zbMATH — the first resource for mathematics

Critical \(p\)-adic \(L\)-functions. (English) Zbl 1318.11067
There is a large body of work on the subject of \(p\)-adic \(L\)-functions of modular forms. See the classical work of B. Mazur and P. Swinnerton-Dyer [Invent. Math. 25, 1–61 (1974; Zbl 0281.14016)], and also see the papers [Yu. I. Manin, Mat. Sb., Nov. Ser. 92 (134), 378–401 (1973; Zbl 0293.14007); M. M. Vishik, Mat. Sb., Nov. Ser. 99 (141), 248–260 (1976; Zbl 0358.14014); Y. Amice and J. Vélu, Astérisque 24–25, 119–131 (1975; Zbl 0332.14010); B. Mazur et al., Invent. Math. 84, 1–48 (1986; Zbl 0699.14028); G. Stevens [“Rigid analytic modular symbols”, Preprint; with R. Pollack, Ann. Sci. Éc. Norm. Supér. (4) 44, No. 1, 1–42 (2011; Zbl 1268.11075); J. Lond. Math. Soc., II. Ser. 87, No. 2, 428–452 (2013; Zbl 1317.11051)].
By building on their work, the author constructs \(p\)-adic \(L\)-functions for modular forms which were not known to have associated \(p\)-adic \(L\)-functions. The work most closely associated with the paper under review seems to be [Zbl 1317.11051], in which Pollack and Stevens constructed a \(p\)-adic \(L\)-function for the \(p\)-stabilized eigenform of the \(p\)-ordinary new form \(f\) so that the stabilized eigenform has the slope \(k+1\) where \(f\) has the weight \(k+2\) (i.e., its eigenvalue for \(U_p\) has \(p\)-adic norm \(k+1\), which is as large as possible). They call this case the “critical slope” one. (Actually, they assume that the eigenform is not in the image of \(\theta^{k+1}\), which the author of the paper under review calls “\(\theta\)-critical”.)
Now, to discuss the result of this paper, let us define the following term: if \(f\) is a newform of level \(\Gamma_1(N)\) and weight \(k_2\) with \((p,N)=1\), a \(p\)-stabilized eigenform \(f_{\beta}\) is defined by \(f_{\beta}(z)=f(z)-\alpha f(pz)\) where \(\alpha, \beta\) are the roots of the polynomial \(X^2-a_p(f) X+\varepsilon(p)p^{k+1}\) for the character \(\varepsilon\) of \(f\). This form is an eigenform of level \(\Gamma_1(pN)\), character \(\varepsilon\). We will say \(f_{\beta}\) is “decent” if it satisfies at least one of the following conditions:
\(f\) is Eisenstein, and if \(f=E_{2, \chi, \psi}\), there is no prime \(l\) dividing \(N\) such that the \(l\)-components of \(\chi\) and \(\psi\) are equal.
\(f_{\beta}\) is non-critical.
\(f\) is cuspidal and \(H_g^1(G_{\mathbb Q}, \operatorname{ad} \rho_f)=0\).
These are technical conditions that cannot be easily explained in this review, and the interested reader can find the detailed explanation in the paper. What is important, is that the paper (Sec. 2.2.4) has criteria to determine whether \(f\) satisfies Condition 3, and it seems that many forms (probably most forms) do.
The author then proves that a certain modular symbols associated to \(f_{\beta}\) have dimension 1, thus proving the existence of associated \(p\)-adic \(L\)-functions. He also defines \(L_p(k, s)\) with weight variable \(k\) so that it interpolates his \(p\)-adic \(L\)-functions.

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F85 \(p\)-adic theory, local fields
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F03 Modular and automorphic functions
Full Text: DOI
[1] Ash, A., Stevens, G.: Modular forms in characteristic l and special values of their L-functions. Duke Math. J. 53(3), 849–868 (1986) · Zbl 0618.10026
[2] Amice, Y., Vélu, J.: Distributions p-adiques associées aux séries de Hecke. In: Journées Arithmétiques de Bordeaux, Conf. Univ. Bordeaux, Bordeaux, 1974. Asterisque, vol. 24–25, pp. 119–131. Soc. Math. France, Paris (1975)
[3] Bellaïche, J.: Non-smooth classical points on eigenvarieties. Duke Math. J. 145(1), 71–90 (2008) · Zbl 1206.11057
[4] Bellaïche, J.: Eigenvarieties, families of Galois representations, p-adic L-functions. Course at Brandeis University (Fall 2010), notes in preparation
[5] Bellaïche, J., Chenevier, G.: Formes non tempérées pour ${\(\backslash\)text {U}}(3)$ et conjectures de Bloch–Kato. Ann. Enseign. Sup. Grenoble 37(4), 611–662 (2004) · Zbl 1201.11051
[6] Bellaïche, J., Chenevier, G.: Lissité de la courbe de Hecke de ${\(\backslash\)text {GL}}(2)$ aux points Eisenstein critiques. J. Inst. Math. Jussieu 5(2), 333–349 (2006) · Zbl 1095.11025
[7] Bellaïche, J., Chenevier, G.: Families of Galois Representations and Selmer Groups. Astérisque, vol. 324. Soc. Math. France, Paris (2009) · Zbl 1192.11035
[8] Bosch, S., Guntzer, U., Remmert, R.: Non-archimedian analysis. Grundlehren der Math., vol. 261. Springer, Berlin (1984)
[9] Breuil, C., Emerton, M.: Représentations p-adiques ordinaires de ${\(\backslash\)text {GL}}_{2}(\(\backslash\)mathbb{Q})$ et compatibilité locale globale. Astérisque 331, 255–315 (2010) · Zbl 1251.11043
[10] Buzzard, K.: Eigenvarieties. In: Proc. of the LMS Durham Conference on L-Functions and Arithmetic (2007)
[11] Sur les représentations l-adiques associées aux formes modulaires de Hilbert. Ann. Sci. Ecole Norm. Sup. (4), 409–468 (1986)
[12] Chenevier, G.: Familles p-adiques de formes automorphes pour ${\(\backslash\)text {GL}}\(\backslash\)sb n$ . J. Reine Angew. Math. 570, 143–217 (2004) · Zbl 1093.11036
[13] Chenevier, G.: Une correspondance de Jacquet–Langlands p-adique. Duke Math. J. 126(1), 161–194 (2005) · Zbl 1070.11016
[14] Coleman, R., Gouvêa, F., Jochnowitz, N.: E 2, {\(\Theta\)}, and overconvergence. Int. Math. Res. Not. 1, 23–41 (1995) · Zbl 0846.11027
[15] Coleman, R.: Classical and overconvergent modular forms. Invent. Math. 124(1–3), 215–241 (1996) · Zbl 0851.11030
[16] Coleman, R.: p-adic Banach spaces & families of modular forms. Invent. Math. 127, 417–479 (1997) · Zbl 0918.11026
[17] Coleman, R., Mazur, B.: The eigencurve. In: Galois Representation in Arithmetic Algebraic Geometry, Durham, 1996. London Math. Soc. Lecture Note. Ser., vol. 254. Cambridge University Press, Cambridge (1996)
[18] Emerton, M.: On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms. Invent. Math. 164(1), 1–84 (2006) · Zbl 1090.22008
[19] Greenberg, R., Stevens, G.: p-adic L-functions and p-adic periods of modular forms. Invent. Math. 111(2), 407–447 (1993) · Zbl 0778.11034
[20] Hida, H.: Elementary Theory of L-Functions and Eisenstein Series. Cambridge University Press 26 · Zbl 0942.11024
[21] Jannsen, U.: On the l-adic cohomology of varieties over number fields and its Galois cohomology. In: Galois Groups over \(\mathbb{Q}\), Berkeley, CA, 1987. Math. Sci. Res. Inst. Publ., vol. 16, pp. 315–360. Springer, New York (1989)
[22] Kitagawa, K.: On standard p-adic L-functions of families of elliptic cusp forms. In: p-Adic Monodromy and the Birch and Swinnerton–Dyer Conjecture, Boston, MA. Contemp. Math., vol. 165, pp. 81–110. Am. Math. Soc., Providence (1994) · Zbl 0841.11028
[23] Kisin, M.: Overconvergent modular forms and the Fontaine–Mazur conjecture. Invent. Math. 153(2), 373–454 (2003) · Zbl 1045.11029
[24] Kisin, M.: Geometric deformations of modular Galois representations. Invent. Math. 157(2), 275–328 (2004) · Zbl 1150.11020
[25] Lang, S.: Algebraic Number Theory, 2nd edn. Graduate Texts in Math., vol. 110. Springer, Berlin (1994) · Zbl 0811.11001
[26] Livne, R.: On the conductors of mod Galois representations coming from modular forms. J. Number Theory 31, 133–141 (1989) · Zbl 0674.10024
[27] Manin, J.: Periods of cusp forms, and p-adic Hecke series. Mat. Sb. (N.S.) 92(134), 378–401 (1973) · Zbl 0293.14007
[28] Mazur, B., Tate, J., Teitelbaum, J.: On p-adic analogues of the conjectures of Birch and Swinnerton–Dyer. Invent. Math. 84(1), 1–48 (1986) · Zbl 0699.14028
[29] Mazur, B., Swinnerton-Dyer, P.: Arithmetic of Weil curves. Invent. Math. 25, 1–61 (1974) · Zbl 0281.14016
[30] Merel, L.: Universal Fourier expansions of modular forms. In: On Artin’s Conjecture for Odd 2-Dimensional Representations. Lecture Notes in Math., vol. 1585, pp. 59–94. Springer, Berlin (1994) · Zbl 0844.11033
[31] Miyake, T.: Modular Forms. Springer, Berlin (1976) · Zbl 0466.10012
[32] Nyssen, L.: Pseudo-representations. Math. Ann. 306, 257–283 (1996) · Zbl 0863.16012
[33] Park, J.: p-Adic family of half-integral weight modular forms via overconvergent Shintani lifting. Manuscr. Math. 131, 355–384 (2010) · Zbl 1221.11115
[34] Panchishkin, A.: Two variable p-adic L functions attached to eigenfamilies of positive slope. Invent. Math. 154(3), 551–615 (2003) · Zbl 1065.11025
[35] Pollack, R.: On the p-adic L-function of a modular form at a supersingular prime. Duke Math. J. 118(3), 523–558 (2003) · Zbl 1074.11061
[36] Rouquier, R.: Caractérisation des caractères et pseudo-caractères. J. Algebra 180(2), 571–586 (1996) · Zbl 0857.16013
[37] Rubin, K.: The ”main conjectures” of Iwasawa theory for imaginary quadratic fields. Invent. Math. 103(1), 25–68 (1991) · Zbl 0737.11030
[38] Schneider, P.: Nonarchimedean Functional Analysis. Springer Monographs in Mathematics. Springer, Berlin (2002) · Zbl 0998.46044
[39] Serre, J.-P.: Endomorphismes complètement continus des espaces de Banach p-adiques. Publ. Math. IHÉS 12, 69–85 (1962) · Zbl 0104.33601
[40] Shokurov, V.: Shimura integrals of cusp forms. Izv. Akad. Nauk SSSR, Ser. Mat. 44(3), 670–718, 720 (1980) · Zbl 0444.14030
[41] Skinner, C., Urban, E.: Sur les déformations p-adiques de certaines représentations automorphes. J. Inst. Math. Jussieu 5(4), 629–698 (2006) · Zbl 1169.11314
[42] Skinner, C., Urban, E.: Vanishing of L-functions and ranks of Selmer groups. In: International Congress of Mathematicians, vol. II, pp. 473–500. Eur. Math. Soc., Zürich (2006) · Zbl 1157.11020
[43] Soulé, C.: On Higher p-Adic Regulator. Lect. Notes Math. 854, 372–401 (1981)
[44] Stein, W.: Explicit approaches to modular abelian varieties. PhD thesis, Berkeley 2000, available on http://www.wstein.org/thesis
[45] Stein, W.: Modular Forms, A Computational Approach. Book, available on http://modular.math.washington.edu/books/modform/modform/index.html · Zbl 1110.11015
[46] Stevens, G.: Rigid analytic modular symbols. Preprint, available on http://math.bu.edu/people/ghs/research.d
[47] Stevens, G.: Family of overconvergent modular symbols. Unpublished
[48] Stevens, G.: The 3-adic L-function of an evil Eisenstein series. Unpublished (transmitted to the author in 2007)
[49] Stevens, G., Pollack, R.: Overconvergent modular symbols and p-adic L-functions. Preprint, available on http://math.bu.edu/people/rpollack/ . Ann. Sci. Ec. Norm. Sup., to appear · Zbl 1268.11075
[50] Stevens, G., Pollack, R.: Critical slope p-adic L-functions. Preprint, available on http://math.bu.edu/people/rpollack · Zbl 1317.11051
[51] Visik, M.: Nonarchimedean measures connected with Dirichlet series. Math USSR Sb. 28, 216–218 · Zbl 0369.14010
[52] Weston, T.: Geometric Euler systems for locally isotropic motives. Compos. Math. 140, 317–332 (2004) · Zbl 1133.11038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.