## 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

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, upto 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 autormorphic 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

### MSC:

 11F11 Holomorphic modular forms of integral weight 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 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols

Zbl 0453.10024
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### References:

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