Hilbert modular forms and the Gross-Stark conjecture

• 成果类型: Article
• 署名作者: Dasgupta, Samit; Darmon, Henri; Pollack, Robert
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2011.174.1.12
• 发表日期: 2011
• 页码: 439-484
• 关键词: totally-real fields; adic zeta-functions; REPRESENTATIONS; VALUES

摘要

Let F be a totally real field and x 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 x and the p-adic logarithm of a p-unit in the extension of F cut out by x. In this paper we prove Gross's conjecture when F is a real quadratic field and x 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 x and x(-1) holds. This condition on L-invariants is always satisfied when x is quadratic.