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The vanishing of certain Rankin-Selberg convolutions. (English) Zbl 0737.11014

Automorphic forms and analytic number theory, Proc. Conf., Montréal/Can. 1989, 123-133 (1990).
[For the entire collection Zbl 0728.00008.]
Let \(L(s,f,g)\) denote the Rankin-Selberg \(L\)-function obtained by convolution of a holomorphic cusp form \(f\) for \(\Gamma_ 1(N)\) of weight \(k=2\) or 4, and a cuspidal Maass form \(g\) for \(\Gamma_ 1(M)\) with eigenvalue \(w(1-w)\geq 1/4\). The vanishing of \(L(s,f,g)\) at \(s=w+{k-1\over 2}\) is a necessary condition for survival of \(g\) as a cusp form under perturbation in the direction specified by \(f\). In the case \(k=4\) the cusp form \(f\) describes a family \(t\mapsto\Gamma_ t\) of discrete groups with value \(\Gamma_ 1(M)\) at \(t=0\). The question is whether \(g\) occurs as the value at \(t=0\) of a family \(t\mapsto g_ t\) of cuspidal Maass forms for this family of groups. The vanishing condition is derived in theorem 3.1 of R. S. Phillips and P. Sarnak [Invent. Math. 80, 339-364 (1985; Zbl 0558.10017)[. If \(f\) has weight 2 it describes a family of characters of the fixed group \(\Gamma_ 1(M)\). A similar vanishing condition can be derived.
The author considers the case \(w=1/2\). He shows that there exist cuspidal Maass forms \(g\) for which \(L(k/2,f,g)=0\) for infinitely many holomorphic cusp forms \(f\) of weight 2 or 4. The Maass form \(g\) is obtained from a dihedral Galois representation \(\rho:\text{Gal}(\overline\mathbb{Q}/\mathbb{Q})\to GL_ 2(\mathbb{C})\) with even determinant \(\chi\) by means of the Langlands correspondence. The order of \(L(s,f,g)\) at the center \(s=k/2\) of the functional equation turns out to have the same parity as \({1\over 2}(1- \chi(N))\). This makes it possible to choose as many \(f\) as indicated.Note that this does not answer the question whether \(g\) really escapes “annihilation” under perturbation in the direction specified by \(f\).
The paper proceeds with a discussion of other approaches to a vanishing result at \(s=k/2\). For other Galois representations \(\rho\) one might get the order of the zero at \(s=k/2\) with help of various conjectures from the existence of an elliptic curve \(E\) such that \(\rho\) occurs in the representation of \(\text{Gal}(\overline\mathbb{Q}/\mathbb{Q})\) on \(\mathbb{C}\otimes E(\overline\mathbb{Q})\). Examples of such elliptic curves are given.
For the entire collection see [Zbl 0764.11036].

MSC:

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
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11F37 Forms of half-integer weight; nonholomorphic modular forms
11F11 Holomorphic modular forms of integral weight
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