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  "body": "\n## Summary\n\nTate's thesis is the 1950 doctoral thesis of John Tate, written under Emil Artin.\nIt proves the meromorphic continuation and the functional equation of the L-functions of [[Hecke characters|R65XKHN4D1]].\nThe tool is harmonic analysis on the [[adele ring|SEH68BA1QQ]] and on the idele group.\nThe L-function appears as a zeta integral of a test function against a character over the ideles.\nThe functional equation then follows from the Poisson summation formula for the discrete subgroup $K$ of $\\mathbb{A}_K$.\nThe method made adeles a standard tool and is a model for the theory of automorphic L-functions.\n\n## Explanations\n\n### L1 human\n\nThe [[Riemann zeta function|1XWBY4N0VT]] has a symmetry that relates its values at $s$ and $1 - s$.\nTate found one uniform reason for this symmetry and for its analogues over every number field.\nHe split the zeta function into local pieces, one for each completion, and studied each piece with Fourier analysis on that completion.\nThe pieces are glued with the Poisson summation formula, and the symmetry drops out.\nFor the rational numbers, this gives a new proof of the functional equation of the Riemann zeta function.\n",
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