On the rank one Gross-Stark conjecture for quadratic extensions and the Deligne-Ribet \(q\)-expansion principle
Let \(F\) be a totally real algebraic number field, let \(\chi\) be a totally odd character of \(F\) and let \(\omega\) be the Teichmüller character for a prime \(p\). \textit{B. Gross} [J. Fac. Sci Tokyo, sect. IA Math. 28, 979--994 (1981; Zbl 0507.12010)] conjectured for \(r=1,2,\dots \) a formula for the ratio \[ \frac{L_p^{(r)}(\chi\omega,0)}{L(\chi,0)}, \] where \(L_p\) is the \(p\)-adic \(L\)-function (see [\textit{P. Cassou-Noguès}, Inv. Math, 51, 29--59 (1979; Zbl 0408.12015); \textit{P. Deligne} and \textit{K. Ribet}, Inv. Math. 59, 227--286 (1980; Zbl 0434.12009)]. In the case \(r=1\) this conjecture has been established by the work of the first author et al. [Ann. Math. (2) 174, 439--484 (2011; Zbl 1250.11099); \textit{K. Ventullo}, Comm. Math. Helv. 90, 939--963 (2015; Zbl 1377.11113)] and in the general case a proof has been given by the authors and \textit{K. Ventullo} [Ann. Math. (2) 188, 833--870 (2018; Zbl 1416.11160)]. Now the authors present a simpler proof in the case \(r=1\) when \(F\) is the maximal real subfield of a \(CM\) field \(K\), \(\chi\) is the nontrivial character of the Galois group of \(K/F\) and there is only one prime ideal over \(p\) in \(F\). The novelty of the proof is described by the authors in the abstract in the following way: ``The proof given in this note is much simpler as it does not use the theory of \(p\)-adic Galois cohomology and Galois representations associated to \(p\)-adic modular forms. Instead, the proof relies on a certain explicit construction using Theta series, congruences with Eisenstein series and the \(q\)-expansion principle of Deligne-Ribet''. For the entire collection see [Zbl 1462.11006].
scientific article; zbMATH DE number 7373072