Algebra & Number Theory

Extended eigenvarieties for overconvergent cohomology

Christian Johansson and James Newton

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/ant.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Recently, Andreatta, Iovita and Pilloni constructed spaces of overconvergent modular forms in characteristic p, together with a natural extension of the Coleman–Mazur eigencurve over a compactified (adic) weight space. Similar ideas have also been used by Liu, Wan and Xiao to study the boundary of the eigencurve. This all goes back to an idea of Coleman.

In this article, we construct natural extensions of eigenvarieties for arbitrary reductive groups G over a number field which are split at all places above p. If G is GL2, then we obtain a new construction of the extended eigencurve of Andreatta–Iovita–Pilloni. If G is an inner form of GL2 associated to a definite quaternion algebra, our work gives a new perspective on some of the results of Liu–Wan–Xiao.

We build our extended eigenvarieties following Hansen’s construction using overconvergent cohomology. One key ingredient is a definition of locally analytic distribution modules which permits coefficients of characteristic p (and mixed characteristic). When G is GLn over a totally real or CM number field, we also construct a family of Galois representations over the reduced extended eigenvariety.

Article information

Source
Algebra Number Theory, Volume 13, Number 1 (2019), 93-158.

Dates
Received: 15 June 2017
Revised: 28 June 2018
Accepted: 25 September 2018
First available in Project Euclid: 27 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.ant/1553652022

Digital Object Identifier
doi:10.2140/ant.2019.13.93

Mathematical Reviews number (MathSciNet)
MR3917916

Zentralblatt MATH identifier
07041707

Subjects
Primary: 11F33: Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50]
Secondary: 11F80: Galois representations

Keywords
$p$-adic automorphic forms $p$-adic modular forms eigenvarieties Galois representations

Citation

Johansson, Christian; Newton, James. Extended eigenvarieties for overconvergent cohomology. Algebra Number Theory 13 (2019), no. 1, 93--158. doi:10.2140/ant.2019.13.93. https://projecteuclid.org/euclid.ant/1553652022


Export citation

References

  • A. Abbes, Éléments de géométrie rigide, I: Construction et étude géométrique des espaces rigides, Progr. Math. 286, Springer, 2010.
  • F. Andreatta, A. Iovita, and V. Pilloni, “The adic, cuspidal, Hilbert eigenvarieties”, Res. Math. Sci. 3 (2016), art. id. 34.
  • F. Andreatta, A. Iovita, and V. Pilloni, “Le halo spectral”, Ann. Sci. Éc. Norm. Supér. $(4)$ 51:3 (2018), 603–655.
  • K. Ardakov and S. Wadsley, “On irreducible representations of compact $p$-adic analytic groups”, Ann. of Math. $(2)$ 178:2 (2013), 453–557.
  • A. Ash and G. Stevens, “$p$-adic deformations of arithmetic cohomology”, preprint, 2008, http://math.bu.edu/people/ghs/preprints/Ash-Stevens-02-08.pdf.
  • J. Bellaïche, “Eigenvarieties and $p$-adic $L$-functions”, preprint, 2010, http://people.brandeis.edu/~jbellaic/preprint/coursebook.pdf.
  • J. Bellaïche and G. Chenevier, Families of Galois representations and Selmer groups, Astérisque 324, Société Mathématique de France, Paris, 2009.
  • J. Bergdall and R. Pollack, “Arithmetic properties of Fredholm series for $p$-adic modular forms”, Proc. Lond. Math. Soc. $(3)$ 113:4 (2016), 419–444.
  • V. G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Math. Surveys and Monographs 33, Amer. Math. Soc., Providence, RI, 1990.
  • S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean analysis: a systematic approach to rigid analytic geometry, Grundlehren der Math. Wissenschaften 261, Springer, 1984.
  • C. Breuil, E. Hellmann, and B. Schraen, “Une interprétation modulaire de la variété trianguline”, Math. Ann. 367:3-4 (2017), 1587–1645.
  • K. Buzzard, “On $p$-adic families of automorphic forms”, pp. 23–44 in Modular curves and abelian varieties (Bellaterra, Spain, 2002), edited by J. Cremona et al., Progr. Math. 224, Birkhäuser, Basel, 2004.
  • K. Buzzard, “Eigenvarieties”, pp. 59–120 in $L$-functions and Galois representations (Durham, 2004), edited by D. Burns et al., London Math. Soc. Lecture Note Ser. 320, Cambridge Univ. Press, 2007.
  • K. Buzzard and L. J. P. Kilford, “The 2-adic eigencurve at the boundary of weight space”, Compos. Math. 141:3 (2005), 605–619.
  • G. Chenevier, “Familles $p$-adiques de formes automorphes pour ${\rm GL}_n$”, J. Reine Angew. Math. 570 (2004), 143–217.
  • G. Chenevier, “Une correspondance de Jacquet–Langlands $p$-adique”, Duke Math. J. 126:1 (2005), 161–194.
  • G. Chenevier, “The $p$-adic analytic space of pseudocharacters of a profinite group and pseudorepresentations over arbitrary rings”, pp. 221–285 in Automorphic forms and Galois representations, I (Durham, 2011), edited by F. Diamond et al., London Math. Soc. Lecture Note Ser. 414, Cambridge Univ. Press, 2014.
  • P. Chojecki, D. Hansen, and C. Johansson, “Overconvergent modular forms and perfectoid Shimura curves”, Doc. Math. 22 (2017), 191–262.
  • R. F. Coleman, “$p$-adic Banach spaces and families of modular forms”, Invent. Math. 127:3 (1997), 417–479.
  • R. Coleman and B. Mazur, “The eigencurve”, pp. 1–113 in Galois representations in arithmetic algebraic geometry (Durham, 1996), edited by A. J. Scholl and R. L. Taylor, London Math. Soc. Lecture Note Ser. 254, Cambridge Univ. Press, 1998.
  • P. Colmez, “Fonctions d'une variable $p$-adique”, pp. 13–59 Astérisque 330, Société Mathématique de France, Paris, 2010.
  • B. Conrad, “Irreducible components of rigid spaces”, Ann. Inst. Fourier $($Grenoble$)$ 49:2 (1999), 473–541.
  • J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic pro-$p$-groups, 2nd ed., Cambridge Studies in Adv. Math. 61, Cambridge Univ. Press, 1999.
  • A. Grothendieck, “Eléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas, II”, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 5–231.
  • A. Grothendieck, “Eléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas, III”, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 5–255.
  • D. R. Gulotta, Equidimensional adic eigenvarietes for groups with discrete series, Ph.D. thesis, Columbia University, 2018, https://search.proquest.com/docview/2031119725.
  • D. Hansen, “Iwasawa theory of overconvergent modular forms, I: Critical $p$-adic $L$-functions”, preprint, 2015.
  • D. Hansen, “Universal eigenvarieties, trianguline Galois representations, and $p$-adic Langlands functoriality”, J. Reine Angew. Math. 730 (2017), 1–64.
  • D. Hansen and C. Johansson, “Completed and overconvergent cohomology”, in preparation.
  • R. Huber, “Continuous valuations”, Math. Z. 212:3 (1993), 455–477.
  • R. Huber, “A generalization of formal schemes and rigid analytic varieties”, Math. Z. 217:4 (1994), 513–551.
  • R. Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Math. E30, Vieweg, Wiesbaden, 1996.
  • C. Johansson and J. Newton, “Irreducible components of extended eigenvarieties and interpolating Langlands functoriality”, 2017. To appear in Math. Res. Lett.
  • C. Johansson and J. Newton, “Parallel weight $2$ points on Hilbert modular eigenvarieties and the parity conjecture”, preprint, 2018.
  • K. S. Kedlaya and R. Liu, Relative $p$-adic Hodge theory: foundations, Astérisque 371, Société Mathématique de France, Paris, 2015.
  • R. Liu, D. Wan, and L. Xiao, “The eigencurve over the boundary of weight space”, Duke Math. J. 166:9 (2017), 1739–1787.
  • D. Loeffler, “Overconvergent algebraic automorphic forms”, Proc. Lond. Math. Soc. $(3)$ 102:2 (2011), 193–228.
  • J. Marot, “Sur les anneaux universellement japonais”, Bull. Soc. Math. France 103:1 (1975), 103–111.
  • J. Neukirch, Algebraic number theory, Grundlehren der Math. Wissenschaften 322, Springer, 1999.
  • J. Pottharst and L. Xiao, “On the parity conjecture in finite-slope families”, preprint, 2014.
  • P. Schneider and J. Teitelbaum, “Algebras of $p$-adic distributions and admissible representations”, Invent. Math. 153:1 (2003), 145–196.
  • P. Scholze, “On torsion in the cohomology of locally symmetric varieties”, Ann. of Math. $(2)$ 182:3 (2015), 945–1066.
  • J.-P. Serre, “Endomorphismes complètement continus des espaces de Banach $p$-adiques”, Inst. Hautes Études Sci. Publ. Math. 12 (1962), 69–85.
  • G. Stevens, “Rigid analytic modular symbols”, preprint, 1994, http://math.bu.edu/people/ghs/research.d/RigidSymbs.pdf.
  • E. Urban, “Eigenvarieties for reductive groups”, Ann. of Math. $(2)$ 174:3 (2011), 1685–1784.
  • P. Valabrega, “On the excellent property for power series rings over polynomial rings”, J. Math. Kyoto Univ. 15:2 (1975), 387–395.
  • P. Valabrega, “A few theorems on completion of excellent rings”, Nagoya Math. J. 61 (1976), 127–133.
  • C. A. Weibel, An introduction to homological algebra, Cambridge Studies in Adv. Math. 38, Cambridge Univ. Press, 1994.
  • Z. Xiang, “A construction of the full eigenvariety of a reductive group”, J. Number Theory 132:5 (2012), 938–952.