chan h h, chua k s, solé p - seven-modular lattices and a septic base jacobi identity - j. number...

8/9/2019 Chan H H, Chua K S, Sol P - Seven-modular lattices and a septic base Jacobi identity - J. Number Theory 99 (2003

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Journal of Number Theory 99 (2003) 361372

Seven-modular lattices and a septic base

Jacobi identity

Heng Huat Chan,a,* Kok Seng Chua,b and Patrick Sole c

a

Department of Mathematics, National University of Singapore, Singapore 117543, Singaporeb Institute of High Performance Computing, 1 Science Park Road, # 01-01, The Capricorn,

Singapore 117528, SingaporecCNRS-I3S, ESSI, Route des Colles, 06 903 Sophia Antipolis, France

Received 19 December 2001; revised 10 July 2002

Communicated by D. Goss

Abstract

A quadratic Jacobi identity to the septic base is introduced and proved by means of modularlattices and codes over rings. As an application the theta series of all the 6-dimensional 7-

modular lattices with an Hermitian structure over Q ffiffiffiffiffiffiffi7p are derived.r 2002 Elsevier Science (USA). All rights reserved.

1. Introduction

For jqjo1; let

W2q XN

nNqn1=2

2;

W3q XN

nNqn

2

*Corresponding author.

E-mail addresses: [email protected] (H.H. Chan), [email protected] (K.S. Chua),

[email protected] (P. Sole ).

0022-314X/03/$ - see front matterr 2002 Elsevier Science (USA). All rights reserved.

PII: S 0 0 2 2 - 3 1 4 X ( 0 2 ) 0 0 0 6 9 - 0


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and

W4q X

N

nN 1n

q

n2

:

One of Jacobis famous identities states that

W4

3q W4

2q W4

4q: 1:1

This identity has been generalized in various ways, in particular in the context of

Vertex Operator Algebras [10].

Around 1991, Borwein and Borwein [4] discovered a cubic analogue of (1.1),

namely,

a3q b3q c3q;

where aq; bq; cq are the bi-dimensional theta series

aq X

m;nAZ

qm2mnn2 ;

b

q

Xm;nAZ omnqm

2mnn2 ; o

e2pi=3

and

cq X

m;nAZ

qm1=32m1=3n1=3n1=32 :

The aim of this note is to introduce another bi-dimensional generalization of

Jacobis identity (1.1). Our main result is

Theorem 1.1. Let

Aq X

m;nAZ

q2m2mn2n2;

Bq X

m;nAZ

1mnqm2mn2n2

and

Cq X

m;nAZ

q2m1=22m1=2n2n2:

H.H. Chan et al. / Journal of Number Theory 99 (2003) 361372362


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Then

A2

q

B2

q

C2

q

:

The proof techniques as in [19] are a combination of codes and lattices techniques.

In particular, we will follow Bachocs approach [1] for the construction of 7-modular

lattices from codes over the ring F2 F2:Like the Borweins cubic identity, our base 7 identity was unknown to Ramanujan

[2]. However, when calculating the theta series of a famous 7-modular lattice (related

to the polarization of the Klein curve) we encounter a formula akin to the ones in

[6,17]. It appears that our identity belongs to the septic analogue of Ramanujans

theories of elliptic functions to the alternative bases [3].

2. Notations and definitions

2.1. Codes

Let R4 denote the ring with 4 elements F2 vF2 where v2 v: This ring containstwo maximal ideals v and v 1: Observe that both ofR4=v and R4=v 1 are F2:The Chinese Remainder Theorem tells us that

R4 v"v 1;

so that ring R4DF2 F2: An explicit isomorphism is

va v 1b/a; b; a; bAF2:

A code over R4 is an R4-submodule of Rn4:

Let K : Qffiffiffiffiffiffiffi7p be the quadratic number field with ring of integers O Za;

with a2

a

2

0: Then we can regard R4 as O=

2

when v is the image ofa under

reduction modulo 2. Denote by a bar the conjugation which fixes F2 and swaps v and1 v: The natural scalar product induced by the hermitian scalar product of Cn isthen given by X

i

xiyi:

The Bachoc weight as defined in [1, Definition 3.1] is of course wB0 0: But moresurprisingly wBv wB1 v 2 and wB1 1: Define the Bachoc composition ofx say, ni

x

; i

0; 1; 2; as the number of entries in x of Bachoc weight i: In terms of

Bachoc composition, we have

wB n1 2n2:

H.H. Chan et al. / Journal of Number Theory 99 (2003) 361372 363


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The symmetrized weight enumerator (swe) is then the polynomial of three

variables X; Y; Z; defined by

sweCX; Y; Z XxAC

Xn0xYn1xZn2x:

2.2. Lattices

An n-dimensional lattice L is a discrete subgroup of Rn: Its theta series is

yLq XxAL

qxx;

where x x Pni1 x

2i; and q exppiz; zAH; where

H : fzAC j Im z > 0g:

The lattice L is called integral if it is contained in its dual Ln defined as

Ln : fyARn; 8xAL; x yAZg:

A topic of current interest in research is the study of modular lattices. The salient

property of these lattices introduced by Quebbemann [16] is that their theta series is amodular form for a suitable subgroup of the modular group. Specifically, an integral

lattice L is said to be c-modular [1,13] for some prime c if L is isometric toffiffiffic

pLn:

Theorem 2.1 (Quebbemann [16, Theorem 7]). Let

D6q : q2YNn1

1 q2n31 q14n3:

The theta series of an even 7-modular lattice is an isobaric polynomial in the two

variables A ffiffiffiqp andD6:The special cusp form D6 is called a CM-form in [15, (3.b)] where an expansion as a

twisted theta series attached to the quadratic form 1; 0; 7 is given.

3. Preliminaries

Define the construction AKC as the preimage in On of CDRn4 under reductionmodulo 2. Specifically,

AKC : fyAOn j y mod 2ACg:



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Theorem 3.1. If CDRn4

is a self-dual code then the lattice AKC=ffiffiffi

2p

is even 7-

modular.

Proof. The assertion follows by Bachoc [1, Proposition 3.6] and can alternatively be

derived directly by checking that O is 7-modular for the bilinear form

x;y/TrKx %y:

To compute the theta series of a lattice AKC as a function of sweC we need todefine some auxiliary theta series. Following [1], we introduce

y0q XxA2O qx %x;

y1q X

xA12Oqx %x

and

y2q X

xAa2Oqx %x:

We quote [1, Proposition 4.2] in the case at hand. &

Theorem 3.2 (Bachoc). If CDRn4 is a code of length n; then the theta series of thelattice AKC satisfies

yAKC sweCy0q; y1q; y2q:

4. Proof of Theorem 1.1

We first express Aq; Bq and Cq in term of Jacobis functions Wiq; i1; 2; 3:

Lemma 4.1. Let Aq; Bq and Cq be given as in Theorem 1.1. Then

Aq W3q2W3q14 W2q2W2q14; 4:1

Bq W4qW4q7 4:2



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and

C

q

W2

q2

W3

q14

W3

q2

W2

q14

:

4:3

Proof. Identity (4.1) can be found, for example, in [5, p. 1738]. For the proofs of the

subsequent identities, we will need the following simple identity, namely, for any odd

integer n;

XNmN

1mqmn22 0: 4:4

This identity follows immediately from [20, p. 464].

Next, note that

Bq XN

m;nN1mnqm2mn2n2

XN

m;nN1mnqmn22D4 n2 ;

with D 7; and that

Bq XNnN

1nqDn2=4 XNmN

1mqmn22

XnA2Z

1nqDn2=4XN

mN1mqmn22

X

nA2Z11nqDn2=4

XNmN

1mqmn22

XnA2Z 1nqDn

2=4 XN

mN 1

m

qmn22

W4

qD

W4

q

by (4.4).

Finally, rewrite Cq as

Cq XN

m;nNq2m

n22D

4n2

XmAZ;nA2Z q2mn

22D

4n2 XmAZ;nA2Z1

q2mn22D

4n2

W2q2W3q14 W3q2W2q14: &



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We proceed to express y0q; y1q; and y2q as functions ofAq; Bq; and Cq:

Proposition 4.2. For all zAH we have

y0q Aq2

and

y1q Cq2:

Proof. If we set x m na; then the norm form becomes

x %x m2 mn 2n2 : NFm; n: 4:5

If we set x 1 2m na; then the norm form becomes

x %x 4m 1=22 m 1=2n 2n2: &

In view of these expressions it is natural to look for an expression for y2 involving

q2: To that end, we shall require the following duplication formulas:

Lemma 4.3. For all zAH we have

Aq 2Aq2 Bq2

and

Bq Aq2 Cq2:

Proof. From (4.1) and (4.3), we find, after some simplification, that

Aq2 Cq2 W3q4 W2q4W3q28 W2q28W4qW4q7 Bq:

This completes the proof of the second assertion. To derive the first assertion, we

claim that

Aq Bq2 2XN

m;nNq22m

22mn4n2: 4:6



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Indeed

A

q

B

q2

XN

m;nN 1

1

mn

q2m

2mn2n2

2X

m;nA2Z

X

m;nA2Z1

" #q2m

2mn2n2

2X

mA2Z;nAZ

X

m;nA2Z1

XmA2Z;nA2Z1

" #q2m

2mn2n2:

From (4.4), we deduce that

Xm;nA2Z1

X

mA2Z;nA2Z1

!q2m

2mn2n2

X

mAZ;nA2Z1q2m

2mn2n2

X

mAZ;nA2Z1q2m

n22D

4n2: &

Proposition 4.4. For all zAH; we have

y2q Aq2 Bq2:

Proof. Writing x a 2m na we see that

x %x 4m2 2n 1=22 mn 1=2

or in other words,

y2q Xm;nAZ

m even;n odd

qNF

m;n

;

where NFm; n is given by (4.5). Introduce for convenience

y3q X

m;nAZm odd;n odd

qNFm;n:

Since

Z Z fm; nAZ j m even; n eveng,fm; nAZ j m odd; n oddg,fm; nAZ j m odd; n eveng,fm; nAZ j m even; n oddg;



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we may split the sum over m; nAZ Z in A and B into four sums and obtain thefollowing system of two equations in y2q and y3q:

Aq1=2 y2q Aq2 Cq2 y3q

and

Bq y2q Aq2 Cq2 y3q:

Solving for y2 we find that

2y2q A ffiffiffiq

p Bq 2Cq2:

The result follows by Lemma 4.3. &

We shall require the following lemma.

Lemma 4.5. For all zAH we have

yffiffi

2pO2q 2Aq2 Bq22:

Proof. By definition

yffiffi

2pO2q A2q:

Applying Lemma 4.3 for Aq; we complete the proof of the lemma.We now complete the proof of Theorem 1.1.

Proof of Theorem 1.1. By Scharlau [18] there existsup to isometrya unique 7-

modular lattice of dimension 4 over Z: An immediate candidate is O2: By Theorem

3.1 another candidate is AK

C2= ffiffiffi2p where C2 is the length 2 self-dual code with

generator matrix 1; 1:Now the swe of that code is computed in [1, p. 102] and evaluated as

sweC2X; Y; Z X2 Y2 2Z2:

The theta series of AKC2 can then be computed on applying Theorem 3.2. By thepreceding discussion it should equal y

ffiffi2

pO2q computed in Lemma 4.5 as a function

of Aq; Bq; and Cq: This yields

2Aq2 Bq22

A2q2 C2q2 2Aq2 Bq22

;

which reduces to the desired identity. &



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5. Seven-modular lattices in dimension 6

5.1. Extremal case

The lattice called

* A26

in Craigs notation [8, Chapter 8, p. 223],* P6 in Barnes notation [13, p. 131],* J in Cohens notation [7]

is registered as a perfect 7-modular lattice in dimension 3 over O of determinant 73;kissing number 42 and norm 4 in NebeSloane Catalogue of lattices [11]. According

to [18] an extremal (i.e. norm 4) 7-modular lattice in dimension 3 over O is unique.

The construction of [14, Section 3, p. 237] attributed to Serre shows that A2

6 AKC3: Here C3 vR3 1 vR>3 where R3 stands for the binary linear code ofgenerator 1; 1; 1: It occurred [14] in relation to the Jacobian of the Klein curve.Another geometric construction, using MordellWeil lattices can be found in [9]

where the theta series is computed (using Quebbemanns theorem) as

yA2

6q Aq3 6D6q:

Combining this information with [1] where it is shown that sweC3X; Y; Z X3

Z3

3XZ2

3ZY2; we obtain

Theorem 5.1. The theta series of A26

is

yA2

6q 5A3q2 9A2q2Bq2 6Aq2B2q2

3C2q2Aq2 3C2q2Bq2 B3q2:

A third evaluation of yA26

can be obtained by using Ramanujan modular

equations to the base 7. Let f

q

:

QN

j

1

1

qj

: On applying Lemma 2.2 of[5] we

get

yA2

6q f

7q2fq14 7D6q 49q

2 f7q14

fq2 :

On the other hand, according to Ranghachari [17, p. 370] the theta series of the

lattice An6

admits the similar expression

yAn6q f

7q2f

q14

7D6q 7q2 f

7q14f

q2

:

This comes from the fact that both theta series are invariant under G07 with thesame quadratic character [6]. The corresponding space of weight 3 modular forms is



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three dimensional and spanned by the three functions

f7

q2

fq14;D

6q and q2 f

7

q14

fq2 :

5.2. Norm 2

There are two other self-dual R4-codes in length 3; namely C3;2 : v/0; 1; 1Sv 1/0; 1; 1S> with weight enumerator

X3 X2Z XY2 2XZ2 Y2Z 2Z3

and C3;3 : v/0; 0; 1S v 1/0; 0; 1S> with weight enumerator

X3 3X2Z 3XZ2 Z3:

Since AKC3;2 and AKC3;3 seem to have the same theta series

1 6q2 24q4 56q6 114q8 168q10 280q12

294q14 444q16 Oq18

we are led to conjecture the cubic relation

2A3q2 A2q2Bq2 2Aq2C2q2 2Aq2B2q2 C2q2Bq2 B3q2 0

which is equivalent to

2Aq2 Bq2A2q2 B2q2 C2q2 0;

which is certainly true.

Proposition 5.2. The lattices AKC3;2 and AKC3;3 are isometric. Their theta series is

8A3q2 12A2q2Bq2 6Aq2B2q2 B3q2:

Proof. The first assertion follows by inspection of SchulzePillots database of

Hermitian lattices [12]. The second assertion follows on applying Theorem 2.2 to

sweC3;3 :To conclude there are exactly two classes ofO lattices in (real) dimension 6 and

they can both be constructed by using codes over F2 F2: &



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References

[1] C. Bachoc, Application of coding theory to the construction of modular lattices, J. Combin. Theory

Ser. A 78 (1997) 92119.[2] B.C. Berndt, email to Heng Huat Chan, September 2001.

[3] B.C. Berndt, S. Bhargava, F.G. Garvan, Ramanujans theories of elliptic functions to alternative

bases, Trans. Amer. Math. Soc. 347 (11) (1995) 41634244.

[4] J.M. Borwein, P.B. Borwein, A cubic counterpart of Jacobis identity and the AGM, Trans. Amer.

Math. Soc. 323 (1991) 691701.

[5] H.H. Chan, Y.L. Ong, On Eisenstein series andPN

m;nN qm2mn2n2 ; Proc. Amer. Math. Soc. 127 (6)

(1999) 17351744.

[6] K.S. Chua, The root lattice An; and Ramanujans circular summation of theta functions, Proc. Amer.Math. Soc. 130 (1) (2002) 18.

[7] A.M. Cohen, Finite complex reflection groups, Ann. Sci. E cole Norm. Sup. (4) 9 (3) (1976) 379436.

[8] J.H. Conway, N.J.A. Sloane, Sphere packings, lattices and groups, third edition, in: Grundlehren der

Mathematischen Wissenschaften, Vol. 290, Springer, New York, 1999.

[9] N.D. Elkies, The Klein quartic in coding theory, in: S. Levy (Ed.), The Eightfold Way, Vol. 35,

Mathematical Sciences Research Institute Publications, Cambridge University Press, Cambridge,

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[10] I. Frenkel, J. Lepowsky, A. Meurman, Vertex Operator Algebras and the Monster, in: Pure and

Applied Mathematics, Vol. 134, Academic Press, Boston, MA, 1988.

[11] http://www.research.att.com/~njas/lattices/P6.5.html

[12] http://www.math.uni-sb.de/~ag-schulze/Hermitian-lattices/D-7/

[13] J. Martinet, Les re seaux parfaits des espaces euclidiens, Mathe matiques, Masson, Paris, 1996.

[14] B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986) 207259.

[15] K. Ono, On the circular summation of the eleventh powers of Ramanujans theta function, J. Number

Theory 76 (1999) 6265.[16] H-G. Quebbemann, Modular lattices in Euclidean spaces, J. Number Theory 54 (2) (1995) 190202.

[17] S.S. Rangachari, On a result of Ramanujan on theta functions, J. Number Theory 48 (1994) 364372.

[18] R. Scharlau, B. Hemkemeier, Classification of integral lattices with large class number, Math.

Comput. 67 (1998) 737749.

[19] P. Sole , P. Loyer, Un lattices, Construction B; and AGM iterations, European J. Combin. 19 (1998)227236.

[20] E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, Cambridge University Press,

Cambridge, 1996.

http://www.research.att.com/~njas/lattices/P6.5.htmlhttp://www.math.uni-sb.de/~ag-schulze/Hermitian-lattices/D-7/http://www.math.uni-sb.de/~ag-schulze/Hermitian-lattices/D-7/http://www.research.att.com/~njas/lattices/P6.5.html

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