D4, E6, E8 and the AGM

We derive Jacobi's quartic identity and the Borweins' cubic identity related to Ramanujan's quadratic modular equation on theta series by lattice enumerative methods. Both identities are instrumental in recent work of the Borweins on the Arithmetic Geometric Mean. Of great use are the constructions of the root lattices D4 and E6 by binary and ternary codes respectively. A third identity, equally due to the Borweins is also derived in relation to the root lattice E8.

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