Admissible \(p\)-adic \(L\)-functions of automorphic forms
Let \(M\) be a motive over \(\mathbb{Q}\), together with its associated \(\ell\)- adic realizations \(H_ \ell (M)\) and its \(p\)-adic \(L\)-functions \(L_ p (M,s)\). These \(L\)-functions are defined, for all but a finite number of primes \(p\), via the geometric Frobenius \(F_ p\in \text{Gal} (\overline {\mathbb{Q}}/ \mathbb{Q})\), and they are independent of the prime number \(\ell\) chosen for a specific realization. In the present note, the author formulates a conjectural statement according to which there exist certain \(p\)-adic \(L\)-functions of logarithmic growth for \(M\). These \(L\)-functions interpolate special values of the global \(L\)-function \(L(M, s)= \prod_ p L_ p (M,s)\) of \(M\) at points within the so-called critical band. In the special case of a motif \(M(f)\) attached to a modular new form \(f\), this conjectural statement also allows special values of the automorphic form \(f\) itself to be interpolated. After the formulation of the hypothetical criterion, in Section 1 of the paper, the author investigates and confirms its validity in some concrete cases. Section 2 treats the case of the symmetric degree of \(\text{Sym}^ rM(f)\) of a motive \(M(f)\), thereby recovering earlier reuslts \textit{M. M. Vishik} [Math. USSR, Sb. 28 (1976), 216-228 (1978); translation from Mat. Sb., Nov. Ser. 99(141), 248-260 (1976; Zbl 0358.14014)], \textit{A. A. Panchishkin} [Math. USSR, Izv. 32, No. 2, 339-358 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, 336-354 (1988; Zbl 0656.10020)] and \textit{P. B. Garrett} and \textit{M. Harris} [Am. J. Math. 115, 161-240 (1993; Zbl 0776.11027)], and Section 3 deals with the case of motives \(\text{Sym}^ r M(f)\) of CM-type. In Section 4, the author returns to the setup of Section 2 and proves the validity of his conjecture for the motif \(M(\text{Sym}^ 2 (f))\) in full generality.
\(p\)-adic \(L\)-functions of logarithmic growth
scientific article; zbMATH DE number 486781