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Le produit de Petersson et de Rankin p-adique. (Petersson and p-adic Rankin products). (French) Zbl 0721.11024
Sémin. Théor. Nombres, Paris/Fr. 1988-89, Prog. Math. 91, 87-102 (1990).
[For the entire collection see Zbl 0711.00009.]
Let p be a prime \(\geq 5\) and let \(\Lambda ={\mathcal O}[[X]]\) where \({\mathcal O}\) is the ring of p-adic integers of a finite extension of \({\mathbb{Q}}_ p\). Let F and G be two ordinary \(\Lambda\)-adic modular forms. In his previous two papers [Invent. Math. 79, 159-195 (1985; Zbl 0573.10020); Ann. Inst. Fourier 38, No.3, 1-83 (1988; Zbl 0645.10028)] the author constructed a p-adic Rankin product \({\mathcal D}_ p\) which p-adically interpolates (essentially) the critical special values \(D(m,f_ k,g_{\ell}| \omega^{-m})/(f_ k,f_ k)\) where \(f_ k\) resp. \(g_{\ell}\) are specializations of F resp. G in weight k resp. \(\ell\), \(\omega\) is the Teichmüller character, D(s,f,g) is the usual Rankin L- series and (,) denotes the usual Petersson scalar product. As the author remarks, although the underlying ideas are simple the actual construction of \({\mathcal D}_ p\) is rather involved due to the fact that one works in the general case of I-adic modular forms for every finite extension I of \(\Lambda\).
In the present paper the author restricts to the special case of \(\Lambda\)-adic modular forms which have primitive “Nebentypus” and shows that in this case \({\mathcal D}_ p\) can be constructed in a rather simple way.

11F85 \(p\)-adic theory, local fields
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols