Period relations for automorphic forms on unitary groups and critical values of \(L\)-functions (Q504297)

Consider a CM field \(L\) with maximal totally real field \(K\). Let \(G\) be a similitude unitary group associated to an hermitian space of dimension \(n\) with respect to \(L/K\), with similitude character \(\nu\). Fix a CM type \(\Phi\) and assume that \(G\) has signature \((r_\tau, s_\tau)\) for each \(\tau \in \Phi\). We parameterize the weights of a cuspidal automorphic representation \(\pi\) as \(((a_{\tau,1}, \ldots, a_{\tau, n})_{\tau \in \Phi} ; a_0)\), and assume that the corresponding algebraic representation of \(G_{\mathbb{C}}\) is actually defined over \(\mathbb{Q}\). Assume furthermore that \(\pi^\vee \simeq \pi \otimes \| \nu \|^{a_0}\). Let \(\psi\) be an algebraic character of \(L\) with infinite type \(m_\tau\), where \(\tau \in \mathrm{Hom}(L, \mathbb{C})\). The main result (Theorem 1) of this paper asserts that \[ L \left( m - \frac{n-1}{2}, \pi \otimes \psi, \mathrm{std} \right) \sim (2\pi i)^{[K: \mathbb{Q}] \cdot (mn - \frac{n(n-1)}{2}) - 2a_0 } D_K^{\lfloor \frac{n+1}{2} \rfloor /2 } P(\psi) Q^{\mathrm{hol}}(\psi) \] under the assumption that \[ m \leq a_{\tau, r_\tau} + s_\tau + m_\tau - m_{\bar{\tau}}, \quad m \leq a_{\tau, s_\tau} + r_\tau + m_{\bar{\tau}} - m_\tau, \quad \text{for all } \tau \in \Phi, \] and that \(\pi\) contributes to anti-holomorphic cohomology, together with Hypothesis 4.5.1 in the paper. Both sides belong to \(E(\pi) \otimes E(\psi) \otimes \mathbb{C}\) and \(\sim\) means equality up to multiplication by an element of \(E(\pi) \otimes E(\psi) \otimes L'\), where \(E(\pi)\) (resp.\ \(E(\psi)\)) denotes the field of definition of \(\pi_{\mathrm{fin}}\) (resp.\ \(\psi\)) and \(L'\) denotes the normal closure of \(L\) in \(\mathbb{C}\). The factor \(P(\psi)\) is an explicit expression involving the CM period of \(\psi\), and \(Q^{\mathrm{hol}}(\pi)\) is some quadratic automorphic period attached to \(\pi\). This generalizes [\textit{M.~Harris}, J. Reine Angew. Math. 483, 75--161 (1997; Zbl 0859.11032)], which concerns the case \(K = \mathbb{Q}\). The method of the proof is based on the doubling method of Piatetski-Shapiro and Rallis. A motivic interpretation is also given in the Section 1.2, which relates the main theorem to Deligne's conjecture on critical \(L\)-values.