Sujatha Ramdorai

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Fontaine's Rings and p-adic L-functions (2015)

In the first part, we introduce theory of p-adic analysis for one variable p-adic functions and then use them to construct Kubota-Leopoldt p-adic L-functions.In the second part, we give a description of the Iwasawa modules attached to p-adic Galois representations of the absolute Galois group of K in terms of the theory of (φ,Γ)-modules of Fontaine. When the representation is de Rham when K be finite extension of Qp. This gives a natural construction of the exponential map of Perrin-Riou which is used in the construction and the study of p-adic L-functions.In the third part, we give formulas for Bloch-Kato’s exponential map and its dual for an alsolutely crystalline p-adic representation V . As a corollary of these computation, we can give a improved description of Perrin-Riou’s exponential map, which interpolates Bloch-Kato’s exponentials for the twists of V. Finally we use this map to reconstruct Kubota-Leopoldt p-adic L-functions.

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