## P-adic L-functions and P-adic RepresentationsTraditionally, $p$-adic $L$-functions have been constructed from complex $L$-functions via special values and Iwasawa theory. In this volume, Perrin-Riou presents a theory of $p$-adic $L$-functions coming directly from $p$-adic Galois representations (or, more generally, from motives). This theory encompasses, in particular, a construction of the module of $p$-adic $L$-functions via the arithmetic theory and a conjectural definition of the $p$-adic $L$-function via its special values. Since the original publication of this book in French (see ""Asterisque"" 229, 1995), the field has undergone significant progress. These advances are noted in this English edition. Also, some minor improvements have been made to the text. |

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### Contents

Preface | xi |

Construction of the module of padic Lfunctions without factors at infinity | 1 |

Modules of padic Lfunctions of V | 19 |

Values of the module of adic Lfunctions | 41 |

The padic Lfunction of a motive | 79 |

### Common terms and phrases

A-submodule algebra associated assume basis Bcris bilinear form Bloch-Kato conjectures canonical isomorphism character 77 characteristic series commutative diagram comparison isomorphism complex conjugation computation congruent modulo crystalline deduce defined definition denote detQp dimension d+(V dimQp Dirichlet character Dp(V eigenvalues elliptic curve equal equivalent Euler factors example exists extension of Qp exterior power factor at infinity Fil°Dp(V finite order finite set finite type follows formula Frac(A functional equation Galois cohomology geometric character Gs.F H(Gx Hlf(F holds homomorphism hypotheses implies injects integer isomorphism Iwasawa Iwasawa theory kernel lemma Leop(V module of p-adic multiplicity non-zero element notation number field p-adic L-functions p-adic representation places of F Pontryagin dual prime number projective limit PROOF proposition Qp-vector space Recall representation of GF resp satisfies set of places subspace Suppose surjective Tamagawa numbers theorem torsion type at infinity unramified values vector write zero