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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 Teichm├╝ller 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 \(r>1\).

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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[1] Beilinson, Alexander; Kings, Guido; Levin, Andrey, Topological polylogarithms and \(p\)-adic interpolation of {\(L\)}-values of totally real fields, Math. Ann.. Mathematische Annalen, 371, 1449-1495, (2018) · Zbl 1448.11122
[2] Burns, D., On derivatives of \(p\)-adic {\(L\)}-series at \(s=0\)
[3] Cassou-Nogu\`“es, Pierrette, Valeurs aux entiers n\'”egatifs des fonctions z\^eta et fonctions z\^eta {\(p\)}-adiques, Invent. Math.. Inventiones Mathematicae, 51, 29-59, (1979) · Zbl 0408.12015
[4] Charollois, Pierre; Dasgupta, Samit, Integral {E}isenstein cocycles on {\(\textbf{GL}_n\)}, {I}: {S}czech’s cocycle and {\(p\)}-adic {\(L\)}-functions of totally real fields, Camb. J. Math.. Cambridge Journal of Mathematics, 2, 49-90, (2014) · Zbl 1353.11074
[5] Dasgupta, Samit; Darmon, Henri; Pollack, Robert, Hilbert modular forms and the {G}ross-{S}tark conjecture, Ann. of Math. (2). Annals of Mathematics. Second Series, 174, 439-484, (2011) · Zbl 1250.11099
[6] Dasgupta, Samit; Spiess, M., On the characteristic polynomial of the {G}ross regulator matrix, (2020) · Zbl 1441.11284
[7] Deligne, Pierre; Ribet, Kenneth A., Values of abelian {\(L\)}-functions at negative integers over totally real fields, Invent. Math.. Inventiones Mathematicae, 59, 227-286, (1980) · Zbl 0434.12009
[8] Federer, Leslie Jane; Gross, Benedict H., Regulators and {I}wasawa modules, Invent. Math.. Inventiones Mathematicae, 62, 443-457, (1981) · Zbl 0468.12005
[9] Ferrero, Bruce; Greenberg, Ralph, On the behavior of {\(p\)}-adic {\(L\)}-functions at {\(s=0\)}, Invent. Math.. Inventiones Mathematicae, 50, 91-102, (1978/79) · Zbl 0441.12003
[10] Greenberg, Ralph, Trivial zeros of {\(p\)}-adic {\(L\)}-functions. {\(p\)}-adic Monodromy and the {B}irch and {S}winnerton-{D}yer Conjecture, Contemp. Math., 165, 149-174, (1994) · Zbl 0838.11070
[11] Gross, Benedict H., {\(p\)}-adic {\(L\)}-series at {\(s=0\)}, J. Fac. Sci. Univ. Tokyo Sect. IA Math.. Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics, 28, 979-994, (1981) · Zbl 0507.12010
[12] Gross, Benedict H.; Koblitz, Neal, Gauss sums and the {\(p\)}-adic {\(\Gamma \)}-function, Ann. of Math. (2). Annals of Mathematics. Second Series, 109, 569-581, (1979) · Zbl 0406.12010
[13] Hida, Haruzo, On {\(p\)}-adic {H}ecke algebras for {\({\rm GL}_2\)} over totally real fields, Ann. of Math. (2). Annals of Mathematics. Second Series, 128, 295-384, (1988) · Zbl 0658.10034
[14] Mazur, Barry, How can we construct abelian {G}alois extensions of basic number fields?, Bull. Amer. Math. Soc. (N.S.). American Mathematical Society. Bulletin. New Series, 48, 155-209, (2011) · Zbl 1228.11163
[15] Mazur, Barry; Wiles, A., Class fields of abelian extensions of {\(\textbf{Q}\)}, Invent. Math.. Inventiones Mathematicae, 76, 179-330, (1984) · Zbl 0545.12005
[16] Ribet, Kenneth A., A modular construction of unramified {\(p\)}-extensions of{\(Q(\mu \sb{p})\)}, Invent. Math.. Inventiones Mathematicae, 34, 151-162, (1976) · Zbl 0338.12003
[17] Serre, {\relax J-P}, Corps locaux, Publications de l’Institut de Math\'ematique de l’Universit\'e de Nancago, VIII, 243 pp., (1962)
[18] Skinner, C., Galois representations, {I}wasawa theory and special values of \({L}\)-functions
[19] Spie\ss, Michael, On special zeros of {\(p\)}-adic {\(L\)}-functions of {H}ilbert modular forms, Invent. Math.. Inventiones Mathematicae, 196, 69-138, (2014) · Zbl 1392.11027
[20] Spiess, Michael, Shintani cocycles and the order of vanishing of {\(p\)}-adic {H}ecke {\(L\)}-series at {\(s=0\)}, Math. Ann.. Mathematische Annalen, 359, 239-265, (2014) · Zbl 1307.11125
[21] Tate, John, Les Conjectures de {S}tark sur les Fonctions {\(L\)} d’{A}rtin en {\(s=0\)}, Progr. Math., 47, 143 pp., (1984) · Zbl 0545.12009
[22] Ventullo, Kevin, On the rank one abelian {G}ross-{S}tark conjecture, Comment. Math. Helv.. Commentarii Mathematici Helvetici. A Journal of the Swiss Mathematical Society, 90, 939-963, (2015) · Zbl 1377.11113
[23] Wiles, A., On ordinary {\(\lambda\)}-adic representations associated to modular forms, Invent. Math.. Inventiones Mathematicae, 94, 529-573, (1988) · Zbl 0664.10013
[24] Wiles, A., The {I}wasawa conjecture for totally real fields, Ann. of Math. (2). Annals of Mathematics. Second Series, 131, 493-540, (1990) · Zbl 0719.11071
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