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  "body": "\n## Summary\n\nThe Jacobi theta function is $\\vartheta(z, \\tau) = \\sum_{n \\in \\Z} e^{\\pi i n^2 \\tau + 2 \\pi i n z}$ for $z \\in \\C$ and $\\tau$ in the [[upper half-plane|X6XGGH5FPW]].\nIt is periodic in $z$ with period $1$ and quasi-periodic with period $\\tau$.\nTheta functions are the building blocks of elliptic functions and of functions on abelian varieties.\nIn the variable $\\tau$, the theta series of quadratic forms are [[modular forms|GWKN5VG78A]] that count representations of numbers.\n\n## Explanations\n\n### L1 human\n\nA theta function is a sum of exponentials whose exponents grow like squares, so it converges very fast.\nShifting $z$ by $1$ changes nothing, and shifting $z$ by $\\tau$ multiplies the function by a simple exponential factor.\nSetting $z = 0$ gives $\\theta(\\tau) = \\sum_{n \\in \\Z} e^{\\pi i n^2 \\tau}$.\nRaising it to the $k$-th power counts the ways to write numbers as sums of $k$ squares.\nThis is how [[Jacobi's four-square theorem|CG17WSS64W]] can be proved.\n",
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