On adjoint Bloch–Kato Selmer groups for $$\textrm{GSp}_{2g}$$

Ju-Feng Wu1
1Mathematics Institute, University of Warwick, Coventry, UK

Tóm tắt

We study the adjoint Bloch–Kato Selmer groups attached to a classical point in the cuspidal eigenvariety associated with $$\textrm{GSp}_{2g}$$ . Our strategy is based on the study of families of Galois representations on the eigenvariety, which is inspired by the book of J. Bellaiche and G. Chenevier.

Tài liệu tham khảo

Avner Ash and Glenn Stevens. p-adic deformations of arithmetic cohomology. Preprint. Available at: http://math.bu.edu/people/ghs/preprints/Ash-Stevens-02-08.pdf. 2008. Joël Bellaïche and Gaëtan Chenevier. Families of Galois representations and Selmer groups. Astéérisque 324. Société mathématique de France, 2009. Joël Bellaïche. Critical p-adic L-functions. In: Inventiones mathematicae 189 (2012), pp. 1-60. https://doi.org/10.1007/s00222-011-0358-z. Thomas Barnet-Lamb, Toby Gee, David Geraghty and Richard Taylor. Potential automorphy and change of weight. In: Annals of Mathematics (2014). https://doi.org/10.4007/annals.2014.1 Gaëtan Chenevier. Familles p-adiques de formes automorphes pour GLn. In: Journal für die reine und angewandte Mathematik 2004.570 (2004), pp. 143-217. https://doi.org/10.1515/crll.2004.031/html Gaëtan Chenevier. The p-adic analytic space of pseudocharacters of a profinite group and pseudorepresentations over arbitrary rings. In: Automorphic Forms and Galois Representations. Ed. by Fred Diamond, Payman L. Kassaei and Minhyong Kim. Vol. 1. London Mathematical Society Lecture Note Series. Cambridge University Press, 2014, pp. 221-285. https://doi.org/10.1017/CBO9781107446335.008. Laurent Clozel, Michael Harris and Richard Taylor. Automorphy for some l-adic lifts of automorphic mod l Galois representations. In: Publications Mathématiques de lIHÉS 108 (2008), pp. 1-181. https://doi.org/10.1007/s10240-008-0016-1. Pierre Colmez. Représentations triangulines de dimension 2. In: Représentations p-adiques de groupes p-adiques I : représentations galoisiennes et (?, G)-modules. Ed. by Berger Laurent, Breuil Christophe and Colmez Pierre. Astérisque 319. Société mathématique de France, 2008. Andrea Conti. Galois level and congruence ideal for p-adic families of finite slope Siegel modular forms. In: Compositio Mathematica 155.4 (2019), pp. 776-831. https://doi.org/10.1112/S0010437X19007048. Fred Diamond, Matthias Flach and Li Guo. The Tamagawa number conjecture of adjoint motives of modular forms. In: Annales scientifiques de lÉcole Normale Supérieure Ser. 4, 37.5 (2004), pp. 663-727. https://doi.org/10.1016/j.ansens.2004.09.001. Luis Dieulefait and Adrián Zenteno. On the images of the Galois representations attached to generic automorphic representations of GSp(4). In: The Annali della Scuola Normale Superiore di Pisa, Classe di Scienze XX (2 2020), pp. 635-655. https://doi.org/10.2422/2036-2145.201609_01 Gerd Faltings. Hodge-Tate structures and modular forms. In: Math. Ann. 278.1-4 (1987), pp. 133-149. issn: 0025-5831. https://doi.org/10.1007/BF01458064. https://doi-org.lib-ezproxy.concordia.ca/10. Gred Faltings and Ching-Li Chai. Degeneration of Abelian Varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag Berlin Heidelberg, 1990. https://doi.org/10.1007/978-3-662-02632-8. William Fulton and Joe Harris. Representation Theory: a first course. Vol. 129. Graduate Texts in Mathematics. Springer, New York, 1991. https://doi.org/10.1007/978-1-4612-0979-9. Alain Genestier and Jacques Tilouine. Systèmes de Taylor-Wiles pour GSp4 . In: Formes automorphes (II) - Le cas du groupe GSp(4). Ed. by Jacques Tilouine, Henri Carayol, Michael Harris and Marie-France Vignéras. Astérisque 302. Société mathématique de France, 2005, pp. 177-290. David Hansen. Universal eigenvarieties, trianguline Galois representations, and p-adic Langlands functoriality. In: Journal für die reine und angewandte Mathematik (730 2017), pp. 1-64. https://doi.org/10.1515/crelle-2014-0130. Robert Harron and Andrei Jorza. On symmetric power L-invariants of Iwahori level Hilbert modular forms. In: American Journal of Mathematics 139.6 (2017), pp. 1605-1647. https://doi.org/10.1353/ajm.2017.0040. Haruzo Hida and Jacques Tilouine. Big image of Galois representations and congruence ideals. In: Arithmetic and Geometry. Ed. by Luis Dieulefait, Gerd Faltings, D. R. HeathBrown, Yu. V. Manin, B. Z. Moroz and Jean-PierreEditors Wintenberger. London Mathematical Society Lecture Note Series. Cambridge University Press, 2015, pp. 217-254. https://doi.org/10.1017/CBO9781316106877.014. David Hansen and Jack Thorne. On the GLn-eigenvariety and a conjecture of Venkatesh. In: Selecta Mathematica 23 (2017), pp. 1205-1234. https://doi.org/10.1007/s00029-017-0303-0. Christian Johansson and James Newton. Extended eigenvarieties for overconvergent cohomology. In: Algebra and Number Theory 13.1 (Feb. 2019), pp. 93-158. issn: 1937-0652. https://doi.org/10.2140/ant.2019.13.93. Mark Kisin. Overconvergent modular forms and the Fontaine-Mazur conjecture. In: Inventiones mathematicae 153 (2003). Arno Kret and Sug Woo Shin. Galois representations for the general symplectic group. To appear in Journal of the European Mathematical Society. Preprint available at: arXiv:1609.0422 2020. Chandrashekhar B. Khare and Jack A. Thorne. In: American Journal of Mathematics 139.5 (2017), pp. 1205-1273. https://doi.org/10.1353/ajm.2017.0030. Ruochuan Liu. Triangulation of refined families. In: Commentarii Mathematici Helvetici 90.4 (2015), pp. 831-904. https://doi.org/10.4171/CMH/372. James Newton and Jack Thorne. Adjoint Selmer groups of automorphic Galois representations of unitary type. In: Journal of the European Mathematical Society (2020). https://doi.org/10.17863/CAM.55603. James Newton and Jack Thorne. Symmetric power functoriality for holomorphic modular forms. In: Publications mathématiques de lIHÉS 134 (2021), pp. 1-116. https://doi.org/10.1007/s10240-021-00127-3. Stefan Patrikis and Richard Taylor. Automorphy and irreducibility of some l-adic representations. In: Compositio Mathematica 151.2 (2015), pp. 207-229. https://doi.org/10.1112/S0010437X14007519. Anthony J. Scholl. Motives for modular forms. In: Inventiones mathematicae 100.2 (1990), pp. 419-430. Jean-Pierre Serre. Local Fields. Vol. 67. Graduate Texts in Mathematics. Springer-Verlag New York, 1979. https://doi.org/10.1007/978-1-4757-5673-9. The Stacks project authors. The Stacks project. https://stacks.math.columbia.edu. 2022. Richard Taylor. Galois representations associated to Siegel modular forms of low weight. In: Duke Math. J. 63.2 (1991), pp. 281-332. https://doi.org/10.1215/S0012-7094-91-06312-X. Richard Taylor and Andrew Wiles. Ring-Theoretic Properties of Certain Hecke Algebras. In: Annals of Mathematics 141.3 (1995). Eric Urban. Sur les représentations p-adiques associées aux représentations cuspidales de GSp4/Q. In: Formes automorphes (II) - Le cas du groupe GSp(4). Ed. by Jacques Tilouine, Henri Carayol, Michael Harris and Marie-France Vignéras. Astérisque 302. Société mathématique de France, 2005, pp. 151-176. Rainer Weissauer. Four dimensional Galois representations. In: Formes automorphes (II) - Le cas du groupe GSp(4). Ed. by Jacques Tilouine, Henri Carayol, Michael Harris and Marie-France Vignéras. Astérisque 302. Société mathématique de France, 2005. Ju-Feng Wu. A pairing on the cuspidal eigenvariety for GSp2g and the ramification locus. In: Documenta Mathematica 26 (2021), pp. 675-711. https://doi.org/10.25537/dm.2021v26.675-711.