Algebra & Number Theory

$(\varphi,\Gamma)$-modules de de Rham et fonctions $L$ $p$-adiques

Joaquín Rodrigues Jacinto

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

On développe une variante des méthodes de Coleman et Perrin-Riou permettant, pour une représentation galoisienne de de Rham, construire des fonctions L p -adiques à partir d’un système compatible d’éléments globaux. On obtient de la sorte des fonctions analytiques sur un ouvert de l’espace des poids contenant les caractères localement algébriques de conducteur assez grand. Appliqué au système d’Euler de Kato, cela fournit des fonctions L p -adiques pour les courbes elliptiques à mauvaise réduction additive et, plus généralement, pour les formes modulaires supercuspidales en p . En dimension 2 , nous prouvons une équation fonctionnelle pour nos fonctions L p -adiques.

We develop a variant of Coleman and Perrin-Riou’s methods giving, for a de Rham p -adic Galois representation, a construction of p -adic L -functions from a compatible system of global elements. As a result, we construct analytic functions on an open set of the p -adic weight space containing all locally algebraic characters of large enough conductor. Applied to Kato’s Euler system, this gives p -adic L -functions for elliptic curves with additive bad reduction and, more generally, for modular forms which are supercuspidal at p . In the case of dimension 2 , we prove a functional equation for our p -adic L -functions.

Article information

Source
Algebra Number Theory, Volume 12, Number 4 (2018), 885-934.

Dates
Received: 18 February 2017
Revised: 10 January 2018
Accepted: 23 February 2018
First available in Project Euclid: 28 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.ant/1532743325

Digital Object Identifier
doi:10.2140/ant.2018.12.885

Mathematical Reviews number (MathSciNet)
MR3830206

Zentralblatt MATH identifier
06911689

Subjects
Primary: 11S40: Zeta functions and $L$-functions [See also 11M41, 19F27]
Secondary: 11R23: Iwasawa theory 11S37: Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E50]

Keywords
$p$-adic $L$-functions modular forms $(\varphi,\Gamma)$-modules $p$-adic Hodge theory

Citation

Rodrigues Jacinto, Joaquín. $(\varphi,\Gamma)$-modules de de Rham et fonctions $L$ $p$-adiques. Algebra Number Theory 12 (2018), no. 4, 885--934. doi:10.2140/ant.2018.12.885. https://projecteuclid.org/euclid.ant/1532743325


Export citation

References

  • Y. Amice and J. Vélu, “Distributions $p$-adiques associées aux séries de Hecke”, pp. 119–131 in Journées Arithmétiques de Bordeaux, Astérisque 24-25, Soc. Math. France, Paris, 1975.
  • J. Bellaïche, “Critical $p$-adic $L$-functions”, Invent. Math. 189:1 (2012), 1–60.
  • L. Berger, “Représentations $p$-adiques et équations différentielles”, Invent. Math. 148:2 (2002), 219–284.
  • L. Berger, “Bloch and Kato's exponential map: three explicit formulas”, Doc. Math. Kazuya Kato's Sixtieth Birthday volume (2003), 99–129.
  • L. Berger, “Équations différentielles $p$-adiques et $(\phi,N)$-modules filtrés”, pp. 13–38 in Représentations $p$-adiques de groupes $p$-adiques, I: Représentations galoisiennes et $(\phi,\Gamma)$-modules, Astérisque 319, Soc. Math. France, Paris, 2008.
  • L. Berger and C. Breuil, “Sur quelques représentations potentiellement cristallines de ${\rm GL}_2(\mathbb{Q}_p)$”, pp. 155–211 in Astérisque 330, Soc. Math. France, Paris, 2010.
  • A. Besser, “Syntomic regulators and $p$-adic integration, I: Rigid syntomic regulators”, Israel J. Math. 120:2 (2000), 291–334.
  • A. A. Beĭlinson, “Higher regulators of modular curves”, pp. 1–34 in Applications of algebraic $K$-theory to algebraic geometry and number theory, I (Boulder, CO, 1983), edited by S. J. Bloch et al., Contemp. Math. 55, Amer. Math. Soc., Providence, RI, 1986.
  • C. J. Bushnell and G. Henniart, The local Langlands conjecture for $\rm GL(2)$, Grundlehren der Math. Wissenschaften 335, Springer, 2006.
  • F. Cherbonnier and P. Colmez, “Représentations $p$-adiques surconvergentes”, Invent. Math. 133:3 (1998), 581–611.
  • F. Cherbonnier and P. Colmez, “Théorie d'Iwasawa des représentations $p$-adiques d'un corps local”, J. Amer. Math. Soc. 12:1 (1999), 241–268.
  • P. Colmez, “Théorie d'Iwasawa des représentations de de Rham d'un corps local”, Ann. of Math. $(2)$ 148:2 (1998), 485–571.
  • P. Colmez, “La conjecture de Birch et Swinnerton-Dyer $p$-adique”, pp. 251–319 in Astérisque 294, Soc. Math. France, Paris, 2004.
  • P. Colmez, “Espaces vectoriels de dimension finie et représentations de de Rham”, pp. 117–186 in Représentations $p$-adiques de groupes $p$-adiques, I: Représentations galoisiennes et $(\phi,\Gamma)$-modules, Astérisque 319, Soc. Math. France, Paris, 2008.
  • P. Colmez, “Représentations de ${\rm GL}_2(\mathbb{Q}_p)$ et $(\phi,\Gamma)$-modules”, pp. 281–509 in Astérisque 330, Soc. Math. France, Paris, 2010.
  • P. Colmez and G. Dospinescu, “Complétés universels de représentations de ${\rm GL}_2(\mathbb{Q}_p)$”, Algebra Number Theory 8:6 (2014), 1447–1519.
  • P. Colmez and W. Nizioł, “Syntomic complexes and $p$-adic nearby cycles”, Invent. Math. 208:1 (2017), 1–108.
  • D. Delbourgo, “On the $p$-adic Birch, Swinnerton-Dyer conjecture for non-semistable reduction”, J. Number Theory 95:1 (2002), 38–71.
  • C. Deninger and A. J. Scholl, “The Beĭlinson conjectures”, pp. 173–209 in $L$-functions and arithmetic (Durham, UK, 1989), edited by J. Coates and M. J. Taylor, London Math. Soc. Lecture Note Ser. 153, Cambridge Univ. Press, 1991.
  • J.-M. Fontaine, “Représentations $p$-adiques des corps locaux, I”, pp. 249–309 in The Grothendieck Festschrift, II, edited by P. Cartier et al., Progr. Math. 87, Birkhäuser, Boston, 1990.
  • M. T. Gealy, “Special values of $p$-adic $l$ functions associated to modular forms”, unpublished manuscript, 2003.
  • M. T. Gealy, On the Tamagawa number conjecture for motives attached to modular forms, Ph.D. thesis, California Institute of Technology, 2006, https://search.proquest.com/docview/305351198.
  • M. Gros, “Régulateurs syntomiques et valeurs de fonctions $L$ $p$-adiques, I”, Invent. Math. 99:2 (1990), 293–320.
  • L. Herr, “Une approche nouvelle de la dualité locale de Tate”, Math. Ann. 320:2 (2001), 307–337.
  • K. Kato, “$p$-adic Hodge theory and values of zeta functions of modular forms”, pp. 117–290 in Cohomologies $p$-adiques et applications arithmétiques, III, Astérisque 295, Soc. Math. France, Paris, 2004.
  • K. S. Kedlaya, “A $p$-adic local monodromy theorem”, Ann. of Math. $(2)$ 160:1 (2004), 93–184.
  • K. S. Kedlaya, J. Pottharst, and L. Xiao, “Cohomology of arithmetic families of $(\varphi,\Gamma)$-modules”, J. Amer. Math. Soc. 27:4 (2014), 1043–1115.
  • R. Liu, “Cohomology and duality for $(\phi,\Gamma)$-modules over the Robba ring”, Int. Math. Res. Not. 2008:3 (2008), art. id. rnm150.
  • D. Loeffler and J. Weinstein, “On the computation of local components of a newform”, Math. Comp. 81:278 (2012), 1179–1200.
  • J. I. Manin, “Periods of cusp forms, and $p$-adic Hecke series”, Mat. Sb. $($N.S.$)$ 92(134):3 (1973), 378–401. In Russian; translated in Math. USSR-Sb. 21:3 (1973), 371–393.
  • B. Mazur, J. Tate, and J. Teitelbaum, “On $p$-adic analogues of the conjectures of Birch and Swinnerton-Dyer”, Invent. Math. 84:1 (1986), 1–48.
  • K. Nakamura, “Iwasawa theory of de Rham $(\varphi,\Gamma)$-modules over the Robba ring”, J. Inst. Math. Jussieu 13:1 (2014), 65–118.
  • K. Nakamura, “Local $\varepsilon$-isomorphisms for rank two $p$-adic representations of ${\rm Gal}(\overline{\mathbb{Q}_p/\mathbb{Q}_p)}$ and a functional equation of Kato's Euler system”, Camb. J. Math. 5:3 (2017), 281–368.
  • J. Nekovář and W. Nizioł, “Syntomic cohomology and $p$-adic regulators for varieties over $p$-adic fields”, Algebra Number Theory 10:8 (2016), 1695–1790.
  • W. Nizioł, “On the image of $p$-adic regulators”, Invent. Math. 127:2 (1997), 375–400.
  • B. Perrin-Riou, “Théorie d'Iwasawa des représentations $p$-adiques sur un corps local”, Invent. Math. 115:1 (1994), 81–161.
  • B. Perrin-Riou, Fonctions $L$ $p$-adiques des représentations $p$-adiques, Astérisque 229, Soc. Math. France, Paris, 1995.
  • R. Pollack and G. Stevens, “Critical slope $p$-adic $L$-functions”, J. Lond. Math. Soc. $(2)$ 87:2 (2013), 428–452.
  • J. Pottharst, “Cyclotomic Iwasawa theory of motives”, preprint, 2012, https://vbrt.org/writings/cyc.pdf.
  • J. Rodrigues Jacinto, “La conjecture $\epsilon$ locale de Kato en dimension $2$”, Math. Ann. (online publication March 2018).
  • R. Schmidt, “Some remarks on local newforms for $\rm GL(2)$”, J. Ramanujan Math. Soc. 17:2 (2002), 115–147.
  • P. Schneider and J. Teitelbaum, “Algebras of $p$-adic distributions and admissible representations”, Invent. Math. 153:1 (2003), 145–196.
  • A. J. Scholl, “Motives for modular forms”, Invent. Math. 100:2 (1990), 419–430.
  • M. M. Višik, “Nonarchimedean measures associated with Dirichlet series”, Mat. Sb. $($N.S.$)$ 99(141):2 (1976), 248–260. In Russian; translated in Math. USSR-Sb. 28:2 (1976), 216–228.