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MathGroup Archive 2006

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Typo in DedekindEta function definition?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71194] Typo in DedekindEta function definition?
  • From: titus_piezas at yahoo.com
  • Date: Fri, 10 Nov 2006 06:37:53 -0500 (EST)

Hello all,

There's something puzzling me about Mathematica's definition of the
Dedekind eta function in terms of the Weierstrass invariants g2, g3.
Recall that the function WeierstrassInvariants[{w, w' }] involve the
half-periods {w, w'} and KleinInvariantJ[z] and DedekindEta[z] the
half-period ratio z = w' /w. To illustrate, (Notes: For brevity, values
have been rounded off. The symbol "i" is the imaginary unit.):

In[1]: = {g2,g3} = N[WeierstrassInvariants[{3, 5i}], 30]
Out[1] = {0.10089, 0.00601}

In[2]:= p1 = N[KleinInvariantJ[5i/3], 30]
Out[2] = 20.86892

In[3]:= p2 = g2^3/(g2^3-27g3^2)
Out[3] = 20.86892

As expected, p1 and p2 are equal. However,

In[4]:= r1 = N[(2Pi)^12 DedekindEta[5i/3]^24, 30]
Out[4] = 107137.63536

In[5]:= r2 = g2^3-27g3^2
Out[5] = 0.0000492

Shouldn't r1 and r2 be equal? After some experimentation, I found they
can be if the equality is modified to r1 = r2 (2w)^12 where "w" is the
first half-period.

In[6]:= r2 (6)^12
Out[6] = 107137.63536

and we now get the same value.  In the help section, the Klein
invariant J[z] is defined as g2^3 /(g2^3-27g3^2) which as we saw
worked.  But the Dedekind eta, or n[z], is defined as satisfying d =
(2Pi)^12 n[z]^24 where d is the discriminant and given in terms of the
Weierstrass invariants by g2^3-27g3^2.  In the section "Elliptic
Modular Functions", the same equality is stated.  But it does not work,
unless d is given the factor (2w)^12.

So can anyone can explain the situation, why it works for J[z] but not
for n[z]? (I have a nagging feeling I'm using a wrong assumption
somewhere...)

-Titus


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