The algebraicity of special values has been a standing topic in the study of \(L\)-functions, both for the analytic reasons and for the arithmetic reasons. Many results have been obtained with fruitful applications, and even more have been conjectured; for example, P. Deligne [Proc. Symp. Pure Math. 33, No. 2, 313–346 (1979; Zbl 0449.10022)] describes how these special values are conjecturally related to certain geometric periods. In the case of symmetric power \(L\)-functions, A. Raghuram and F. Shahidi [Prog. Math. 258, 271–293 (2008; Zbl 1229.11080)] give a “simplistic” form of Deligne’s conjecture.
Conjecture. Let \(F / \mathbb{Q}\) be a totally real number field of degree \(d\) and with discriminant \(D_F\), let \(\mathfrak{c}\) be an integral ideal of \(F\), let \(\mathbf{k} \in (2 \mathbb{N})^d\), let \(\mathbf{f} \in \mathcal{S}_{\mathbf{k}}(\mathfrak{c}, 1)\) be a primitive cuspform, and let \(\pi = \pi(\mathbf{f})\) be the irreducible cuspidal automorphic representation of \(\mathrm{GL}_2(\mathbb{A}_F)\) associated to \(\mathbf{f}\) with trivial central character. Also, let \(r \geq 1\) and \(1 \leq m \leq \mathbf{k}^0 - 1\), where \[ \mathbf{k}^0 = \min\{k_1, \dots, k_d\} \] Further, let \(\chi\) be a finite order Hecke character of \(\mathbb{A}_F^{\times}\) such that \(\chi_v = (\operatorname{sgn})^{r+m}\) at every archimedean place \(v\) of \(F\). Then we have \[ \frac{L_{\mathrm{fin}}(m, \pi(\mathbf{f}), \operatorname{Sym}^{2r} \times \chi)} {(2 \pi i)^{\frac{(r+1)(r|\mathbf{k}|+d(2m-r))}{2}} |D_F|^{\frac{1}{2}} \mathfrak{g}(\chi)^{r+1} \langle \mathbf{f}, \mathbf{f}\rangle^{r(r+1)}} \in \overline{\mathbb{Q}} \qquad \qquad (*) \] where \(\mathfrak{g}(\chi)\) denotes the Gauss sum of \(\chi\).
The focus of the paper under review is the algebraicity of special values of symmetric 4-th and 6-th power \(L\)-functions of Hilbert cusp forms. More precisely, the author shows that

1.

if \(r = 2\) and \(\mathbf{k}^0 \geq 6\), then the algebraicity (*) holds for every \(\chi\) and every \(m\);

2.

if \(r = 3\) and \(\mathbf{k}^0 \geq 6\), then the algebraicity (*) holds for every \(\chi\) and every \(m\) when \(F = \mathbb{Q}\), for every \(\chi\) and certain critical points \(m\) when \(F \cap \mathbb{Q}(\zeta_5) = \mathbb{Q}\), and for \(m = 1\) and \(\chi = 1\) when \(F\) is a general totally real field.

The main approach of the proof is to use symmetric power liftings of \(\mathrm{GL}(2)\) to connect the symmetric power \(L\)-function with the standard \(L\)-functions of these liftings and their twists, so that the expected algebraicity can be deduced from the previously established results of these standard \(L\)-functions with certain periods.
The author also considers the lifts introduced by D. Ramakrishnan and F. Shahidi [Math. Res. Lett. 14, No. 2, 315–332 (2007; Zbl 1132.11023)] with \(F = \mathbb{Q}\) and by R. Harron and A. Jorza [Am. J. Math. 139, No. 6, 1605–1647 (2017; Zbl 1425.11089)] in general. As a consequence of his algebraicity result, the author establishes a period relation for the Ramakrishnan-Shahidi lifts, proving a conjecture of T. Ibukiyama and H. Katsurada [J. Math. Soc. Japan 66, No. 1, 139–160 (2014; Zbl 1291.11087); Conjecture 3.3] over totally real fields.
This paper is well written. In particular, the author carefully explains the reasons for the introduction of those restrictions in his results, helping the audience clearly understand the obstacles and the opportunities therein.