Spherical varieties and norm relations in Iwasawa theory

Loeffler, David

Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 3.2, pp. 1021-1043.

Les familles des classes de cohomologie compatibles pour l’application norme, définies pour les variétés de Shimura et pour d’autres espaces arithmétiques symétriques, jouent un rôle important dans la théorie d’Iwasawa des formes automorphes. Dans cette note, nous développons une approche systématique pour établir la compatibilité dans le cas « vertical » (c’est-à-dire dans le cas où le niveau ne change qu’en un nombre premier fixé $p$), à la fois pour la cohomologie de Betti et la cohomologie étale, en révélant une relation inattendue avec la théorie des variétés sphériques. Cette machinerie peut être utilisée pour construire de nouveaux exemples de telles familles, éventuellement donnant naissance à la fois à de nouvelles constructions des systèmes d’Euler et à de nouvelles fonctions $L$ $p$-adiques : par example, nous obtenons des familles anticyclotomiques de cycles algébriques sur les variétés de Shimura pour les groupes $U\left(n\right)\times U(n+1)$ et $U\left(2n\right)$.

Norm-compatible families of cohomology classes for Shimura varieties, and other arithmetic symmetric spaces, play an important role in Iwasawa theory of automorphic forms. The aim of this note is to give a systematic approach to proving “vertical” norm-compatibility relations for such classes (where the level varies at a fixed prime $p$), treating the case of Betti and étale cohomology at once, and revealing an unexpected relation to the theory of spherical varieties. This machinery can be used to construct many new examples of norm-compatible families, potentially giving rise to new constructions of both Euler systems and $p$-adic $L$-functions: examples include families of algebraic cycles on Shimura varieties for $U\left(n\right)\times U(n+1)$ and $U\left(2n\right)$ over the $p$-adic anticyclotomic tower.

Classification : 11F67, 11R23, 14M17
Mots-clés : Euler systems, norm relations, spherical varieties

[1] Bertolini, Massimo; Castella, Francesc; Darmon, Henri; Dasgupta, Samit; Prasanna, Kartik; Rotger, Victor $p$-adic $L$-functions and Euler systems: a tale in two trilogies, Automorphic forms and Galois representations. Vol. 1 (London Mathematical Society Lecture Note Series), Volume 414, Cambridge University Press, 2014, pp. 52-101 | DOI | MR | Zbl

[2] Boumasmoud, Réda; Brooks, Ernest Hunter; Jetchev, Dimitar Vertical distribution relations for special cycles on unitary Shimura varieties, Int. Math. Res. Not., Volume 2020 (2020) no. 13, pp. 3902-3926 | DOI | MR | Zbl

[3] Brion, Michel Classification des espaces homogènes sphériques, Compos. Math., Volume 63 (1987) no. 2, pp. 189-208 | Numdam

[4] Cauchi, Antonio Norm-compatible systems of cohomology classes for $GU(2,2)$, Int. J. Number Theory, Volume 16 (2020) no. 3, pp. 461-510 | DOI | MR | Zbl

[5] Cauchi, Antonio; Rodrigues Jacinto, Joaquín Norm-compatible systems of Galois cohomology classes for ${GSp}_6$ (2018) (https://arxiv.org/abs/1807.06512)

[6] Cornut, Christophe An Euler system of Heegner type (2018) (https://webusers.imj-prg.fr/~christophe.cornut/papers/ESHT.pdf)

[7] Darmon, Henri; Rotger, Victor Diagonal cycles and Euler systems I: a $p$-adic Gross–Zagier formula, Ann. Sci. Éc. Norm. Supér., Volume 47 (2014) no. 4, pp. 779-832 | DOI | MR | Zbl

[8] Deligne, Pierre Travaux de Shimura (Exp. No. 389), Séminaire Bourbaki, 23ème année (1970/71) (Lecture Notes in Mathematics), Volume 244, Springer, 1970, pp. 123-165 | DOI | Numdam

[9] Dimitrov, Mladen; Januszewski, Fabian; Raghuram, A. $L$-functions of $\mathrm{GL}\left(2n\right)$: $p$-adic properties and non-vanishing of twists, Compos. Math., Volume 156 (2020) no. 12, pp. 2347-2468 | MR | Zbl

[10] Emerton, Matthew On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms, Invent. Math., Volume 164 (2006) no. 1, pp. 1-84 | DOI | MR | Zbl

[11] Faltings, Gerd Arithmetic Eisenstein classes on the Siegel space: some computations, Number fields and function fields—two parallel worlds (Progress in Mathematics), Volume 239, Birkhäuser, 2005, pp. 0-8176 | MR | Zbl

[12] Graham, Andrew; Shah, Syed Waqar Ali Anticyclotomic Euler systems for unitary groups (2020) (https://arxiv.org/abs/2001.07825)

[13] Hsu, Chi-Yun; Jin, Zhaorong; Sakamoto, Ryotaro Euler systems for $\mathrm{GSp}\left(4\right)\times \mathrm{GL}\left(2\right)$ (2020) (https://arxiv.org/abs/2011.12894)

[14] Januszewski, Fabian Modular symbols for reductive groups and $p$-adic Rankin–Selberg convolutions over number fields, J. Reine Angew. Math., Volume 653 (2011), pp. 1-45 | DOI | MR | Zbl

[15] Jetchev, Dimitar; Loeffler, David; Zerbes, Sarah Livia Heegner points in Coleman families, Proc. Lond. Math. Soc., Volume 122 (2021) no. 1, pp. 124-152 | DOI | MR | Zbl

[16] Kato, Kazuya $p$-adic Hodge theory and values of zeta functions of modular forms, Cohomologies $p$-adiques et applications arithmétiques (III) (Astérisque), Volume 295, Société Mathématique de France, 2004, pp. 117-290 | Numdam | Zbl

[17] Kazhdan, David; Mazur, Barry; Schmidt, Claus-Günther Relative modular symbols and Rankin–Selberg convolutions, J. Reine Angew. Math., Volume 519 (2000), pp. 97-141 | MR | Zbl

[18] Lei, Antonio; Loeffler, David; Zerbes, Sarah Livia Euler systems for Rankin–Selberg convolutions of modular forms, Ann. Math., Volume 180 (2014) no. 2, pp. 653-771 | DOI | MR | Zbl

[19] Lei, Antonio; Loeffler, David; Zerbes, Sarah Livia Euler systems for Hilbert modular surfaces, Forum Math. Sigma, Volume 6 (2018), e23, 67 pages | DOI | MR | Zbl

[20] Loeffler, David; Skinner, Chris; Zerbes, Sarah Livia Euler systems for $\mathrm{GSp}\left(4\right)$, J. Eur. Math. Soc. (2021) (electronically published, https://doi.org/10.4171/JEMS/1124, print version to appear) | DOI

[21] Loeffler, David; Williams, Chris $p$-adic Asai $L$-functions of Bianchi modular forms, Algebra Number Theory, Volume 14 (2020) no. 7, pp. 1669-1710 | DOI | MR | Zbl

[22] Loeffler, David; Zerbes, Sarah Livia Rankin–Eisenstein classes in Coleman families, Res. Math. Sci., Volume 3 (2016), 29, 53 pages | DOI | MR | Zbl

[23] Milne, James S. Introduction to Shimura varieties, Harmonic analysis, the trace formula, and Shimura varieties (Clay Mathematics Proceedings), Volume 4, American Mathematical Society, 2005, pp. 265-378 | MR | Zbl

[24] Webb, Peter A guide to Mackey functors, Handbook of Algebra. Vol. 2, Volume 2, North-Holland, 2000, pp. 1570-7954 | DOI | MR | Zbl