Hilbert modular forms and the Gross-Stark conjecture. (English) Zbl 1250.11099

Dasgupta, Samit; Darmon, Henri; Pollack, Robert

Ann. Math. (2) 174, No. 1, 439-484 (2011).

This paper concerns B. H. Gross’s \(p\)-adic analogue of the Stark conjecture, introduced in [J. Fac. Sci., Univ. Tokyo, Sect. I A 28, 979–994 (1981; Zbl 0507.12010)]. This conjecture describes the derivatives at \(s = 0\) of \(p\)-adic \(L\)-functions attached to totally odd ideal class characters \(\chi\) of totally real number fields \(F\), in cases where \(\chi(\mathfrak{p}) = 1\) for some prime \(\mathfrak{p} \mid p\) (in which case the value of the \(p\)-adic \(L\)-function at \(s = 0\) is forced to vanish, because of the form of the interpolating factor at \(p\)). Gross’s conjecture predicts that the derivative of the \(p\)-adic \(L\)-function is equal to the value of the complex \(L\)-function multiplied by an “\(\mathcal{L}\)-invariant” defined in terms of a certain \(p\)-unit of the splitting field of \(\chi\). It was shown in the original paper of Gross that the conjecture holds when \(F = \mathbb{Q}\).
This paper proves this conjecture for a wide class of \(F\) and \(\chi\). More specifically, the conjecture is proved under the assumption that (i) Leopoldt’s conjecture holds for \(F\), and either (ii) there are at least two primes of \(F\) above \(p\) or (ii’) a certain condition relating the \(L\)-invariants of \(\chi\) and \(\chi^{-1}\) holds.
For a sketch of the argument, the reader is referred to the paper’s very clear and informative introduction.

Reviewer: David Loeffler (Coventry)

MSC:

11R42	Zeta functions and \(L\)-functions of number fields
11F41	Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F85	\(p\)-adic theory, local fields

Keywords:

p-adic L-functions; p-adic families of modular forms; Stark’s conjectures

Citations:

Zbl 0507.12010

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