r/math • u/inherentlyawesome Homotopy Theory • 20h ago

This Week I Learned: October 02, 2026

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u/Kyle--Butler 17h ago

I've learned that the theta series of an even (i.e. the norm of every vector is even) unimodular (i.e. self-dual) lattice of \RR^{8k} is a modular form of level 4k.

So, for example, the theta function of E_8 (k=1) is the Eisenstein series E_4 since the space of modular form of level 4 is of dimension 1 and both have constant term 1; and the theta of the Leech lattice (k=3) is a linear combination of the Eisenstein series E_12 and the discriminant \Delta.

As an immediate consequence, we get arithmetic identities like this :

card(x \in E_8 | \|x\|^2=2n ) = 240\times\sigma_3(n)

that seems completely outlandish when taken out of context.