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The Jacobi triple product identity gives as a special case a product formula for the theta function of a 1-dimensional lattice. Is there a more general product formula for the theta function of an arbitrary lattice?

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  • $\begingroup$ Have you heard of MacDonald identities? $\endgroup$
    – S. Carnahan
    Mar 23, 2014 at 15:36
  • $\begingroup$ I know a little bit about them, but it's not clear to me whether I get a formula for the theta function of an arbitrary lattice from them. For one, I don't see a way of associating an affine root system to an arbitrary lattice. $\endgroup$
    – Tom Price
    Mar 23, 2014 at 18:29
  • $\begingroup$ Yes, that is a valid point. I think I spoke too soon. You can get product formulas for theta functions of a more general class of positive definite lattices using a Borcherds-Harvey-Moore lift of a suitable weight 1/2 weakly holomorphic modular form (i.e., with a pole at a cusp). $\endgroup$
    – S. Carnahan
    Mar 23, 2014 at 20:03
  • $\begingroup$ I'm not familiar with the Borcherds-Harvey-Moore lift; is there any particular introduction to it you would recommend? Also, do you know which positive-definite lattices this gives a formula for? $\endgroup$
    – Tom Price
    Mar 23, 2014 at 22:43
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    $\begingroup$ I don't know a good reference, but a search for "Borcherds products" should yield something fruitful. $\endgroup$
    – S. Carnahan
    Mar 28, 2014 at 8:47

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