) → ∞
It seems that all known results in this second family concern only specific G-functions,
essentially polylogarithms. This is not the case of our main result, Theorem 3 below, which
is very general. Starting from a G-function F(z) = ∞k=0Akzk with radius of convergence
R, we define for any integers n 1 and s 0 the G-functions
≥ ≥ P
∞ A
F[s](z) = k zk+n (1.4)
n (k +n)s
k=0
X
which all have R as radius of convergence.
Let K be a number field that contains all the Taylor coefficients A of F. For any
k
integer S and any α K such that 0 < α < R, let Φ denote the K-vector space
α,S
∈ | |
3
[s]
spanned by the numbers F (α) for n 1 and 0 s S; of course Φ depends also
n α,S
≥ ≤ ≤
implicitly on F and K. We shall obtain lower and upper bounds on dim (Φ ) but to
K α,S
state them precisely, we need to introduce some notations.
We consider a differential operator L = µ P (z)( d )j Q[z, d ] such that LF(z) = 0
j=0 j dz ∈ dz
and L is of minimal order for F; then L is a G-operator and in particular it is fuchsian by a
P
result ofChudnovsky [13,14]. Wedenoteby δ thedegreeofLandbyω 0themultiplicity
≥
of 0 as a singularity of L, i.e. the order of vanishing of P at 0. We have δ = deg(P )
µ µ
because is a regular singularity of L. We let ℓ = δ ω, and ℓ = max(ℓ,f ,...,f )
0 1 η
∞ −
where f , ..., f are the integer exponents of L at (so that ℓ = ℓ if no exponent at
1 η 0
∞ ∞
is an integer). We refer to [24] for the definitions and properties of these classicabl notionbs,
and tob[4, 3] fobr those of G-operators.
§
Theorem 3. If F is not a polynomial, then there exists an effective constant C(F) > 0
such that for any α K, 0 < α < R, we have
∈ | |
1+o(1)
log(S) dim (Φ ) ℓ S +µ. (1.5)
[K : Q]C(F) ≤ K α,S ≤ 0
The second inequality holds for all S 0 while in the first one, o(1) is for S + .
≥ → ∞
The upper bound in (1.5) depends only on F. The constant C(F) is independent from
the number field K, which is assumed to contain α and all the Taylor coefficients A of F;
k
its expression involves certain quantities introduced in Proposition 1 in 5.1.
§
We have the following corollary, in a case where ℓ = 1. The proof is given in 2,
0
§
together with many examples and other applications of Theorem 3.
Corollary 1. Let us fix some rational numbers a ,...,a and b ,...,b such that a
1 p+1 1 p i
6∈
Z 1 and b N for any i, j. Then for any α Q such that 0 < α < 1, infinitely
j
\ { } 6∈ − ∈ | |
many of the hypergeometric values
∞ (a ) (a ) (a ) αk
1 k 2 k p+1 k
··· , s 0 (1.6)
(1) (b ) (b ) (k +1)s ≥
k 1 k p k
k=0 ···
X
are linearly independent over Q(α).
The numbers in (1.6) are hypergeometric because they are equal to
a ,a ,...,a ,1,...,1
F 1 2 p+1 ;α
p+s+1 p+s b ,b ,...,b ,2,...,2
1 2 p
(cid:20) (cid:21)
where 1 and2 are bothrepeated s times. It seems to be the first general Diophantine result
of this type for values of hypergeometric functions. Of course the conclusion of Corollary 1
can be stated more precisely as
∞ (a ) (a ) (a ) αk 1+o(1)
1 k 2 k p+1 k
dim Span ··· , 0 s S log(S),
Q(α) Q(α) (1) (b ) (b ) (k +1)s ≤ ≤ ≥ [Q(α) : Q]C
k 1 k p k
nXk=0 ··· o
4
where C > 0 depends on a ,...,a and b ,...,b . The special case p = 0, a = 1
1 p+1 1 p 1
corresponds to Theorem 2 stated above, except that log(2e) is replaced with C. An ad hoc
analysis in this special case would give C = log(2e), thereby providing Theorem 2 again
(with a new proof, see below).
The strategy to prove Theorem 3 is as follows. First, we construct certain algebraic
numbers κ K and polynomials K (z) K[z] such that for any s,n 1:
j,t,s,n j,s,n
∈ ∈ ≥
s ℓ0 µ−1 d j
F[s](z) = κ F[t](z)+ K (z) z F(z), (1.7)
n j,t,s,n j j,s,n dz
Xt=1 Xj=1 Xj=0 (cid:16) (cid:17)
withgeometricboundsondenominatorsandmoduliofGaloisconjugates(seeProposition1
in 5.1 for a precise statement). Eq. (1.7) is a far reaching generalization of a property
§
trivially satisfied by polylogarithms: for any n 1,
≥
∞ zk+n n−1 zk
= Li (z) .
(k +n)s s − ks
k=0 k=1
X X
Toobtainthisresult westudylinearrecurrences associatedwithG-operators, andmakeuse
in a crucial way of the results of Andr´e, Chudnovsky and Katz [4, 19]. With z = α, (1.7)
proves the inequality on the right-hand side of (1.5). This part of the proof of Theorem 3
uses only methods with an algebraic flavor.
To prove the inequality on the left-hand side of (1.5), we use methods with a more
Diophantine flavor. We consider the series
∞ k(k 1) (k rn+1)
T (z) = n!S r − ··· − A z k
S,r,n − (k +1)S(k +2)S (k +n+1)S k −
k=0 ···
X
where z > 1/R, r and n are integer parameters such that r S and n + . If A = 1
k
| | ≤ → ∞
for any k, this is essentially the series used in [31] and [25] to prove Theorem 2. Using
(1.7) again, we prove that T (1/α) is a K-linear combination of the numbers F[t](α)
S,r,n j
(1 t S, 1 j ℓ ) and (z d )jF(α) (0 j µ 1). In fact, the series T (z) can
≤ ≤ ≤ ≤ 0 dz ≤ ≤ − S,r,n
[t]
be interpreted has an explicit Pad´e-type approximant at z = for the functions F (1/z)
∞ j
and (z d )jF(1/z).
dz
We apply singularity analysis and the saddle point method to prove that
Q
T (1/α) = annκlog(n)λ c ζn +o(1) as n , (1.8)
S,r,n q q → ∞
(cid:16)Xq=1 (cid:17)
for some integers Q 1 and λ 0, real numbers a > 0 and κ, non-zero complex numbers
≥ ≥
c ,..., c and pairwise distinct complex numbers ζ , ..., ζ such that ζ = 1 for any q.
1 Q 1 Q q
| |
These parameters are effectively computed in terms of the finite singularities of F.
5
To conclude the proof we apply a linear independence criterion, as for all results of the
second family mentioned above. Such a criterion enables one to deduce a lower bound on
the dimension of the K-vector space spanned by complex numbers ϑ , ..., ϑ from the
1 J
existence of linear forms T = J p ϑ with coefficients p . This lower bound
n j=1 j,n j j,n ∈ OK
is non-trivial if T is very small, and p is not too large. However one more assumption
n j,n
| | P
is needed. In Siegel-type criteria this assumption is the non-vanishing of a determinant;
(k)
Theorem 2 is proved in this way in [25], by constructing several sequences (T ). On the
n
opposite, Nesterenko’scriterion[26](anditsgeneralizations[35,9]tonumberfields)enables
onetoconstruct onlyonesequence (T ), butitrequiresalowerboundon T 1/n; thisishow
n n
| |
Theorem 2 is proved in [31] if α is real. If liminf T 1/n is smaller than limsup T 1/n,
n| n| n| n|
this lower bound is weaker. In fact, in our situation, namely with the asymptotics (1.8),
it is not even clear that liminf T 1/n is positive so that these criteria do not apply. We
n n
| |
solve this problem by generalizing Nesterenko’s criterion (over any number field) to linear
forms (T ) with asymptotics given by (1.8); our lower bound is best possible (see 3 for
n
§
precise statements). In the special case of polylogarithms, this provides a new proof of
Theorem 2 when α is not real.
The structure of this paper is as follows. In 2 we deduce Corollary 1 from Theorem
§
3, and give applications of these results. In 3 we state and prove the generalization of
§
Nesterenko’s linear independence criterion to linear forms with asymptotics given by (1.8).
Then in 4 we prove a general result, of independent interest, on linear recurrences related
§
to G-operators (using in a crucial way the Andr´e-Chudnovsky-Katz theorem). This result
allows ustoprove (1.7)in 5, withgeometric boundsondenominatorsandmoduli ofGalois
§
conjugates. We conclude the proof of Theorem 3 in 6, except for the asymptotic estimate
§
(1.8) that we obtain in 7 using singularity analysis and the saddle point method. At last,
§
we mention in 8 how to simplify the proof in the special case where A 0 for any k, and
k
§ ≥
α > 0.
2 Examples
The generalized hypergeometric series defined by (1.3), if b N for any j, is solution of
j
6∈ −
the differential equation L y(z) = 0 where
h
d
L = θ(θ+b 1) (θ+b 1) z(θ +a ) (θ+a ), θ = z .
h 1 p 1 p+1
− ··· − − ··· dz
It is a G-function if and only if the a ’s and b ’s are rational numbers, in which case L is
j j h
a G-operator. Assuming a N, it is not a polynomial. We now compute the quantities
i
6∈ −
defined before Theorem 3, especially ℓ . The degree δ of L is p+2 and the multiplicity ω
0 h
of 0 as a singularity of L is p+1. Hence, ℓ = δ ω = 1 (consistently with the expression
h
−
of L and Lemma 1 below). Moreover, the exponents of L at 0 are 0,1 b ,...,1 b ,
h h 1 p
− −
while those at are a ,...,a , so that ℓ = max(1,a ,...,a ) where the a are the
1 p+1 0 1 η j
∞
integer parameters amongst a , ..., a . If none of the a ’s is an integer greater than 1
1 p+1 j
then ℓ = 1. This proves Corollary 1.
0 b b b
6
We now list the hypergeometric parameters of the examples stated in the Introduction:
1 1,1 2k 1,1 3k 1, 2
k 2 3 3
k +1 ←→ 2 k +1 ←→ 2 2k ←→ 1
(cid:20) (cid:21) (cid:0) (cid:1) (cid:20) (cid:21) (cid:18) (cid:19) (cid:20) 2 (cid:21)
4k 1, 3 2k 1,1,1 1 1,1,1
4 4 k 2
2k ←→ 1 (k +1)2 ←→ 2,2 (k +1)2 2k+2 ←→ 3,2
(cid:18) (cid:19) (cid:20) 2 (cid:21) (cid:0) (cid:1) (cid:20) (cid:21) k+1 (cid:20) 2 (cid:21)
2k 1, 1 (30k)!k! 1 , 7 , 11(cid:0), 13, 1(cid:1)7, 19, 23, 29
k 2 2 30 30 30 30 30 30 30 30 .
2k +1 ←→ 3 (15k)!(10k)!(6k)! ←→ 1, 1, 2, 1, 3, 2, 4
(cid:0) (cid:1) (cid:20) 2 (cid:21) (cid:20) 5 3 5 2 5 3 5 (cid:21)
In these eight cases, we have ℓ = 1 so that Corollary 1 applies (separately) to them.
0
Let us now compute ℓ for four non-hypergeometric examples. The function 1 =
0 √1 6z+z2
∞ ( k k k+j )zk is solution of the differential equation −
k=0 j=0 j j
P P (cid:0) (cid:1)(cid:0) (cid:1)
(z2 6z +1)y (z)+(z 3)y(z) = 0
′
− −
which is minimal forthis function; its exponent at is1. Hence ℓ = ℓ = 2 and Theorem 3
0
∞
provides 1+o(1) log(S) K-linearly independent numbers amongst the numbers
[K:Q]C
∞ k k k +j αk ∞ k k k +j αk
and , 0 s S.
j j (k +1)s j j (k +2)s ≤ ≤
Xk=0(cid:16)Xj=0 (cid:18) (cid:19)(cid:18) (cid:19)(cid:17) Xk=0(cid:16)Xj=0 (cid:18) (cid:19)(cid:18) (cid:19)(cid:17)
The function 1 log(1 z)2 = ∞ ( 1 k 1)zk+1 is solution of the differential equa-
2 − k=0 k+1 j=1 j
tion
P P
(z 1)2y (z)+3(z 1)y (z)+y (z) = 0
′′′ ′′ ′
− −
which is minimal for this function; its exponents at are 0,0,0. Hence ℓ = ℓ = 2 and
0
∞
Theorem 3 applies in the same way to the numbers
∞ k−1 1 αk ∞ k−1 1 αk
and , s 0.
j ks+1 j k(k +1)s ≥
Xk=1(cid:16)Xj=1 (cid:17) Xk=1(cid:16)Xj=1 (cid:17)
The generating function of the Ap´ery numbers ∞ ( k k 2 k+j 2)zk is solution of
k=0 j=0 j j
the minimal differential equation
P P (cid:0) (cid:1) (cid:0) (cid:1)
z2(1 34z +z2)y (z)+z(3 153z +6z2)y (z)+(1 112z+7z2)y (z)+(z 5)y(z) = 0.
′′′ ′′ ′
− − − −
Its exponents at are 1,1,1. Hence ℓ = ℓ = 2 and Theorem 3 applies again to the
0
∞
numbers
∞ k k 2 k +j 2 αk ∞ k k 2 k +j 2 αk
and , s 0.
j j (k +1)s j j (k +2)s ≥
Xk=0(cid:16)Xj=0 (cid:18) (cid:19) (cid:18) (cid:19) (cid:17) Xk=0(cid:16)Xj=0 (cid:18) (cid:19) (cid:18) (cid:19) (cid:17)
7
We conclude this section with the case of the series Gb(z) = ∞k=1 χk(kb)zk where b
is any fixed positive integer and χ is the unique non-principal character mod 4. Since
Gb(z) = ∞k=0 (2(−k+1)1k)bz2k+1, it is a G-function. Moreover, θ (1+z2)θbPGb(z) = 0, which is
of minimal order for G (z). Hence θ (1+z2)θb is a G-operator: it is such that µ = b+1,
P b (cid:0) (cid:1)
δ = b+3, ω = b+2, ℓ = 1 and its exponents at infinity are 0,0,...,0,2, where 0 is repeated
(cid:0) (cid:1)
b times. Hence ℓ = 2 and Theorem 3 applies to the numbers
0
∞ χ(k) ∞ χ(k)
αk and αk, s 0.
kb+s kb(k +1)s ≥
k=1 k=1
X X
More generally, Theorem 3 applies to any G-function of the form ∞ χ(k)zk where χ is
k=1 A(k)
a Dirichlet character and A(X) Q[X] is split over Q and such that A(k) = 0 for any
∈ P 6
positive integer k.
3 Generalization of Nesterenko’s linear independence
criterion
Thefollowing version ofNesterenko’s linear independence criterionwill beusedintheproof
of Theorem 3.
Let K be a number field embedded in C. We let L = R if K R, and L = C otherwise.
⊂
We denote by o(1) any sequence that tends to 0 as n .
→ ∞
Theorem 4. Let (Q ) be an increasing sequence of positive real numbers, with limit + ,
n
∞
1+o(1)
such that Q = Q . Let T 1, c ,..., c be non-zero complex numbers, and ζ , ...,
n+1 n 1 T 1
≥
ζ be pairwise distinct complex numbers such that ζ = 1 for any t.
T t
| |
Consider N numbers ϑ ,...,ϑ L. Assume that for some τ > 0 there exist N se-
1 N
∈
quences (p ) , j = 1,...,N, such that for any j and n, p , all Galois conjugates
j,n n 0 j,n K
≥ 1+o(1) ∈ O
of p have modulus less than Q , and
j,n n
N T
p ϑ = Q τ+o(1) c ζn +o(1) . (3.1)
j,n j n− t t
Xj=1 (cid:16)Xt=1 (cid:17)
Then
τ +1
dim Span (ϑ ,...,ϑ ) .
K K 1 N ≥ [K : Q]
Given 0 < α < 1 < β and κ C, λ C, this theorem can be applied when all Galois
∈ ∈
conjugates of p have modulus less than βn(1+o(1)) and
j,n
N T
p ϑ = αnnκ(logn)λ c ζn +o(1) ; (3.2)
j,n j t t
Xj=1 (cid:16)Xt=1 (cid:17)
8
then the conclusion reads
1 log(α)
dim Span (ϑ ,...,ϑ ) 1 .
K K 1 N ≥ [K : Q] − log(β)
(cid:16) (cid:17)
Nesterenko’s original linear independence criterion [26] is a general quantitative result,
of which Theorem 4 is a special case if K = Q, T = 1, ζ = 1. The case where K = Q,
1
±
T = 2, ζ = ζ and c = c follows using either lower bounds for linear forms in logarithms
2 1 2 1
(if c ,ζ Q, see [34] or [17, 2.2]) or Kronecker-Weyl’s equidistribution theorem [17].
1 1
∈ §
Nesterenko’s criterion has been extended to any number field K by To¨pfer [35] and
Bedulev [9]; their results are similar, but different in several aspects. The case T = 1 of
Theorem 4 follows from T¨opfer’s Korollar 2 [35], but does not seem to follow directly from
Bedulev’s result since he uses the exponential Weil height relative to K instead of the house
of p (i.e., the maximum of the moduli of all Galois conjugates of p ).
j,n j,n
We shall deduce the general case of Theorem 4 from T¨opfer’s result using Vandermonde
determinants (as in the proof of [19, Lemma 6]). This provides also a new and simpler
proof of the above-mentioned case K = Q, T = 2, ζ = ζ and c = c .
2 1 2 1
Even in the special case where T = 1 and K = Q, the lower bound in Theorem 4 is best
possible (see [18]). We have the following corollary, which we shall not use in this paper
but which can be useful in other contexts.
Corollary 2. Let α,β R be such that 0 < α < 1 < β. Consider N numbers ϑ ,...,ϑ
1 N
∈ ∈
L. Assume that there exist N sequences (p ) , j = 1,...,N, such that for any j and
j,n n 0
n, p , all Galois conjugates of p have m≥odulus less than βn(1+o(1)), and
j,n K j,n
∈ O
N
1/n
limsup p ϑ α.
j,n j
≤
n
→∞ (cid:12)Xj=1 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Assume also that N p ϑ = 0 for infinitely many n, and that for any j the function
j=1 j,n j 6
∞n=0pj,nzn is soluPtion of a homogeneous linear differential equation with coefficients in
Q(z). Then
P
1 log(α)
dim Span (ϑ ,...,ϑ ) 1 .
K K 1 N ≥ [K : Q] − log(β)
(cid:16) (cid:17)
The point in Corollary 2 is that no lower bound is needed on N p ϑ . This re-
| j=1 j,n j|
sult fits in the context of G-functions, since its assumptions implyPthat ∞n=0pj,nzn is a
G-function for any j. To deduce Corollary 2 from Theorem 4, it is enough to notice that
∞n=0 Nj=1pj,nϑjzn = Nj=1ϑj ∞n=0pj,nzn is solution of a homogeneousPlinear differen-
tial equation with coefficients in Q(z). We can then apply classical transfer results from
P P P P
Singularity Analysis: an asymptotic estimate like (3.2) holds.
9
Proof of Theorem 4. : For any n 0 we consider the following determinant:
≥
ζn ... ζn
1 T
. .
∆n = (cid:12) .. .. (cid:12).
(cid:12) (cid:12)
(cid:12)(cid:12)ζ1n+T−1 ... ζTn+T−1(cid:12)(cid:12)
(cid:12) (cid:12)
We have ∆ = ζn...ζn∆ = ∆ (cid:12)= 0 since ∆ is the V(cid:12) andermonde determinant built
| n| | 1 T 0| | 0|(cid:12)6 0 (cid:12)
on the pairwise distinct complex numbers ζ , ..., ζ . We claim that for any n 0 there
1 T
≥
exists δ 0,...,T 1 such that
n
∈ { − }
T
c ∆
c ζn+δn | 1 0|. (3.3)
t t ≥ T!
(cid:12)Xt=1 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Indeedifthisequationholdsfornointegerδ 0,...,T 1 thenuponreplacingC with
n 1,n
∈ { − }
1 T c C (where C is the t-th column of the matrix of which ∆ is the determinant)
c1 t=1 t t,n t,n n
we obtain:
P T c ∆
1 0
∆ = ∆ < | |(T 1)! = ∆ ,
0 n 0
| | | | c T! − | |
1
since all minors of size T 1 have modulus less thanor equal to (T 1)!. This contradiction
− −
proves the claim (3.3) for some δ 0,...,T 1 .
n
∈ { − }
1+o(1)
Now let p = p . Since Q = Q and 0 δ T 1 (where T 1 does
′j,n j,n+δn n+1 n ≤ n ≤ − −
1+o(1)
not depend on n), all Galois conjugates of p have modulus less than Q . Moreover
j,n n
(3.3) yields | Nj=1p′j,nϑj| = Qn−τ+o(1). Therefore T¨opfer’s Korollar 2 [35] applies to the
sequences (p ): this concludes the proof of Theorem 4.
′j,n
P
Remark 1. In the proof of Theorem 4 the sequences (p ) may be such that p = p
′j,n ′j,n ′j,n′
for some n < n even if this does not happen with p . This is not a problem since in this
′ j,n
case n n T 1, where T is independent from n.
′
− ≤ −
4 Linear recurrences associated with G-operators
In this section we apply some results of Andr´e, Chudnovsky and Katz to prove a few
general properties of G-operators (stated in 4.1). We recall that for any G-function F,
§
any differential operator L Q[z, d ] of minimal order such that LF = 0 is a G-operator.
∈ dz
We refer to [4, 3] for the definition and properties of G-operators.
§
4.1 Setting and statements
Lemma 1. Let K be a number field, and L = µ P (z) d j a G-operator with P
j=0 j dz j ∈
K[X] and P = 0; denote by δ the degree of L, and by ω 0 the multiplicity of 0 as a
µ
6 P ≥(cid:0) (cid:1)
singularity of L; let ℓ = δ ω.
−
10
