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  "body": "\n## Statement\n\nLet $\\Lambda = \\Z\\omega_1 + \\Z\\omega_2$ be a lattice in $\\C$, with $\\omega_1, \\omega_2$ linearly independent over $\\R$.\nLet $f$ be an [[elliptic function|FF9ZPCZ59K]] with respect to $\\Lambda$.\nLet $P = \\set{a + s\\omega_1 + t\\omega_2 : 0 \\le s, t \u003c 1}$ be a fundamental parallelogram, for some $a \\in \\C$.\n\n1. If $f$ has no poles, then $f$ is constant.\n2. The sum of the [[residues|0JFGKKWWFK]] of $f$ at its poles in $P$ is $0$.\n3. If $f$ is not constant and $c \\in \\C$, then $f$ has as many solutions of $f(z) = c$ in $P$ as [[poles|0WPQYW8MTE]] in $P$, both counted with multiplicity.\n\n## Remarks\n\n1. An elliptic function without poles is continuous on the closure of $P$, hence bounded on $\\C$ by periodicity, hence constant by [[Liouville's theorem|0EHKVRF4J6]].\n2. By the [[residue theorem|032HKYS329]], the sum of the residues is $\\frac{1}{2\\pi i}\\int_{\\partial P} f(z)\\,dz$, and the integrals over opposite sides cancel by periodicity.\n3. Apply part 2 to the elliptic function $f'/(f - c)$ and use the [[argument principle|0PN2BXRS8G]].\n\nThe common number in part 3 is the *order* of $f$.\nPart 2 shows that the order is never $1$: a single simple pole would have a nonzero residue.\nSo a nonconstant elliptic function has order at least $2$; the [[Weierstrass function|361NAK68QK]] $\\wp$ has order $2$, with one double pole in $P$.\nA further result of the same kind says that the sum of the zeros minus the sum of the poles in $P$, with multiplicity, lies in $\\Lambda$.\nEach point of $\\C$ is congruent modulo $\\Lambda$ to exactly one point of $P$, so the counts in parts 2 and 3 do not depend on $a$.\nThe contour arguments use a choice of $a$ for which $f$ has no zeros, poles or $c$-points on the boundary of $P$.\n\n## Explanations\n\n### L1 human\n\nThese three facts constrain what a doubly periodic function can look like.\nIt must have poles unless it is constant.\nIts residues inside one period cell add up to zero.\nAnd it takes every value the same number of times in a period cell.\n\n### L2 human\n\nAll three proofs use one idea: integrate around the boundary of a period parallelogram, where opposite sides cancel by periodicity.\nThe results show that the smallest possible order of a nonconstant elliptic function is $2$.\nThis is why the Weierstrass function has a double pole and the Jacobi functions have two simple poles per cell.\nOn the torus $\\C/\\Lambda$, they are the first instances of general facts on compact Riemann surfaces.\nThere a holomorphic function is constant, and a meromorphic function has as many zeros as poles.\n",
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