Journal de Théorie des Nombres de Bordeaux

Masanari Kida

On metacyclic extensions

Dong Quan Ngoc Nguyen

The arithmetic of certain del Pezzo surfaces and K3 surfaces

Peter J. Cho

Henry H. Kim

Dihedral and cyclic extensions with large class numbers

Aaron Levin

Siegel’s theorem and the Shafarevich conjecture

Andrew V. Sutherland

A local-global principle for rational isogenies of prime degree

Yuichi Kamiya

Leo Murata

Relations among arithmetical functions, automatic sequences, and sum of digits functions induced by certain Gray codes

Alena Pirutka

$R$-équivalence sur les familles de variétés rationnelles et méthode de la descente

Mladen Dimitrov

Eknath Ghate

On classical weight one forms in Hida families

Evgeniy Zorin

Evgeniy ZORIN
Multiplicity estimate for solutions of extended Ramanujan’s system
Tome 24, no 3 (2012), p. 773-781.
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Journal de Théorie des Nombres
de Bordeaux 24 (2012), 773-781
Multiplicity estimate for solutions of extended
Ramanujan’s system
par Evgeniy ZORIN
Résumé. Nous établissons un nouveau lemme de multiplicité
pour les solutions d’un système différentiel généralisant les rela-
tions différentielles classiques de Ramanujan. Ce résultat peut être
utile pour l’étude des propriétés arithmétiques des valeurs de la
fonction zêta de Riemann aux entiers positifs impairs (Nesterenko,
2011).
Abstract. We establish a new multiplicity lemma for solutions
of a differential system extending Ramanujan’s classical differen-
tial relations. This result can be useful in the study of arithmetic
properties of values of Riemann zeta function at odd positive in-
tegers (Nesterenko, 2011).
1. Introduction
In what follows we denote by σk(n), k ∈ Z, n ∈ N the sum of kth powers
of divisors of n:
σk(n) :=
∑
d|n
dk.
In this paper we consider the following sets of functions. First of all, Fourier
series of the Eisenstein functions
(1.1) E2k(z) := 1−
4k
B2k
∞∑
n=1
σ2k−1(n)zn, k ∈ N,
where B2k are Bernoulli numbers. Also we consider
(1.2) gu,v(z) :=
∞∑
n=1
nuσ−v(n)zn, u, v ∈ N, 0 ≤ u < v, v is odd.
We recall the following well-known fact: functions E2, E4 and E6 are
algebraically independent over C(z) and all the other functions E2k, k ≥ 4
belong to Q[E4, E6] (see for instance [8]). More precisely, for all k ≥ 4 there
exists a polynomial Ak ∈ C[X,Y ] such that
E2k(z) = Ak(E4(z), E6(z))
Manuscrit reçu le 13 septembre 2011.
Classification math. 11J81, 11J82, 11J61.
774 Evgeniy Zorin
and these polynomials Ak(X,Y ), k ≥ 4 contain only monomials M of bi-
degrees (degXM, degY M) satisfying 2 degXM + 3 degY M = k.
In [2] Yu.Nesterenko proved the following theorem:
Theorem 1.1 (Nesterenko). For any complex q, 0 < |q| < 1, the set
q, E2(q), E4(q), E6(q)
contains at least three algebraically independent over Q numbers.
In particular, this result allowed him to establish the algebraic indepen-
dence of π, eπ and Γ
(
1
4
)
. Indeed, using modular properties of Eisenstein
series one can easily calculate
E2(e−2π) =
3
π
, E4(e−2π) = 3
Γ(14)
8
(2π)6 , E6(e
−2π) = 0
(see Chapter 1 of [4]). Hence applying Theorem 1.1 with q = e−2π one
obtains algebraic independence of π, eπ and Γ
(
1
4
)
.
Later Yu.Nesterenko [3] showed that considering a bigger set of functions,
notably introducing functions (1.2), we can capture other extremely inter-
esting cases, for instance one has ζ(4n + 3) ∈ Q(E2(e−2π), g0,4n+3(e−2π))
(n ∈ N). So extending Theorem 1.1 for a set of functions like
E2, E4, E6, gu,v, 0 ≤ u < v, v ≡ 3 mod 4
one could potentially illuminate the question of transcendence or irrational-
ity of ζ at odd positive points. See [3] for some more discussion.
An important stage in the proof of Theorem 1.1 is the following multi-
plicity lemma:
Theorem 1.2 (Nesterenko). For any polynomial A(z,X1, X2, X3) ∈
C[z,X1, X2, X3], A 6= 0, the inequality holds
ordz=0A(z, P (z), Q(z), R(z)) < c(degz A+ 1)(degX A+ 1)3,
where c is an absolute constant.
We invite the interested reader to consult Chapters 3 and 10 in [4] as
well as [6, 7, 9] concerning the use of multiplicity estimates in algebraic
independence proofs.
In this paper we adopt the method from [2] and [4, Chapter 10] to es-
tablish (for any fixed odd m ≥ 3) a multiplicity lemma for the whole set of
functions
(1.3) E2(z), E4(z), E6(z), gu,v(z), v is odd, 0 ≤ u < v ≤ m,
see Theorem 2.1 below. Note that algebraic independence over C(z) of the
set of functions (1.3) was already established in 2010 by P. Kozlov ([1],
Multiplicity estimate 775
for a statement see also [3], page 2), we use this result in the proof of
Theorem 2.1.
Proofs of Theorems 1.1 and 1.2 are based on the fact that functions E2,
E4, E6 satisfy the following system of differential equations:
(1.4)
δE2 =
1
12
(
E22 − E4
)
, δE4 =
1
3 (E2E4 − E6) , δE6 =
1
2
(
E2E6 − E24
)
,
where δ := z ddz .
The set of functions (1.3) satisfy the following extended system of differ-
ential equations [3]. We keep the three equations (1.4) and for any odd v ≥ 3
(1.5)
δgu,v(z) = gu+1,v(z), 0 ≤ u < v − 1,
δgv−1,v(z) = Bv+1
Av+1(E4(z), E6(z))− 1
2v + 2 ;
in the case v = 1 we have the equation
(1.6) δg0,1(z) =
1
24 (1− E2(z)) .
2. Multiplicity Lemma
Let m ∈ N be a fixed positive odd integer. We introduce the following
notation:
(2.1) R := C[z,X1, X2, X3, Y0,1, Y0,3, Y1,3, Y2,3, . . . , Ym−1,m].
Theorem 2.1. Let m ≥ 1 be an odd integer. There exists a constant C
depending on m only such that for all non-zero P ∈ R
ordz=0P (z, E2(z), E4(z), E6(z),g0,1(z), g0,3(z), . . . , g0,m(z), . . . , gm−1,m(z))
≤ C (degz P + 1)
(
degX,Y P + 1
)( m+12 )2+3 ,(2.2)
where degX,Y P denotes the total degree of P in the variables X1, X2, X3,
Y0,1, . . . , Ym−1,m, i.e. all the variables appearing in the definition (2.1) of
R but z.
Remark 2.1. The exponent
(
m+1
2
)2
+ 3 in the r.h.s. of (2.2) equals the
number of functions different than z in the l.h.s. of (2.2) (and also the
transcendence degree of R over C(z)). It is an easy exercise from linear
algebra that for any given set f1, . . . , fl of functions analytic at z = 0 and
any N ∈ N, N ≥ 1, one can construct a polynomial PN,f ∈ C[z,X1, . . . , Xl]
of degree degP ≥ N and satisfying
ordz=0PN,f (z, f1(z), . . . , fl(z)) ≥ C0
(
degz PN,f + 1
) (
degX PN,f + 1
)l
,
where C0 = 1(l+1)! .
776 Evgeniy Zorin
Hence Theorem 2.1 provides multiplicity estimate for the set of func-
tions (1.3) with the optimal exponent.
In the sequel we denote
D0 := z
d
dz
+ 112
(
X21 −X2
) d
dX1
+ 13 (X1X2 −X3)
d
dX2
+ 12
(
X1X3 −X22
) d
dX3
,
D1 :=
1
24 (1−X1)
d
dY0,1
,
Dv :=
v−2∑
k=0
Yk+1,v
d
dYk,v
+Bv+1
Av+1(X2, X3)− 1
2v + 2
d
dYv−1,v
,
v = 3, 5, . . . ,m, and
D := D0 +
(m−1)/2∑
k=0
D2k+1.(2.3)
The differential operator D satisfies
(2.4) DP (z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z))
= z d
dz
P (z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z)).
We deduce Theorem 2.1 using Nesterenko’s conditional Multiplicity Lem-
ma (see Theorem 2.2 below). This result deals with differential system of
the following type:
(2.5) f ′i(z) =
Ai(z, f)
A0(z, f)
, i = 1, . . . , n,
where Ai(z,X1, . . . , Xn) ∈ C[z,X1, . . . , Xn] for i = 0, ..., n (we suppose that
A0 is a non-zero polynomial).
Remark 2.2. It is easy to see that the system (1.4)∪(1.5)∪(1.6) is of the
type (2.5).
One associates to the system (2.5) the differential operator
(2.6) DA = A0(z,X1, . . . , Xn)
∂
∂z
+
n∑
i=1
Ai(z,X1, . . . , Xn)
∂
∂Xi
.
In our case (i.e. the case of the system (2.5)) this formula gives exactly the
differential operator D as defined in (2.3).
Multiplicity estimate 777
Theorem 2.2 (Nesterenko, see Theorem 1.1, Chapter 10 [4]). Suppose that
functions
f = (f1(z), . . . , fn(z)) ∈ C[[z]]n
are analytic at the point z = 0 and form a solution of the system (2.5).
If there exists a constant K0 such that every DA-stable prime ideal P ⊂
C[X ′1, X1, . . . , Xn], P 6= (0), satisfies
(2.7) min
P∈P
ordz=0P (z, f) ≤ K0,
then there exists a constant K1 > 0 such that for any polynomial P ∈
C[X ′1, X1, . . . , Xn], P 6= 0, the following inequality holds
(2.8) ordz=0(P (z, f)) ≤ K1(degX′ P + 1)(degX P + 1)n.
To deduce Theorem 2.1 from Theorem 2.2 it is sufficient to prove Propo-
sition 2.1 here below.
Proposition 2.1. If P is a non-zero prime ideal of
R = C[z,X1, X2, X3, Y0,1, . . . , Ym−1,m]
with DP ⊂ P, then either z ∈ P or ∆ = X32 −X23 ∈ P.
Proof of Theorem 2.1 modulo Proposition 2.1. If we have the result an-
nounced in Proposition 2.1, then any prime D-stable ideal P contains the
polynomial
(2.9) Θ := z∆ = z
(
X32 −X23
)
.
In this case we have obviously
(2.10) min
P∈P
ordz=0P (z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z))
≤ ordz=0Θ(z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z)) = 2,
as Θ(z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z)) = z∆(E2(z), E4(z)) by
definition (2.9) and ordz=0∆(E2(z), E4(z)) = 1. The upper bound (2.10)
assures the hypothesis (2.7) of Theorem 2.2, and applying this theorem we
obtain the upper bound (2.8), that is (2.2) in our case. 
To prove Proposition 2.1, we describe at first principal D-stable ideals
of R.
Lemma 2.1. There exist only two non-zero D-invariant principal prime
ideals of R, namely, the ideals generated by z and ∆.
Proof. Suppose that A ∈ R is any irreducible polynomial with the property
that A|DA. Thus
(2.11) DA = AB, B ∈ R.
778 Evgeniy Zorin
We readily verify with the definition of D (see (2.3)) that degY DA ≤
degY A and degzDA ≤ degz A, hence (2.11) implies degY B = degz B = 0,
i.e. B ∈ C[X1, X2, X3].
For any F ∈ R we define the weight of F as φ : R→ N,
φ(F ) := degt F (z, tX1, t2X2, t3X3, tm+1Y ).
Then φ satisfies the following properties:
(1) For any F ∈ R
φ(DF ) ≤ φ(F ) + 1.
(2) For any F,G ∈ R
φ(FG) = φ(F ) + φ(G).
These properties together with (2.11) imply
φ(A) + φ(B) = φ(DA) ≤ φ(A) + 1,
hence φ(B) ≤ 1. Thus B ∈ C[X1] and degB ≤ 1, i.e. B = aX1 +b, a, b ∈ C.
So (2.11) in fact means
(2.12) DA = (aX1 + b)A,
where a, b ∈ C.
We claim that b ∈ N (and later we prove also that a ∈ Z). To prove this
we consider one more weight φ2 : R → Z (in fact we shall use this weight
only for monomials from R). For any F ∈ R, we denote
φ2(F ) := degt F (z, tX1, t2X2, t3X3, t−1Y0,1, . . . , t−mY0,m,
t−m+1Y1,m, . . . , t
−1Ym−1,m)
(i.e. we assign to the variable Yu,v the weight φ2(Yu,v) := u− v).
Let M ∈ R be a monomial. We readily verify that all the monomials of
Av+1(X2, X3)− 1
2v + 2
d
dYv−1,v
M
have the weight φ2 strictly bigger than φ2(M). The same is true for
1
24 (1−X1)
d
dY0,1
M,
as well as for
Yk+1,v
d
dYk,v
M (0 ≤ k < v − 1)
and( 1
12
(
X21 −X2
) d
dX1
+ 13 (X1X2 −X3)
d
dX2
+ 12
(
X1X3 −X22
) d
dX3
)
M.
The monomial z ddzM either is 0 or satisfy φ2(z
d
dzM) = φ2(M).
Multiplicity estimate 779
Thus remembering the definition of D, see (2.3), we readily verify that
every non-zero monomial appearing in DM has the weight φ2 not less than
φ2(M) and the only monomial in DM that eventually has the weight φ2
equal to φ2(M) is z ddzM .
Let C be the sum of monomials of A with minimal weight φ2. Then in
the r.h.s. of (2.12) the sum of monomials of minimal weight is equal to bC
(all the other monomials of (aX1 + b)A obviously have a bigger value of
weight φ2).
Applying to the l.h.s. of (2.12) (that is DA) the above considerations we
conclude
(2.13) z d
dz
C = bC.
Comparing the coefficients on the both sides of (2.13) we find b = degz C,
in particular b ∈ N.
We proceed to the proof a ∈ Z. Substituting X1 = E2(z), X2 = E4(z),
X3 = E6(z), Yu,v = gu,v(z), 0 ≤ u < v ≤ m in (2.12) we obtain
(2.14) (aE2(z) + b)A (z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z))
= DA (z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z)) .
We consider the Taylor expansion (at the point z = 0) of
(2.15) A (z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z)) .
It is a function analytic at z = 0 because all the functions E2, E4, E6, g0,1,
. . . , gm−1,m are. Let M denote the order at z = 0 of the function (2.15),
that is
A (z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z))
= czM + (terms of order > M),
where c 6= 0. Using (2.4) we obtain
DA (z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z))
= cMzM + (terms of order > M).
In view of (1.1) we have E2(z) = 1+terms of order > 1. We readily deduce
from (2.14)
(a+ b)czM + (terms of order > M) = cMzM + (terms of order > M).
Comparing coefficients with zM on both sides in this expression (and sim-
plifying out c) we readily deduce a + b = M . We have already established
b ∈ N. Obviously, M ∈ N (as it is a degree in a Taylor series). We conclude
a ∈ Z.
So we have established that coefficients a, b involved in (2.12) are in fact
integers.
780 Evgeniy Zorin
Note that
(2.16) D(∆−az−b) = (−aX1 − b) ∆−az−b.
We denote
(2.17) S (z, E2, E4, E6, g0,1, . . . , gm−1,m) :=
A (z, E2, E4, E6, g0,1, . . . , gm−1,m)×∆−az−b.
Applying the differential operator D to the r.h.s. of (2.17) and using (2.12),
(2.16) we obtain
DS = 0.
Using (2.4) on the latter equality we conclude
d
dz
S(z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z)) = 0,
hence
S(z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z)) ∈ C.
Recall that functions z, E2, E4, E6, g0,1, . . . , gm−1,m are all algebraically
independent over C, see [3] page 2. For this reason we deduce
S[z,X1, X2, X3, Y ] ∈ C and thereby
A = λ∆azb
with λ ∈ C. If we suppose that A is irreducible, we obtain immediately that
either (a, b) = (1, 0) or (a, b) = (0, 1). 
Proof of Proposition 2.1. We consider the following nested sequence of rings
C[z,X] ⊂ C[z,X, Y0,1](2.18)
⊂ C[z,X, Y0,1, Y2,3]
⊂ C[z,X, Y0,1, Y1,3, Y2,3]
⊂ C[z,X, Y0,1, Y0,3, Y1,3, Y2,3]
...
⊂ C[z,X, Y0,1, . . . , Ym−3,m−2]
⊂ C[z,X, Y0,1, . . . , Ym−3,m−2, Ym−1,m]
⊂ C[z,X, Y0,1, . . . , Ym−3,m−2, Ym−2,m, Ym−1,m]
...
⊂ C[z,X, Y0,m, . . . , Ym−1,m] = R.
We readily verify with the definition of D that every term Ri appearing in
the chain (2.18) satisfies DRi ⊂ Ri.
Let P ⊂ R be a prime ideal of R satisfying DP ⊂ P. If P∩C[z,X] 6= {0},
it contains a polynomial z∆ as shown in [5][Theorem 1.4]. So the claim of
Multiplicity estimate 781
Proposition 2.1 is proved in this case. We suppose henceforth P∩C[z,X] =
{0}.
As we suppose P 6= {0} and P∩C[z,X] = {0}, we find in the chain (2.18)
at some step an extension of rings Ri ⊂ Ri+1 satisfying P ∩ Ri = {0} and
P ∩ Ri+1 6= {0}. In this case the ideal (of the ring Ri+1) P ∩ Ri+1 is
a principal one, because we add exactly one variable at each step in the
chain (2.18), i.e. tr.deg.RiRi+1 = 1. Hence P ∩Ri+1 is a D-stable principal
ideal (of the ring Ri+1, and also this ideal generates a principal D-stable
ideal of the ringR, becauseD-stability of a principal ideal means exactly the
condition Q|DQ on a generator of the ideal). We deduce with Lemma 2.1
that z∆ ∈ P ∩Ri+1 ⊂ P, Q.E.D. 
References
[1] P. Kozlov, On algebraic independence of functions from a certain class. Izvestiya: Mathe-
matics (Izvestiya of the Russian Academy of Sciences, in russian), in press.
[2] Yu. Nesterenko,Modular functions and transcendence questions. Math. Sb. 187/9 (1996),
65–96 (Russian); English translation in Sb. Math. 187/9, 1319–1348.
[3] Yu. Nesterenko, Some identities of Ramanujan type. Moscow Journal of Combinatorics
and Number Theory, Vol. 1 (2011), issue 2.
[4] Yu. Nesterenko, Patrice Philippon (eds.), Introduction to Algebraic Independence The-
ory. Lecture Notes in Mathematics, Vol. 1752, Springer, 2001.
[5] F. Pellarin, La structure différentielle de l’anneau des formes quasi-modulaires pour
SL2(Z). Journal de Théorie des Nombres de Bordeaux 18, no.1 (2006), 241–264.
[6] P. Philippon, Indépendance algébrique et K-fonctions. J. reine angew. Math. 497 (1998),
1–15.
[7] P. Philippon, Une approche méthodique pour la transcendance et l’indépendance algébrique
de valeurs de fonctions analytiques. J. Number Theory 64 (1997), 291–338.
[8] J.-P. Serre, A Course in Arithmetic. Springer-Verlag, New York, 1973.
[9] E. Zorin, Lemmes de zéros et relations fonctionnelles. Thèse de doctorat de l’Université
Paris 6, 2010. Accessible at http://tel.archives-ouvertes.fr/tel-00558073/fr/
Evgeniy Zorin
Department of Mathematics
University of York
York, YO10 5DD, UK
E-mail: evgeniyzorin@yandex.ru


Multiplicity estimate for solutions of extended Ramanujan’s system

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