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modular discriminant
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(Definition)
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Definition 1 Let $\Lambda\subset\Complex$ be a lattice.
- Let $q_{\tau}=e^{2\pi i \tau}$ . The Dedekind eta function is defined to be $$\eta(\tau)=q_{\tau}^{1/24}\prod_{n=1}^{\infty}(1-q_{\tau}^n)$$ The Dedekind eta function should not be confused with the Weierstrass eta function, $\eta(w;\Lambda)$ .
- The $j$ -invariant, as a function of lattices, is defined to be: $$j(\Lambda)=\frac{g_2^3}{g_2^3-27g_3^2}$$ where $g_2$ and $g_3$ are certain multiples of the Eisenstein series of weight $4$ and $6$ (see this entry).
- The $\Delta$ function (delta function or modular discriminant) is defined to be $$\Delta(\Lambda)=g_2^3-27g_3^2$$ Let $\Lambda_{\tau}$ be the lattice generated by $1,\tau$ . The $\Delta$ function for $\Lambda_{\tau}$ has a product expansion $$\Delta(\tau)=\Delta(\Lambda_{\tau})=(2\pi i)^{12}q_{\tau}\prod_{n=1}^{\infty}(1-q_{\tau}^n)^{24}=(2\pi i)^{12}\eta(\tau)^{24}$$
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"modular discriminant" is owned by alozano.
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Cross-references: product, generated by, weight, Eisenstein series, multiples, function, Weierstrass eta function, lattice
There are 7 references to this entry.
This is version 3 of modular discriminant, born on 2003-08-25, modified 2008-02-26.
Object id is 4651, canonical name is ModularDiscriminant.
Accessed 8514 times total.
Classification:
| AMS MSC: | 33E05 (Special functions :: Other special functions :: Elliptic functions and integrals) |
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Pending Errata and Addenda
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