## A Collection of Manuscripts Written in Honour of John H. Coates on the Occasion of His Sixtieth BirthdayThis volume is dedicated to Professor John H. Coates, an outstanding contributor to number theory, both through his pioneering and fundamental mathematical works and through the magnificent mathematical school he has established. It contains 24 articles written by 38 authors on a wide range of topics in the cutting edge of research in number theory, algebraic geometry and analysis: zeta functions and $L$-functions, automorphic and modularity issues, Galois representations,arithmetic of elliptic curves, Iwasawa theory, noncommutative Iwasawa theory, and $p$-adic analysis. This volume will be of interest to researchers and students in these and neighboring fields. Information for our distributors: A publication of the Documenta Mathematica. The AMS distributes this series,beginning with volume 3, in the United States, Canada, and Mexico. |

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### Contents

Preface | 1 |

K Ardakov and K A Brown | 7 |

G Banaszak W Gajda P Krason | 35 |

Copyright | |

21 other sections not shown

### Common terms and phrases

A-module abelian variety action affinoid assume assumption automorphic canonical choose coefficients cohomology cokernel compact complex compute congruence conjecture consider construction Corollary corresponding cusp forms cyclotomic decomposition defined definition denote Dirichlet character dual eigenform Eisenstein series element elliptic curve embedding equal exact sequence factors finite extension finite places follows formal formula Fourier Frobenius function functor Galois representation given group G Hecke algebra hence Hida homomorphism implies induced integral isomorphism Iwasawa theory kernel Krull dimension Lemma Let G locus Math maximal modular forms module morphism multiplication nonzero norm notation Note number field overconvergent p-adic L-functions places of F points polynomial power series prime ideal pro-p group Proof Proposition prove pseudo-null quotient ramified recall reduction reflexive Remark resp result ring satisfies Selmer group semisimple Shimura varieties space subgroup subspace supersingular Suppose surjective Tate Theorem totally real trivial unramified vector weight write