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  "body": "\n## Summary\n\nThe monodromy of the [[hypergeometric differential equation|W8VE25R5A7]] records how a basis of solutions changes under analytic continuation along closed loops.\nThe loops go around the singular points $0$, $1$ and $\\infty$.\nIt is a representation of the [[fundamental group|S1XK3HXHFX]] of the thrice-punctured sphere, a free group on two generators, into $\\mathrm{GL}_2(\\C)$.\nIt is the special case of [[monodromy|C8HD62AB6Q]] that Riemann used to study the hypergeometric function.\nIt decides, for example, when all solutions are algebraic functions.\n\n## Explanations\n\n### L1 human\n\nSolutions of the hypergeometric equation are not single-valued: following a solution once around the point $1$ can bring it back as a different solution.\nSince the solutions form a two-dimensional space, each loop acts on it by a $2 \\times 2$ matrix.\nCollecting these matrices for all loops gives the monodromy group.\nFor example, when $a = b = 1$ and $c = 2$ the solutions are spanned by $1/z$ and $\\log(1 - z)/z$, and going around $z = 1$ adds a multiple of $2\\pi i / z$ to the second one.\nRiemann showed that this matrix data, together with the local exponents, nearly determines the equation and its solutions.\n\n### L2 human\n\nThe local monodromy around each singular point has eigenvalues $e^{2\\pi i \\rho}$, where $\\rho$ runs over the two exponents at that point.\nThe monodromy is irreducible exactly when none of $a$, $b$, $c - a$, $c - b$ is an integer.\nIn the irreducible case, the representation is determined up to conjugacy by the three pairs of local eigenvalues, a fact known as rigidity.\nFinite monodromy groups give algebraic solutions, and these are classified by [[Schwarz's list|W0MJ51242N]].\n",
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