Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-27T20:55:37.782Z Has data issue: false hasContentIssue false
  • English
  • Français

Trivial zeros of $p$-adic $L$-functions at near-central points

Part of: Arithmetic algebraic geometry Algebraic number theory: global fields Algebraic number theory: local and $p$-adic fields Discontinuous groups and automorphic forms (Co)homology theory

Published online by Cambridge University Press:  24 September 2013

Denis Benois
Denis Benois*
Affiliation:
Institut de Mathématiques de Bordeaux, UMR 5251, Université de Bordeaux, 351 cours de la Libération, F-33400 Talence, France ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Using the $\ell $-invariant constructed in our previous paper we prove a Mazur–Tate–Teitelbaum-style formula for derivatives of $p$-adic $L$-functions of modular forms at trivial zeros. The novelty of this result is to cover the near-central point case. In the central point case our formula coincides with the Mazur–Tate–Teitelbaum conjecture proved by Greenberg and Stevens and by Kato, Kurihara and Tsuji at the end of the 1990s.

Keywords

modular formp-adic representation(φ,Γ)-modulep-adic L-function

MSC classification

Secondary: 11R23: Iwasawa theory 11F80: Galois representations 11F85: $p$-adic theory, local fields 11S25: Galois cohomology 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14F30: $p$-adic cohomology, crystalline cohomology
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 13 , Issue 3 , July 2014 , pp. 561 - 598
Copyright
©Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Amice, Y. and Vélu, J., Distributions p-adiques associées aux séries de Hecke, in Journées Arithmétiques de Bordeaux (Bordeaux 1974), Astérisque, Volume 24–25, pp. 119–131 (Société Mathématique de France, Paris, 1975).Google Scholar
Atkin, A. O. L. and Li, W., Twists of newforms and pseudo-eigenvalues of $W$ -operators, Invent. Math. 48 (1978), 221–243.Google Scholar
Bellaïche, J., Critical p-adic $L$ -functions, Invent. Math. 189 (2012), 1–60.Google Scholar
Bellaïche, J. and Chenevier, G., p-adic families of Galois representations and higher rank Selmer groups, Astérisque, Volume 324 (Société Mathématique de France, Paris, 2009).Google Scholar
Benois, D., On Iwasawa theory of crystalline representations, Duke Math. J. 104 (2000), 211–267.CrossRefGoogle Scholar
Benois, D., Infinitesimal deformations and the $\ell $ -invariant, (extra volume: Andrei A. Suslin’s sixtieth birthday) Doc. Math. (2010), 5–31.Google Scholar
Benois, D., A generalization of Greenberg’s $\mathscr{L} $ -invariant, Amer. J. Math. 133 (2011), 1573–1632.Google Scholar
Benois, D. and Berger, L., Théorie d’Iwasawa des représentations cristallines II, Comment. Math. Helv. 83 (2008), 603–677.Google Scholar
Berger, L., Représentations p-adiques et équations différentielles, Invent. Math. 148 (2002), 219–284.Google Scholar
Berger, L., Bloch and Kato’s exponential map: three explicit formulas, (extra volume: Kazuya Kato’s fiftieth birthday) Doc. Math. (2003), 99–129.Google Scholar
Berger, L., Limites de représentations cristallines, Compositio Math. 140 (2004), 1473–1498.Google Scholar
Berger, L., Equations différentielles p-adiques et $(\varphi , N)$ -modules filtrés, in Représentations p-adiques de groupes p-adiques. I. Représentations Galoisiennes et (φ,Γ)-modules, Astérisque, Volume 319, pp. 13–38 (Société Mathématique de France, Paris, 2008).Google Scholar
Bloch, S. and Kato, K., $L$ -functions and Tamagawa numbers of motives, in Grothendieck Festschrift, Volume 1, Progress in Mathematics, Volume 86, pp. 333–400 (Birkhäuser, Boston, MA, 1990).Google Scholar
Carayol, H., Sur les représentations $l$ -adiques associées aux formes modulaires de Hilbert, Ann. Sci. Éc. Norm. Supér. 19 (1986), 409–468.Google Scholar
Cherbonnier, F. and Colmez, P., Représentations p-adiques surconvergentes, Invent. Math. 133 (1998), 581–611.Google Scholar
Cherbonnier, F. and Colmez, P., Théorie d’Iwasawa des représentations p-adiques d’un corps local, J. Amer. Math. Soc. 12 (1999), 241–268.Google Scholar
Coleman, R., A p-adic Shimura isomorphism and p-adic periods of modular forms, in p-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston MA 1991), Contemporary Mathematics, Volume 165, pp. 21–51 (American Mathematical Society, Providence, RI, 1994).Google Scholar
Coleman, R. and Iovita, A., Hidden structures on semi-stable curves, in Représentations p-adiques de groupes p-adiques. III. Global and geometrical methods, Astérisque, Volume 331, pp. 179–254 (Société Mathématique de France, Paris, 2010).Google Scholar
Colmez, P., Théorie d’Iwasawa des représentations de de Rham d’un corps local, Ann. of Math. (2) 148 (1998), 485–571.Google Scholar
Colmez, P., Représentations crystallines et représentations de hauteur finie, J. Reine Angew. Math. 514 (1999), 119–143.Google Scholar
Colmez, P., La conjecture de Birch et Swinnerton-Dyer p-adique, in Séminaire Bourbaki 2002/03, Astérisque, Volume 294, pp. 251–319 (Société Mathématique de France, Paris, 2004).Google Scholar
Colmez, P., Zéros supplémentaires de fonctions $L~p$ -adiques de formes modulaires, in Algebra and Number Theory, pp. 193–210 (Hindustan Book Agency, Delhi, 2005).CrossRefGoogle Scholar
Colmez, P., Représentations triangulines de dimension 2, in Représentations p-adiques de Groupes p-adiques. I. Représentations Galoisiennes et (φ,Γ)-modules, Astérisque, Volume 319, pp. 213–258 (Société Mathématique de France, Paris, 2008).Google Scholar
Colmez, P., Fonctions d’une variable p-adique, in Représentations p-adiques de Groupes p-adiques. II. Représentations de GL 2(Q p) et (φ,Γ)-modules, Astérisque, Volume 330, pp. 13–59 (Société Mathématique de France, Paris, 2010).Google Scholar
Dasgupta, S., Darmon, H. and Pollack, R., Hilbert modular forms and the Gross–Stark conjecture, Ann. of Math. (2) 174 (2011), 439–484.Google Scholar
Deligne, P., Formes modulaires et représentations $l$ -adiques, in Séminaire Bourbaki 1968/69, Lecture Notes in Mathematics, Volume 179, pp. 139–172 (Springer, 1971).Google Scholar
Deligne, P., La conjecture de Weil I, Publ. Math. Inst. Hautes Études Sci. 43 (1974), 273–307.Google Scholar
Ferrero, B. and Greenberg, R., On the behavior of p-adic $L$ -functions at $s= 0$ , Invent. Math. 50 (1978/79), 91–102.Google Scholar
Faltings, G., Hodge–Tate structures and modular forms, Math. Ann. 278 (1–4) (1987), 133–149.Google Scholar
Faltings, G., Crystalline cohomology and p-adic Galois representations, in Algebraic analysis, geometry and number theory (Baltimore, MD, 1988), pp. 25–80 (John Hopkins University Press, Baltimore MD, 1989).Google Scholar
Fontaine, J.-M., Représentations p-adiques des corps locaux, in Grothendieck Festschrift, Volume II, Progress in Mathematics, Volume 87, pp. 249–309 (Birkhäuser, Boston, MA, 1990).Google Scholar
Fontaine, J.-M., Le corps des périodes p-adiques, in Périodes p-adiques (Bures-sur-Yvette, 1988), Astérisque, Volume 223, pp. 59–102 (Société Mathématique de France, 1994).Google Scholar
Fontaine, J.-M., Représentations p-adiques semi-stables, in Périodes p-adiques (Bures-sur-Yvette, 1988), Astérisque, Volume 223, pp. 113–184 (Société Mathématique de France, 1994).Google Scholar
Fontaine, J.-M. and Perrin-Riou, B., Part 1 Autour des conjectures de Bloch et Kato; cohomologie galoisienne et valeurs de fonctions $L$ , in Motives (Seattle, WA, 1991), Proceedings of Symposia in Pure Mathematics, Volume 55, pp. 599–706 (American Mathematical Society, Providence, RI, 1994).Google Scholar
Greenberg, R., Iwasawa theory of p-adic representations, in Algebraic number theory, Advanced Studies in Pure Mathematics, Volume 17, pp. 97–137 (Academic Press, Boston, MA, 1989).Google Scholar
Greenberg, R., Trivial zeros of p-adic $L$ -functions, in p-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston MA 1991), Contemporary Mathematics, Volume 165, pp. 149–174 (American Mathematical Society, Providence, RI, 1994).Google Scholar
Greenberg, R. and Stevens, G., p-adic $L$ -functions and p-adic periods of modular forms, Invent. Math. 111 (1993), 407–447.Google Scholar
Gross, B., p-adic $L$ -series at $s= 0$ , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 979–994.Google Scholar
Gross, B. and Koblitz, N., Gauss sums and the p-adic $\Gamma $ -function, Ann. of Math. 109 (1979), 569–581.Google Scholar
Herr, L., Sur la cohomologie galoisienne des corps p-adiques, Bull. Soc. Math. France 126 (1998), 563–600.CrossRefGoogle Scholar
Herr, L., Une approche nouvelle de la dualité locale de Tate, Math. Ann. 320 (2001), 307–337.Google Scholar
Huber, A. and Kings, G., Bloch–Kato conjecture and main conjecture of Iwasawa theory for Dirichlet characters, Duke Math. J. 119 (2003), 393–464.CrossRefGoogle Scholar
Iwasawa, K., On explicit formulas for the norm residue symbol, J. Math. Soc. Japan 20 (1968), 151–165.Google Scholar
Jacquet, H. and Shalika, J., A non-vanishing theorem for zeta functions on ${\mathrm{GL} }_{n} $ , Invent. Math. 38 (1976), 1–16.Google Scholar
Kato, K., Iwasawa theory and p-adic Hodge theory, Kodai Math. J. 16 (1993), 1–31.Google Scholar
Kato, K., p-adic Hodge theory and values of zeta-functions of modular forms, in Cohomologies p-adiques et applications arithmétiques III, Astérisque, Volume 295, pp. 117–290 (Société Mathématique de France, 2004).Google Scholar
Kato, K., Kurihara, M. and Tsuji, T., Local Iwasawa theory of Perrin-Riou and syntomic complexes, preprint, 1996.Google Scholar
Kedlaya, K., A p-adic monodromy theorem, Ann. of Math. (2) 160 (2004), 93–184.Google Scholar
Kedlaya, K., Pottharst, J. and Xiao, L., Cohomology of arithmetic families of $(\varphi , \Gamma )$ -modules, preprint, 2002, 66 pages (available on http://arxiv.org/abs/1203.5718).Google Scholar
Langlands, R. P., Modular forms and $l$ -adic representations, in Modular forms of one variable II, Lecture Notes in Mathematics, Volume 349, pp. 361–500 (Springer, 1973).Google Scholar
Li, W., Newforms and functional equations, Math. Ann. 212 (1975), 285–315.CrossRefGoogle Scholar
Liu, R., Cohomology and duality for $(\varphi , \Gamma )$ -modules over the Robba ring, Int. Math. Res. Not. 3 (2008), Art. ID rnm150, 32 pages.Google Scholar
Manin, Y., Periods of cusp forms and p-adic Hecke series, Math. USSR Sb. 92 (1973), 371–393.CrossRefGoogle Scholar
Mazur, B., On monodromy invariants occurring in global arithmetic and Fontaine’s theory, in p-adic Monodromy and the Birch and Swinnerton-Dyer conjecture (Boston MA 1991), Contemporary Mathematics, Volume 165, pp. 1–20 (American Mathematical Society, Providence, RI, 1994).Google Scholar
Mazur, B., Tate, J. and Teitelbaum, J., On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), 1–48.Google Scholar
Miyake, T., Modular forms, Springer Monographs in Mathematics (Springer, 2006).Google Scholar
Nakamura, K., Iwasawa theory of de Rham $(\varphi , \Gamma )$ -modules over the Robba ring, J. Inst. Math. Jussieu. 54 pages, to appear, available on CJO2013. doi: 10.1017/S1474748013000078.Google Scholar
Nakamura, K., A generalization of Kato’s local $\varepsilon $ -conjecture for $(\varphi , \Gamma )$ -modules, preprint, 2013, 74 pages (available on http://arxiv.org/abs/1305.0880).Google Scholar
Nekovář, J., On p-adic height pairing, in Séminaire de Théorie des Nombres, Paris 1990/91, Progress in Mathematics, Volume 108, pp. 127–202 (Birkhäuser, Boston, 1993).CrossRefGoogle Scholar
Ogg, A., On the eigenvalues of Hecke operators, Math. Ann. 179 (1969), 101–108.Google Scholar
Orton, L., On exceptional zeros of p-adic $L$ -functions associated to modular forms, PhD thesis, University of Nottingham (2003).Google Scholar
Perrin-Riou, B., Théorie d’Iwasawa et hauteurs p-adiques, Invent. Math. 109 (1992), 137–185.Google Scholar
Perrin-Riou, B., Théorie d’Iwasawa des représentations p-adiques sur un corps local, Invent. Math. 115 (1994), 81–149.Google Scholar
Perrin-Riou, B., La fonction $L~p$ -adique de Kubota–Leopoldt, in Algebraic geometry (Tempe, AZ, 1993), Contemporary Mathematics, Volume 174, pp. 65–93 (American Mathematical Society, Providence, RI, 1994).Google Scholar
Perrin-Riou, B., Fonctions $L~p$ -adiques des représentations p-adiques, Astérisque, Volume 229 (Société Mathématique de France, 1995).Google Scholar
Perrin-Riou, B., Zéros triviaux des fonctions $L~p$ -adiques, Compositio Math. 114 (1998), 37–76.Google Scholar
Perrin-Riou, B., Quelques remarques sur la théorie d’Iwasawa des courbes elliptiques, in Number theory for millennium, III (Urbana, IL, 2000), pp. 119–147 (A K Peters, Natick, MA, 2002).Google Scholar
Perrin-Riou, B., Théorie d’Iwasawa des représentations p-adiques semistables, Mémoires de la Société Mathématique de France (nouvelle série), Volume 84 (Société Mathématique de France, 2001).Google Scholar
Pollack, R. and Stevens, G., Critical slope p-adic $L$ -functions, J. Lond. Math. Soc. (2) 87 (2013), 428–452.Google Scholar
Pottharst, J., Cyclotomic Iwasawa theory of motives, preprint, 2012.Google Scholar
Riedel, A., On Perrin-Riou’s exponential map and reciprocity laws, PhD thesis, University of Heidelberg (2013).Google Scholar
Rubin, K., Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 64 (1981), 455–470.Google Scholar
Saito, T., Modular forms and p-adic Hodge theory, Invent. Math. 129 (1997), 607–620.CrossRefGoogle Scholar
Stevens, G., Coleman’s $\mathscr{L} $ -invariant and families of modular forms, in Représentations p-adiques de groupes p-adiques. III. Global and geometrical methods, Astérisque, Volume 331, pp. 1–12 (Société Mathématique de France, Paris, 2010).Google Scholar
Tsuji, T., p-adic etale cohomology and crystalline cohomology in the semistable reduction case, Invent. Math. 137 (1999), 233–411.Google Scholar
Vishik, M., Non Archimedean measures connected with Dirichlet series, Math. USSR Sb. 28 (1976), 216–228.Google Scholar
Wach, N., Représentations p-adiques potentiellement cristallines, Bull. Soc. Math. France 124 (1996), 375–400.Google Scholar