Special values of anticyclotomic \(L\)-functions modulo \(\lambda\). (English) Zbl 1392.11043
Summary: The purpose of this article is to generalize some results of Vatsal on the special values of Rankin-Selberg \(L\)-functions in an anticyclotomic \(\mathbb Z_p\)-extension. Let \(g\) be a cuspidal Hilbert modular newform of parallel weight \((2,\dots ,2)\) and level \(\mathcal{N}\) over a totally real field \(F\), and let \(K/F\) be a totally imaginary quadratic extension of relative discriminant \(\mathcal{D}\). We study the \(l\)-adic valuation of the special values \(L(g,\chi,\frac{1}{2})\) as \(\chi\) varies over the ring class characters of \(K\) of \(\mathcal{P}\)-power conductor, for some fixed prime ideal \(\mathcal{P}\). We prove our results under the only assumption that the prime to \(\mathcal{P}\) part of \(\mathcal{N}\) is relatively prime to \(\mathcal{D}\).
MSC:
| 11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 11G18 | Arithmetic aspects of modular and Shimura varieties |
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