On p-adic analytic families of Galois representations (original) (raw)
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This paper investigates p-adic analytic families of Galois representations, exploring constructions and properties related to modular forms and their associated group schemes. It discusses the implications of p-adic Hodge theory and the characteristics of various p-adic representations, specifically through the lens of degeneracy operators and eigenvalues, while posing several propositions and remarks on future research directions regarding compatible families of representations and their non-semi-simple structures.
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2016
Let F be a totally real field and χ an abelian totally odd character of F. In 1988, Gross stated a p-adic analogue of Stark's conjecture that relates the value of the derivative of the p-adic L-function associated to χ and the p-adic logarithm of a p-unit in the extension of F cut out by χ. In this paper we prove Gross's conjecture when F is a real quadratic field and χ is a narrow ring class character. The main result also applies to general totally real fields for which Leopoldt's conjecture holds, assuming that either there are at least two primes above p in F , or that a certain condition relating the L-invariants of χ and χ −1 holds. This condition on L-invariants is always satisfied when χ is quadratic. Contents S. DASGUPTA, H. DARMON, and R. POLLACK 4. Galois representations 477 4.1. Representations attached to ordinary eigenforms 477 4.2. Construction of a cocycle 480 References 482
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