# zbMATH — the first resource for mathematics

On automorphic forms on the unitary symplectic group $$\mathrm{Sp}(n)$$ and $$\mathrm{SL}_ 2(\mathbb R)$$. (English) Zbl 0634.10024
In standard notation the authors consider the two natural embeddings $(\mathrm{Sp}(1)\times\mathrm{Sp}(n))/\pm (1,1)\hookrightarrow \mathrm{SO}(4n), \tag{i}$
$\mathrm{SL}(2,\mathbb R)\times \mathrm{SO}(4n)\hookrightarrow \mathrm{Sp}(4n,\mathbb R).\tag{ii}$ They study a correspondence between automorphic forms $$F$$ on $$\mathrm{Sp}(1)\times \mathrm{Sp}(n)$$ and $$\theta_ F$$ on $$\mathrm{SL}(2,\mathbb R)$$. This correspondence arises from (i), (ii) by the use of theta series and is suggested by the theory of Weil representations and Howe’s theory of dual reductive pairs. For $$n\geq 2$$ the correspondence is explicitly given in terms of the associated $$L$$-functions, where the case $$n=2$$ was already dealt with by the second author [J. Math. Soc. Japan 16, 214–225 (1964; Zbl 0126.07101)].
Let $$F$$ be a simultaneous eigenform under all Hecke operators. Theorem 1 describes $$L(s,F)$$ as a finite product involving $$L(s,\theta_ F)$$ and another Dirichlet series of elliptic modular type associated with $$F$$. Theorem 2 expresses the eigenvalues of $$F$$ as a finite sum of special values of $$F$$. Theorem 1 is derived from Theorem 2 by the application of the commutation relation between Hecke operators and the Siegel $$\phi$$-operator.

##### MSC:
 11F27 Theta series; Weil representation; theta correspondences 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations 11F25 Hecke-Petersson operators, differential operators (one variable)
Full Text:
##### References:
 [1] Andrianov, A.N.: Rationality Theorems for Hecke series and zeta functions of the groups GL n and Sp n over local fields. Math. USSR Izv.3, 439-476 (1969) · Zbl 0236.20033 [2] Eichler, M.: Über die Idealklassenzahl total definiter Quaternion Algebren. Math. Z.43, 102-109 (1938) · JFM 63.0093.02 [3] Eichler, M.: Über die Darstellbarkeit von Modulformen durch Thetareihen. J. Reine Angew. Math.195, 159-171 (1956) · Zbl 0068.06601 [4] Hashimoto, K.: On Brandt matrices associated with the positive definite quaternion hermitian forms. J. Fc. Fac. Sci. Univ. Tokyo Sect. IA27, 227-245 (1980) · Zbl 0433.10017 [5] Hashimoto, K.: Class numbers of positive definite ternary quaternion hermitian forms. Proc. Jap. Acad. 59 Ser. A,10, 490-493 (1983) · Zbl 0539.10020 [6] Hashimoto, K., Ibukiyama, T.: On class numbers of positive definite binary quaternion hermitian forms (I). J. Fac. Sci. Univ. Tokyo Sect. IA27, 549-601 (1980), (II)28, 695-699 (1982); (III)30, 393-401 (1983) · Zbl 0452.10029 [7] Hecke, E.: Mathematische Werke. Göttingen: Vandenhoeck and Ruprecht 1970 · Zbl 0205.28902 [8] Howe, R.: ?-series and invariant theory. Proc. Symp. Pure Math. XXXIII, Part1, 275-286 (1979) · Zbl 0423.22016 [9] Ibukiyama, T.: On relations of dimensions between automorphic forms of Sp(2,R) and its compact twist Sp(2) (I). Adv. Stud. Pure Math.7, 7-29 (1985) · Zbl 0609.10018 [10] Ihara, Y.: On certain arithmetical Dirichlet series. J. Math. Soc. Japan16, 214-225 (1964) · Zbl 0126.07101 [11] Kashiwara, M., Vergne, M.: On the Segal-Shale-Weil representation and harmonic polynomials. Invent. Math.44, 1-47 (1978) · Zbl 0375.22009 [12] Krieg, A.: Das Vertauschungsgesetz zwischen Hecke-Operatoren und dem Siegelschen ?-Operator. Arch. Math.46, 323-329 (1986) · Zbl 0566.10019 [13] Kudla, S.S.: Seesaw dual reductive pairs. Progr. Math. 46. Birkhäuser: Basel, Boston, Stuttgart 1984 · Zbl 0549.10017 [14] Satake, I.: Theory of spherical functions on reductive algebraic groups over p-adic fields. I.H.E.S. Publ. Math.18, 5-69 (1963) · Zbl 0122.28501 [15] Shimura, G.: Arithmetic of alternating forms and quaternion hermitian forms. J. Math. Soc. Japan15, 33-65 (1963) · Zbl 0121.28102 [16] Shimura, G.: On modular correspondences for Sp(n, Z) and their congruence relations. Proc. Natl. Acad. Sci. USA49, 824-828 (1963) · Zbl 0122.08803 [17] Shimura, G.: Introduction to the arithmetic theory of automorphic functions. Tokyo Iwanami Shoten and Princeton: Princeton Univ. Press 1971 · Zbl 0221.10029 [18] Tanigawa, Y.: Construction of Siegel modular forms of degree three and commutation relations of Hecke operators. Nagoya Math. J.100, 83-96 (1985) · Zbl 0557.10023 [19] Weyl, H.: Classical groups. Princeton: Princeton University Press 1939 · Zbl 0020.20601 [20] Yoshida, H.: Siegel modular forms and the arithmetic of Quadratic forms. Invent. Math.60, 193-248 (1980) · Zbl 0453.10022 [21] Zharkovskaya, N.A.: The Siegel operator and Hecke operators. Funct. Anal. Appl.8, 113-120 (1974) · Zbl 0314.10021 [22] Zhuravrev, V.G.: Hecke rings for a covering group of the symplectic group. Math. USSR Sb.49, 379-399 (1984) · Zbl 0542.10021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.