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On the Gross-Stark conjecture. (English) Zbl 1416.11160
This article presents a proof of Gross’s conjecture on the leading term at $$s = 0$$ of the $$p$$-adic $$L$$-function $$L_p(\chi\omega,s)$$ associated to a totally real field $$F$$ and a totally odd character $$\chi$$ of the Galois group $$G_F$$ ($$\omega$$ is the Teichmüller character). The case of a simple zero at $$s=0$$ (notation $$r=1$$) was proved in previous articles by Dasgupta, Darmon, Pollack, Ventullo.
Let $$\Lambda = \mathcal{O}_E[[T]]$$, where $$E$$ is a finite extension of $$\mathbb{Q}_p$$ containing the values of $$\chi$$. In the proof for $$r=1$$ certain cuspidal Hida families of $$\Lambda$$-adic Hilbert modular forms for $$F$$ are used. The same Hida families are used for $$r>1$$, but the technique is much more complicated and demands the explicit computation of the Hecke action on explicitly given cuspidal Hida families. Essential is the application of results of Hida and Wiles associating Galois representations $$G_F\rightarrow \text{GL}_2 (L)$$ to cuspidal Hida eigenfamilies (the $$p$$-adic regulator appears in the eigenvalues).
Finally, the proof of the conjecture is deduced from the properties of a 1-cohomology class of $$G_F$$.
At each stage of the proof the authors explain why and how the method used for $$r=1$$ has to be changed for $$r>1$$.

##### MSC:
 11R42 Zeta functions and $$L$$-functions of number fields 11R80 Totally real fields 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F80 Galois representations
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