L-invariants; p-adic L-functions; Galois representations; weight one modular forms
Résumé :
[en] We compute Benois L-invariants of weight 1 cuspforms and of their adjoint representations and show how this extends Gross’ p-adic regulator to Artin motives which are not critical in the sense of Deligne. Benois’ construction depends on the choice of a regular submodule which is well understood when the representation is p-regular, as it then amounts to the choice of a “motivic” p-refinement. The situation is dramatically different in the p-irregular case, where the regular submodules are parametrized by a flag variety and thus depend on continuous parameters. We are nevertheless able to show in some examples, how Hida theory and the geometry of the eigencurve can be used to detect a finite number of choices of arithmetic and “mixed-motivic” significance.
Disciplines :
Mathématiques
Auteur, co-auteur :
Dimitrov, Mladen; University of Lille
MAKSOUD, Alexandre ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)
Co-auteurs externes :
yes
Langue du document :
Anglais
Titre :
L-invariants of Artin motives
Date de publication/diffusion :
27 juillet 2022
Titre du périodique :
Annales mathématiques du Québec
Peer reviewed :
Peer reviewed
Projet FnR :
FNR12589973 - Galois Representations, Automorphic Forms And Their L-functions, 2018 (01/02/2019-31/08/2024) - Gabor Wiese
J. Bellaïche, The eigenbook—eigenvarieties, families of Galois representations, p -adic L -functions, Pathways in Mathematics, Birkhäuser/Springer, Cham, 2021.
J. Bellaïche and M. Dimitrov, On the eigencurve at classical weight 1 points, Duke Math. J., 165 (2016), pp. 245–266. DOI: 10.1215/00127094-3165755
D. Benois, A generalization of Greenberg’s L -invariant, Amer. J. Math., 133 (2011), pp. 1573–1632. DOI: 10.1353/ajm.2011.0043
D. Benois, On extra zeros of p -adic L -functions: the crystalline case, in Iwasawa theory 2012, vol. 7 of Contrib. Math. Comput. Sci., Springer, Heidelberg, 2014, pp. 65–133.
D. Benois and S. Horte, On extra zeros of p-adic Rankin-Selberg L -functions, arXiv:2009.01096, (2020).
L. Berger, Équations différentielles p -adiques et (ϕ, N) -modules filtrés, Astérisque, 319 (2008), pp. 13–38. Représentations p -adiques de groupes p -adiques. I. Représentations galoisiennes et (ϕ, Γ) -modules.
A. Betina and M. Dimitrov, Geometry of the eigencurve at CM points and trivial zeros of Katz p -adic L -functions, Adv. Math., 384 (2021), Article no. 107724.
A. Betina and M. Dimitrov, A geometric view on Iwasawa theory, J. Théorie Nombres Bordeaux, 33 (2021), pp. 703–731. DOI: 10.5802/jtnb.1176
W. Bley, Equivariant Tamagawa number conjecture for abelian extensions of a quadratic imaginary field, Doc. Math., 11 (2006), pp. 73–118. DOI: 10.4171/dm/205
S. Bloch and K. Kato, L -functions and Tamagawa numbers of motives, in The Grothendieck Festschrift, Vol. I, vol. 86 of Progr. Math., Birkhäuser Boston, Boston, MA, 1990, pp. 333–400.
A. Brumer, On the units of algebraic number fields, Mathematika, 14 (1967), pp. 121–124. DOI: 10.1112/S0025579300003703
K. Buyukboduk and R. Sakamoto, On the non-critical exceptional zeros of Katz p -adic L -functions for CM fields, Adv. Math. 406 (2022), Article no. 108478.
F. Calegari and B. Mazur, Nearly ordinary Galois deformations over arbitrary number fields, J. Inst. Math. Jussieu, 8 (2009), pp. 99–177. DOI: 10.1017/S1474748008000327
M. Chida and M.-L. Hsieh The derivative formula of p-adic L-functions for imaginary quadratic fields at trivial zeros, Ann. Math. Québec, (2022). https://doi.org/10.1007/s40316-022-00198-6.
P. Colmez, Fonctions L p -adiques, Astérisque, 266 (2000), pp. Exp. No. 851, 3, 21–58. Séminaire Bourbaki, Vol. 1998/99.
H. Darmon, A. Lauder, and V. Rotger, Gross-Stark units and p -adic iterated integrals attached to modular forms of weight one, Ann. Math. Qué., 40 (2016), pp. 325–354. DOI: 10.1007/s40316-015-0042-6
H. Darmon, A. Lauder, and V. Rotger, First order p -adic deformations of weight one newforms, in L-functions and automorphic forms, vol. 10 of Contrib. Math. Comput. Sci., Springer, Cham, 2017, pp. 39–80.
S. Dasgupta, Factorization of p -adic Rankin L -series, Invent. Math., 205 (2016), pp. 221–268. DOI: 10.1007/s00222-015-0634-4
P. Deligne and K. A. Ribet, Values of abelian L -functions at negative integers over totally real fields, Invent. Math., 59 (1980), pp. 227–286. DOI: 10.1007/BF01453237
L. J. Federer and B. H. Gross, Regulators and Iwasawa modules, Invent. Math., 62 (1981), pp. 443–457. With an appendix by Warren Sinnott.
B. Ferrero and R. Greenberg, On the behavior of p -adic L -functions at s = 0, Invent. Math., 50 (1978/79), pp. 91–102.
J.-M. Fontaine, Sur certains types de représentations p -adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate, Ann. of Math. (2), 115 (1982), pp. 529–577. DOI: 10.2307/2007012
R. Greenberg, On a certain l -adic representation, Invent. Math., 21 (1973), pp. 117–124. DOI: 10.1007/BF01389691
R. Greenberg, On p -adic Artin L -functions II, in Iwasawa theory 2012, vol. 7 of Contrib. Math. Comput. Sci., Springer, Heidelberg, 2014, pp. 227–245.
B. H. Gross, p -adic L -series at s = 0, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28 (1981), pp. 979–994 (1982).
L. Herr, Sur la cohomologie galoisienne des corps p -adiques, Bull. Soc. Math. France, 126 (1998), pp. 563–600. DOI: 10.24033/bsmf.2337
H. Hida, A p -adic measure attached to the zeta functions associated with two elliptic modular forms. II, Ann. Inst. Fourier (Grenoble), 38 (1988), pp. 1–83. DOI: 10.5802/aif.1141
A. Maksoud, Théorie d’Iwasawa des motifs d’Artin et des formes modulaires de poids 1, arXiv:1811.05368, (2019).
B. Perrin-Riou, Fonctions Lp -adiques des représentations p -adiques, Astérisque, 229 (1995), p. 198.
O. Rivero and V. Rotger, Derived Beilinson-Flach elements and the arithmetic of the adjoint of a modular form, J. Eur. Math. Soc. (JEMS), 23 (2021), pp. 2299–2335. DOI: 10.4171/JEMS/1054
K. Rubin, Euler systems, vol. 147 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 2000. Hermann Weyl Lectures. The Institute for Advanced Study.
C.-G. Schmidt, p -adic measures attached to automorphic representations of GL (3), Invent. Math., 92 (1988), pp. 597–631. DOI: 10.1007/BF01393749
J.-P. Serre, Cohomologie galoisienne, vol. 5 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, fifth ed., 1994.
J. Tate, Les conjectures de Stark sur les fonctions L d’Artin en s = 0, vol. 47 of Progress in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 1984. Lecture notes edited by Dominique Bernardi and Norbert Schappacher.
