FUNCTION : ETA[provemodfuncGAMMA0UpETAid] - prove U[p] eta-product identity
CALLING SEQUENCE : provemodfuncGAMMA0UpETAid()
provemodfuncGAMMA0UpETAid(EP,p,etacombo,N)
PARAMETERS :
EP - one eta-product
p - prime
etacombo - sum of modular functions on Gamma[0](N)
Each term in the sum is a eta-quotient to base N.
N - Positive integer multiple of p
GLOBAL VARIABLES :
SYNOPSIS :
This function PROVES the id U[p](EP) = etacombo
global vars (can be used for error-checking):
qcheck, modfunccheck, totcheck, _ORDS, jptmp, jpqd, eptmp,
gltmp, EPRODL, GETAL, COFS, conpres, CONTERMS, mintottmp
EXAMPLES :
> with(qseries):
> with(ETA):
> gpA1:=[1,2,4,3,2,-5,50,5,25,-2,100,-3]:
> epA1:=gp2etaprod(gpA1);
2 3 5
eta(tau) eta(4 tau) eta(50 tau)
epA1 := --------------------------------------
5 2 3
eta(2 tau) eta(25 tau) eta(100 tau)
> A1q:=etaprodtoqseries(epA1,1000):
> gpa1:=[1,2,10,4,2,-4,5,-2]:
gpa1 := [1, 2, 10, 4, 2, -4, 5, -2]
> epa1:=gp2etaprod(gpa1);
2 4
eta(tau) eta(10 tau)
epa1 := -----------------------
4 2
eta(2 tau) eta(5 tau)
> aq1:=etaprodtoqseries(epa1,2000):
> gprho:=[2,2,20,4,4,-4,10,-2]:
> eprho:=gp2etaprod(gprho);
2 4
eta(2 tau) eta(20 tau)
eprho := ------------------------
4 2
eta(4 tau) eta(10 tau)
> G1a:=etaprodtoqseries(eprho,1001):
> G1:=convert(series(1-G1a,q,1020),polynom):
> etamake(G1,q,100);
3
eta(10 tau) eta(2 tau)
-----------------------
3
eta(20 tau) eta(4 tau)
> U01:=sift(A1q,q,5,0,2000):
> etacombo:=findlincombo(U01,[seq( aq1^k,k=-2..5),seq( aq1^k*(G1),k=-2..5)],[seq( etamake((aq1)^k,q,100),k=-2..5),seq( etamake((aq1)^k*G1,q,100),k=-2..5)],q,0);
nx = , 16
# of terms , 37
2 4
53 eta(tau) eta(10 tau)
etacombo := - -------------------------
4 2
eta(2 tau) eta(5 tau)
8 4 12 6
350 eta(10 tau) eta(tau) 1050 eta(10 tau) eta(tau)
+ -------------------------- - ----------------------------
4 8 6 12
eta(5 tau) eta(2 tau) eta(5 tau) eta(2 tau)
16 8 20 10
1375 eta(10 tau) eta(tau) 625 eta(10 tau) eta(tau)
+ ---------------------------- - ----------------------------
8 16 10 20
eta(5 tau) eta(2 tau) eta(5 tau) eta(2 tau)
3
13 eta(10 tau) eta(2 tau)
+ --------------------------
3
eta(20 tau) eta(4 tau)
7 2
75 eta(10 tau) eta(tau)
- -----------------------------------------------
2 3 3
eta(20 tau) eta(5 tau) eta(4 tau) eta(2 tau)
11 4
175 eta(10 tau) eta(tau)
+ -----------------------------------------------
4 3 7
eta(20 tau) eta(5 tau) eta(4 tau) eta(2 tau)
15 6
125 eta(10 tau) eta(tau)
- ------------------------------------------------
6 3 11
eta(20 tau) eta(5 tau) eta(4 tau) eta(2 tau)
;
> provemodfuncGAMMA0UpETAid(epA1,5,etacombo,20);
*** There were NO errors.
*** o EP is an MF on Gamma[0](100)
*** o Each term in the etacombo is a modular function on
Gamma0(20).
*** o We also checked that the total order of
each term etacombo was zero.
*** To prove the identity U[5](EP)=etacombo we need to show
that v[oo](ID) > 9 This means checking up to q^(10).
Do you want to prove the identity? (yes/no)
You entered yes.
We verify the identity to O(q^(49)).
We find that LHS - RHS is
/ 49\
O\q /
RESULT: The identity holds to O(q^(49)).
CONCLUSION: This proves the identity since we had only
to show that v[oo](ID) > 9.
DISCUSSION :
SEE ALSO :