## Base Change for GL(2) |

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### Contents

Introduction | 3 |

Properties of Base Change | 14 |

Applications to Artin Lfunctions | 23 |

Copyright | |

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### Common terms and phrases

absolutely cuspidal admissible representation Artin L-functions automorphic representation base change Cartan subgroup choose compact compactly supported conclude conjugacy classes conjugate constant cyclic extension defined direct sum double coset eigenvalues Eisenstein series element equal F is non-archimedean factor finite-dimensional fixed G(AE G(Ey G(Fy G+(F given global field global liftings HCS family Hecke algebra homomorphism infinite places integrate with respect introduce invariant isomorphic kernel L-group Lemma lies locally integrable function matrices meas G(0 modulo Moreover non-split Cartan subgroup non-trivial character norm Observe obtain orbital integrals paragraph points prime degree Proposition quadratic extension quasi-character replaced representation of G(F representation ring restriction right coset satisfying scalar Shintani smooth function split square-integrable Suppose F Tamagawa measures term theorem tion trivial twisted trace formula unitary unless unramified vector verify yields Z(AE