Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula. (Dimensions des espaces de formes automorphes en niveau un pour les groupes classiques déployés à l’aide de la formule des traces.) (English. French summary) Zbl 1394.11042
There are two different problems motivating the research of this paper. The first one is the derivation of explicit dimension formulas for spaces of vector-valued Siegel modular forms. The second one is the classification of motives of conductor \(1\) and given Hodge weights.
Let \(\mathrm{Sp}_{2n} (\mathbb{Z})\) be the full modular group of genus \(n\). Given integers \(k_1 \geq \ldots \geq k_n\), let \(r\) be the algebraic finite-dimensional representation of \(\mathrm{GL}_n (\mathbb{C})\) with highest weight \(\underline{k} = (k_1, \ldots , k_n)\) and define by \(S_{\underline{k}} ( \mathrm{Sp}_{2n} (\mathbb{Z}))\) the space of Siegel cusp forms of genus \(n\), level \(\mathrm{Sp}_{2n} (\mathbb{Z})\) and weight \(r\). Dimension formulas for spaces of vector-valued Siegel modular forms were known in specific cases of small genus, (see for instance [J. I. Igusa, Am. J. Math. 84, 175–200 (1964; Zbl 0133.33301); D. Petersen, Pac. J. Math. 275, No. 1, 39–61 (2015; Zbl 1394.11041); R. Tsushima, Proc. Japan Acad., Ser. A 59, 139–142 (1983; Zbl 0513.10025)] for \(n=2\) and [J. Bergström et al., Sel. Math., New Ser. 20, No. 1, 83–124 (2014; Zbl 1343.11051); S. Tsuyumine, Am. J. Math. 108, 755–862 (1986; Zbl 0602.10015)] for \(n=3\)).
The first achievement in this paper is a generalization of the above formulas for the dimension of \(S_{\underline{k}} ( \mathrm{Sp}_{2n} (\mathbb{Z}))\) in higher genera. In particular, the dimension formula reads: \[ \dim S_{\underline{k}} ( \mathrm{Sp}_{2n} (\mathbb{Z})) = \sum_{a \in A} \mathrm{tr}_{\mathbb{Q}(\zeta_{m_a}) / \mathbb{Q}} \left( P_{a} (k_1, \ldots, k_n) \zeta_{m_a}^{\Lambda_a (k_1, \ldots , k_n)} \right)\tag{1} \] where \((m_a, P_a, \Lambda_a)_{a \in A}\) is a finite family for any \(n\), and where \(m_a \geq 1\) is an integer, \(P_a\) is a polynomial in \(n\) variables over \(\mathbb{Q}(\zeta_{m_a})\) and \(\Lambda\) is a surjective group morphism from \((\mathbb{Z} / m_{a} \mathbb{Z})^n\) to \(\mathbb{Z} / m_{a} \mathbb{Z}\). The family \(A\) can be very large to compute it explicitly, and indeed, the author indicates how to calculate the formula (1) but his calculations are only for \(n \leq 7\).
The second problem under investigation is the derivation of explicit formulas for the number of level one, regular algebraic and essentially self-dual automorphic cuspidal representations \(\pi = \otimes_{v}^{'} \pi_v\) of the general linear group \(\mathrm{GL}_N\) over the rationals in terms of Hodge weights. When \(\pi_p\) is unramified for all prime \(p\) and under explicit restrictions on the eigenvalues of \(\pi_{\infty}\), this turns out to be a finite set by a result of Harish-Chandra [Automorphic forms on semisimple Lie groups. Berlin etc.: Springer-Verlag (1968; Zbl 0186.04702)], and the main result of the paper is a formula similar to formula (1) for the number of these automorphic representations. In fact, the explicit formula for dimensions of spaces of Siegel modular cusp forms follows as corollary of this main result. Here again, there is an algorithmic limitation of the explicitly of the results, and the author computes the families \(A\) for \(N \leq 13\) and \(N=15\).
The main tools in the proof of the main theorem are the Arthur’s “simple” trace formula [J. Arthur, Invent. Math. 97, No. 2, 257–290 (1989; Zbl 0692.22004)] and Arthur’s endoscopic classification of the discrete automorphic spectrum for symplectic and special orthogonal groups [J. Arthur, The endoscopic classification of representations. Orthogonal and symplectic groups. Providence, RI: AMS (2013; Zbl 1310.22014)]. The deduction of the results is algorithmic in style, and the explicit statements are under the natural limitations of the computer. At the end of the paper, the author also writes down as examples many calculations for Arthur parameters, dimensions of Siegel cusp forms and the “masses” (i.e., the rational coefficients appearing at the right hand side of formula (1)) for special linear groups of small dimension \(N\).
All the above results are conditional on the Arthur’s endoscopic classification, hence conditional on results announced but not yet published. Finally, the author expects that his results can have applications to scalar modular forms, vector-valued modular forms and the geometry of the moduli stack \(\mathcal{A}_{g}\).
Let \(\mathrm{Sp}_{2n} (\mathbb{Z})\) be the full modular group of genus \(n\). Given integers \(k_1 \geq \ldots \geq k_n\), let \(r\) be the algebraic finite-dimensional representation of \(\mathrm{GL}_n (\mathbb{C})\) with highest weight \(\underline{k} = (k_1, \ldots , k_n)\) and define by \(S_{\underline{k}} ( \mathrm{Sp}_{2n} (\mathbb{Z}))\) the space of Siegel cusp forms of genus \(n\), level \(\mathrm{Sp}_{2n} (\mathbb{Z})\) and weight \(r\). Dimension formulas for spaces of vector-valued Siegel modular forms were known in specific cases of small genus, (see for instance [J. I. Igusa, Am. J. Math. 84, 175–200 (1964; Zbl 0133.33301); D. Petersen, Pac. J. Math. 275, No. 1, 39–61 (2015; Zbl 1394.11041); R. Tsushima, Proc. Japan Acad., Ser. A 59, 139–142 (1983; Zbl 0513.10025)] for \(n=2\) and [J. Bergström et al., Sel. Math., New Ser. 20, No. 1, 83–124 (2014; Zbl 1343.11051); S. Tsuyumine, Am. J. Math. 108, 755–862 (1986; Zbl 0602.10015)] for \(n=3\)).
The first achievement in this paper is a generalization of the above formulas for the dimension of \(S_{\underline{k}} ( \mathrm{Sp}_{2n} (\mathbb{Z}))\) in higher genera. In particular, the dimension formula reads: \[ \dim S_{\underline{k}} ( \mathrm{Sp}_{2n} (\mathbb{Z})) = \sum_{a \in A} \mathrm{tr}_{\mathbb{Q}(\zeta_{m_a}) / \mathbb{Q}} \left( P_{a} (k_1, \ldots, k_n) \zeta_{m_a}^{\Lambda_a (k_1, \ldots , k_n)} \right)\tag{1} \] where \((m_a, P_a, \Lambda_a)_{a \in A}\) is a finite family for any \(n\), and where \(m_a \geq 1\) is an integer, \(P_a\) is a polynomial in \(n\) variables over \(\mathbb{Q}(\zeta_{m_a})\) and \(\Lambda\) is a surjective group morphism from \((\mathbb{Z} / m_{a} \mathbb{Z})^n\) to \(\mathbb{Z} / m_{a} \mathbb{Z}\). The family \(A\) can be very large to compute it explicitly, and indeed, the author indicates how to calculate the formula (1) but his calculations are only for \(n \leq 7\).
The second problem under investigation is the derivation of explicit formulas for the number of level one, regular algebraic and essentially self-dual automorphic cuspidal representations \(\pi = \otimes_{v}^{'} \pi_v\) of the general linear group \(\mathrm{GL}_N\) over the rationals in terms of Hodge weights. When \(\pi_p\) is unramified for all prime \(p\) and under explicit restrictions on the eigenvalues of \(\pi_{\infty}\), this turns out to be a finite set by a result of Harish-Chandra [Automorphic forms on semisimple Lie groups. Berlin etc.: Springer-Verlag (1968; Zbl 0186.04702)], and the main result of the paper is a formula similar to formula (1) for the number of these automorphic representations. In fact, the explicit formula for dimensions of spaces of Siegel modular cusp forms follows as corollary of this main result. Here again, there is an algorithmic limitation of the explicitly of the results, and the author computes the families \(A\) for \(N \leq 13\) and \(N=15\).
The main tools in the proof of the main theorem are the Arthur’s “simple” trace formula [J. Arthur, Invent. Math. 97, No. 2, 257–290 (1989; Zbl 0692.22004)] and Arthur’s endoscopic classification of the discrete automorphic spectrum for symplectic and special orthogonal groups [J. Arthur, The endoscopic classification of representations. Orthogonal and symplectic groups. Providence, RI: AMS (2013; Zbl 1310.22014)]. The deduction of the results is algorithmic in style, and the explicit statements are under the natural limitations of the computer. At the end of the paper, the author also writes down as examples many calculations for Arthur parameters, dimensions of Siegel cusp forms and the “masses” (i.e., the rational coefficients appearing at the right hand side of formula (1)) for special linear groups of small dimension \(N\).
All the above results are conditional on the Arthur’s endoscopic classification, hence conditional on results announced but not yet published. Finally, the author expects that his results can have applications to scalar modular forms, vector-valued modular forms and the geometry of the moduli stack \(\mathcal{A}_{g}\).
Reviewer: Dimitrios Chatzakos (Villeneuve d’Ascq)
MSC:
| 11F46 | Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms |
| 22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |
| 22E46 | Semisimple Lie groups and their representations |
