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:
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.
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:
- 1.
- \(f\) 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.
- 2.
- \(f_{\beta}\) is non-critical.
- 3.
- \(f\) is cuspidal and \(H_g^1(G_{\mathbb Q}, \operatorname{ad} \rho_f)=0\).
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.
Reviewer: Byoung Du Kim (Wellington)
MSC:
| 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 |
Citations:
Zbl 0281.14016; Zbl 0358.14014; Zbl 0332.14010; Zbl 0699.14028; Zbl 1268.11075; Zbl 1317.11051; Zbl 0293.14007References:
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