\(p\)-adic \(L\)-functions for base change lifts of \(\mathrm{GL}_ 2\) to \(\mathrm{GL}_ 3\). (English) Zbl 0705.11033

Hida, Haruzo

Automorphic forms, Shimura varieties, and L-functions. Vol. II, Proc. Conf., Ann Arbor/MI (USA) 1988, Perspect. Math. 11, 95-142 (1990).

[For the entire collection see Zbl 0684.00004.]
The author constructs a two-variable \(p\)-adic \(L\)-function which \(p\)-adically interpolates critical values of the symmetric square \(L\)-functions attached to cusp forms of integral weights. The proof uses generalization to the case of half-integral weight of the author’s results on \(p\)-adic modular forms of integral weight and \(p\)-adic Hecke algebras and the method of \(p\)-adic Rankin convolutions as developed by the author [cf. Invent. Math. 79, 159–195 (1985; Zbl 0573.10020); Ann. Inst. Fourier 38, No. 3, 1–83 (1988; Zbl 0645.10028)]. To prove that the \(L\)-functions obtained are holomorphic an idea of C.-G. Schmidt [Invent. Math. 92, 597–631 (1988; Zbl 0656.10023)] is adopted.

Reviewer: W.Kohnen

MSC:

11F85	\(p\)-adic theory, local fields
11F67	Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11S40	Zeta functions and \(L\)-functions

Keywords:

p-adic interpolation; spectrum of nearly ordinary Hecke algebra; two- variable p-adic L-function; symmetric square L-functions; cusp forms of integral weights; half-integral weight; p-adic modular forms; p-adic Hecke algebras; p-adic Rankin convolutions