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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:
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 $$H_g^1(G_{\mathbb Q}, \operatorname{ad} \rho_f)=0$$.
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.

##### 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
##### Keywords:
$$p$$-adic $$L$$-functions; modular symbols
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##### References:
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