On \(p\)-adic \(L\)-functions attached to motives over \(\mathbb Q\).

*(English)*Zbl 0783.11039
Algebraic number theory - in honor of K. Iwasawa, Proc. Workshop Iwasawa Theory Spec. Values \(L\)-Funct., Berkeley/CA (USA) 1987, Adv. Stud. Pure Math. 17, 23-54 (1989).

[For the entire collection see Zbl 0721.00006.]

The authors conjecture the existence of certain \(p\)-adic measures attached to a motif \(V\) over \(\mathbb Q\) which admits at least one critical point \(s\in\mathbb Z\) and which is ordinary at \(p\). A. A. Panchishkin [C. R. Acad. Sci., Paris, Sér. I 295, 51–53 and 227–230 (1982; Zbl 0501.10028 and Zbl 0505.10014)] constructed \(p\)-adic bounded distributions for the Rankin \(L\)-series associated to two Hecke modular forms of different weights, under the assumption that they satisfy some “ordinarity conditions”.

In §1, the authors elaborate Panchishkin’s idea in order to derive canonical distributions from general Euler products. The purpose only requires the use of some elementary harmonic \(p\)-adic analysis.

§2 is devoted to reformulate basic conjectures, due to Deligne, about the algebraic nature of the values at critical points of the \(L\)-function of a motif over \(\mathbb Q\). The behaviour of such values under twists by Dirichlet characters is discussed in §3.

The notion of ordinarity at \(p\) for a motif is formulated in §4. When the motif is ordinary at \(p\), a conjecture relates the Galois structure of the \(p\)-adic cohomology \(H_ p(V)\) with the Hodge decomposition of the Betti cohomology \(H_ B(V)\otimes\mathbb C\) of the motif. Under the assumption that such a conjecture is fulfilled and that \(s=1\) is critical for \(V\), the authors formulate the conjecture about the existence of canonical \(p\)-adic measures attached to \(V\). When the weight of the motif \(m(V)\) is even, \(V\) must not contain the motif \(\mathbb Q(-m(V)/2)\) as a direct summand; otherwise, the general theory runs into pseudo- measures.

The formulation of these conjectures were slightly modified by the first author in a subsequent paper [see Bol. Soc. Bras. Mat., Nova Sér. 20, 201–112 (1989; Zbl 0783.11040)].

The authors conjecture the existence of certain \(p\)-adic measures attached to a motif \(V\) over \(\mathbb Q\) which admits at least one critical point \(s\in\mathbb Z\) and which is ordinary at \(p\). A. A. Panchishkin [C. R. Acad. Sci., Paris, Sér. I 295, 51–53 and 227–230 (1982; Zbl 0501.10028 and Zbl 0505.10014)] constructed \(p\)-adic bounded distributions for the Rankin \(L\)-series associated to two Hecke modular forms of different weights, under the assumption that they satisfy some “ordinarity conditions”.

In §1, the authors elaborate Panchishkin’s idea in order to derive canonical distributions from general Euler products. The purpose only requires the use of some elementary harmonic \(p\)-adic analysis.

§2 is devoted to reformulate basic conjectures, due to Deligne, about the algebraic nature of the values at critical points of the \(L\)-function of a motif over \(\mathbb Q\). The behaviour of such values under twists by Dirichlet characters is discussed in §3.

The notion of ordinarity at \(p\) for a motif is formulated in §4. When the motif is ordinary at \(p\), a conjecture relates the Galois structure of the \(p\)-adic cohomology \(H_ p(V)\) with the Hodge decomposition of the Betti cohomology \(H_ B(V)\otimes\mathbb C\) of the motif. Under the assumption that such a conjecture is fulfilled and that \(s=1\) is critical for \(V\), the authors formulate the conjecture about the existence of canonical \(p\)-adic measures attached to \(V\). When the weight of the motif \(m(V)\) is even, \(V\) must not contain the motif \(\mathbb Q(-m(V)/2)\) as a direct summand; otherwise, the general theory runs into pseudo- measures.

The formulation of these conjectures were slightly modified by the first author in a subsequent paper [see Bol. Soc. Bras. Mat., Nova Sér. 20, 201–112 (1989; Zbl 0783.11040)].

Reviewer: P.Bayer (Barcelona)

##### MSC:

11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |

11F33 | Congruences for modular and \(p\)-adic modular forms |

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

11G09 | Drinfel’d modules; higher-dimensional motives, etc. |

11R23 | Iwasawa theory |