Critical slope \(p\)-adic \(L\)-functions.

*(English)*Zbl 1317.11051“Let \(p\) be a prime number, and let \(f=\sum_n a_nq^n\) denote a normalized cuspidal eigenform of weight \(k+2\) on \(\Gamma_1(N)\) with nebentype \(\varepsilon\) and with \(p\neq N\). If \(f\) is a \(p\)-ordinary form, then, by Y. Amice and J. Vélu [Astérisque 24–25, 119–131 (1975; Zbl 0332.14010)] and M. M. Vishik [Mat. Sb., N. Ser. 99(141), 248–260 (1976; Zbl 0358.14014)], we can attach a \(p\)-adic \(L\)-function to \(f\) which interpolates special values of its \(L\)-series. On the other hand, if \(f\) is non-ordinary at \(p\), then we have two \(p\)-adic \(L\)-functions attached to \(f\), one for each root of \(x^2- a_px+ \varepsilon(p) p^{k+1}\). These two roots correspond to the two \(p\)-stabilization of \(f\) to level \(\Gamma_0:= \Gamma_1(N)\cap \Gamma_0(p)\), and, more precisely, we are attaching a \(p\)-adic \(L\)-function to each of these forms.

In the case when \(f\) is \(p\)-ordinary, one of these \(p\)-stabilizations is \(p\)-ordinary and other has slope \(k+1\) (critical slope). The methods of Amice and Vélu and Vishik [loc. cit.] only apply to forms of slope strictly less than \(k+1\), which is why in this cse we only have one \(p\)-adic \(L\)-function.”

“It is the goal of this paper to give a natural construction of \(p\)-adic \(L\)-functions of critical slope forms, and thus to construct the ‘missing’ \(p\)-adic \(L\)-function in the ordinary case.”

“The basic starting point of our method is the theory of overconvergent modular forms developed by the second author.”

If the modular form is not in the image of theta, then the critical slop \(p\)-adic \(L\)-function satisfies the standard interpolation property (Proposition 9.3).

“Many examples of these critical slope \(p\)-adic \(L\)-functions are computed by the authors [Ann. Sci. Éc. Norm. Supér. (4) 44, No. 1, 1–42 (2011; Zbl 1268.11075)]. Their zeroes appear to contain interesting patterns that encode the classical \(\mu\)- and \(\lambda\)-invariants of the corresponding ordinary \(p\)-adic \(L\)-function.”

“Since the writing of this paper, D. Loeffler and S. L. Zerbes [J. Reine Angew. Math. 679, 181–206 (2013; Zbl 1276.11192)] proved that analogous formulas involving Iwasawa invariants hold for the critical slope \(p\)-adic \(L\)-function defined via Kato’s Euler system (see Remark 9.5).”

In the case when \(f\) is \(p\)-ordinary, one of these \(p\)-stabilizations is \(p\)-ordinary and other has slope \(k+1\) (critical slope). The methods of Amice and Vélu and Vishik [loc. cit.] only apply to forms of slope strictly less than \(k+1\), which is why in this cse we only have one \(p\)-adic \(L\)-function.”

“It is the goal of this paper to give a natural construction of \(p\)-adic \(L\)-functions of critical slope forms, and thus to construct the ‘missing’ \(p\)-adic \(L\)-function in the ordinary case.”

“The basic starting point of our method is the theory of overconvergent modular forms developed by the second author.”

If the modular form is not in the image of theta, then the critical slop \(p\)-adic \(L\)-function satisfies the standard interpolation property (Proposition 9.3).

“Many examples of these critical slope \(p\)-adic \(L\)-functions are computed by the authors [Ann. Sci. Éc. Norm. Supér. (4) 44, No. 1, 1–42 (2011; Zbl 1268.11075)]. Their zeroes appear to contain interesting patterns that encode the classical \(\mu\)- and \(\lambda\)-invariants of the corresponding ordinary \(p\)-adic \(L\)-function.”

“Since the writing of this paper, D. Loeffler and S. L. Zerbes [J. Reine Angew. Math. 679, 181–206 (2013; Zbl 1276.11192)] proved that analogous formulas involving Iwasawa invariants hold for the critical slope \(p\)-adic \(L\)-function defined via Kato’s Euler system (see Remark 9.5).”

Reviewer: Andrzej Dąbrowski (Szczecin)

##### MSC:

11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |

11F85 | \(p\)-adic theory, local fields |

11R23 | Iwasawa theory |