On the values of certain automorphic \(L\)-functions at their center of symmetry. (Sur les valeurs de certaines fonctions \(L\) automorphes en leur centre de symétrie.) (French) Zbl 0567.10021

Waldspurger, J.-L.

Compos. Math. 54, 173-242 (1985).

Let \(f\) be a (holomorphic) cusp form of (even) weight \(k\) which is a newform for the congruence modular group \(\Gamma_0(N)\) and let \(\mathbb{Q}(f)\) be the number field generated by the Hecke eigenvalues \(a_n\) of \(f\). Let \(f'\) be another newform of weight \(kf\) for \(\Gamma_0(N)\) with eigenvalues under Hecke operators \(T_p\) equal to \(\chi(p)a_p\) for almost all primes \(p\), where \(\chi\) is a real Dirichlet character with conductor prime to \(N\) and \(\chi(-1)=1\). For the associated \(L\)-functions \(L(\cdot,s)\), suppose that \(L(f,k/2) L(f',k/2) \ne 0\). A theorem of M.-F. Vigneras [Prog. Math. 12, 331–356 (1981; Zbl 0453.10024)] now asserts that, up to an explicit multiplicative factor, the ratio \(L(f,k/2)/L(f',k/2)\) is a square in \(\mathbb{Q}(f)\). The object of the author is to establish the same result in a more general form and by a different method.
More specifically, the author proves the following, in particular. Let \(F\) be a number field, \(\mathbb{A}\) the ring of \(F\)-adèles and \(\pi\) an automorphic representation of \(\mathrm{GL}_2(\mathbb{A})\) with trivial central character and local components \(\pi_v\) at archimedean primes \(v\) of \(F\) satisfying some special conditions (such as \(\pi_v\) being in the discrete series for real \(v\)). Let \(\chi_1,\chi_2\) be quadratic characters of \(\mathbb{A}^{\times}/F^{\times}\) such that their local components coincide for every archimedean prime \(v\) of \(F\) and every prime \(v\) at which the local component \(\pi_v\) is ramified. Let \(L(\pi \otimes \chi_2, 1/2)\ne 0\) for the \(L\)-function associated to \(\pi \otimes \chi_2\). (There exists a quaternion algebra \(M\) over \(F\), depending on \(\chi\) and an automorphic representation \(\pi'\) of the adelic group \(G(\mathbb{A})\), corresponding to \(G=M^{\times}\), in an irreducible submodule \(E'\) of the space of automorphic (cusp) forms on \(G(F)\setminus G(\mathbb{A})\) such that \(\pi\) corresponds to \(\pi'\) under the Jacquet-Langlands correspondence.) Then there exist constants \(p(\chi_i)\) depending only on \(\chi_i\) \((i=1,2)\) and on the components \(\pi_v\) of \(\pi\) at archimedean primes \(v\) of \(F\) such that \[ (L(\pi \otimes \chi_1, 1/2)/L(\pi \otimes \chi_2, 1/2))\cdot p(\chi_2)/p(\chi_1) \] is a square in \(\mathbb{Q}(\pi)\), the field of rationality of \(\pi\).

Reviewer: S. Raghavan (Mumbai)

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
22E55	Representations of Lie and linear algebraic groups over global fields and adèle rings

Keywords:

values of L-functions at center of symmetry; squares in field of rationality; newform; automorphic representation

Citations:

Zbl 0453.10024

References:

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