Class numbers of positive definite Ternary quaternion Hermitian forms

Hashimoto, Ki-ichiro

doi:10.3792/pjaa.59.490

Cited by 46 publications

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“…On the other hand, we have computed in [4,6] the dimension of the spaces of automorphic forms attached to the positive definite binary quaternion hermitian ! This work has been partially supported by SFB 40 "Theoretische Mathematik" and Max-Planck-Institut fiir Mathematik, Bonn forms, motivated by a conjecture of Ihara on a possible relation between the automorphic forms on definite and indefinite quaternion unitary groups [-7, 8].…”

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confidence: 99%

The dimension of the spaces of cusp forms on Siegel upper half plane of degree two

Hashimoto

1984

Math. Ann.

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The dimension of the spaces of cusp forms on Siegel upper half plane of degree two

Hashimoto

1984

Math. Ann.

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“…When n=3, D=2, and (DI,D2)=(2,1), we have h=H=l[2,5] and dimM2, 2(2, 1) = 1. For any positive integers k and d, we denote by Ak(Fo(d)) [resp.…”

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confidence: 99%

On automorphic forms on the unitary symplectic group Sp(n) andSL 2(R)

Ibukiyama

Ihara

1987

Math. Ann.

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We denote by Sp(n) [resp. Sp(n, lR)] the compact (resp. split) real form of the symplectic group of degree n (size 2n). Then there are two natural embeddings: (i) (Sp(1) x Sp(n))/+ (I, 1)c. SO(4n) (the compact orthogonal group) and (ii) Sp(r, IR) x SO(4n) % Sp(4rn, IR) (a dual reductive pair [8]; r= 1,2, ...).From the theory of the Weft representations and that of Howe's dual reductive pairs, one anticipates a global correspondence between automorphic forms, F on Sp(1) x Sp(n) and OF on Sp(r, IR), "defined" by the above two embeddings via theta series. The aim of this paper is to study such a correspondence, giving explicit relations between their L-functions L(s, F) and L(s, 0F) for the case n > 2 and r = 1. This generalizes [10] which was for (n, r) = (2, 1). We note that the Saito-Kurokawa lifting can be regarded as a split symplectic version of [10]. We also add that Yoshida [20] studied the case (n, r) = (1, 2), and Tanigawa [18], the case (n, r) = (1,3). Thc global algebraic group that we consider here iswhere B is a definite quaternion algebra over the rationals Q and GR,, is the group of similitudes of the standard positive quaternion hermitian metric on B': G n,, = { g e G L,( B); g-tg = n(g)l, with some n(g) ~ Q • }.* The first named author was partially supported by Max Planck Institute fiir Mathematik and SFB 40, during the preparation of this paper Incidentally, this restriction on the weight follows automatically from the assumption r= 1 [! 1]. For each such automorphic form F on G A we associate its "theta series" Ov which is a form on GL(2)A. Our first main result (Theorem 1, Sect. 2) gives an explicit relation between two L-functions L(s, F) and L(s, Or).There are examples of F with 0 v + 0 even for n > 3 (cf. Sect. 3). Our standard choice of the level makes it easy to treat automorphic forms in a "classical manner". For example, the study of Hecke operators acting on forms (spherical functions) is reduced to analysis of their dual actions on "invariant 0-cycles" on some B-spaces. Our second result (Theorem 2, Sect. 2) describes the eigenvalues of Hecke operators for the eigenfunction F by the sum of special values of F on invariant 0-cycles. The Proof of Theorem 2 will be given in Sect. 4. In Sect. 5, we first give some refinement of Theorem 2, and then we prove Theorem I by using this and applying the commutation relation of Hecke operators and the Siegel ,b-operators (at good non-archimedean places). We note that the pair (Sp(l) • Sp(n), Sp(r, lR)) is not a dual reductive pair for n>3, even in the weak sense of Kudla [13]. The existence of a "good" global correspondence even in such a case may suggest that we can use the theory of the Weil representations for a wider class of reductive groups. It is the authors' desire to draw attentions to this by showing such an example, that motivated us to publish these results. It seems to be an interesting problem to generalize our results starting from other subgroups G' C SO(n') [instead of the above embedding (i)]. In this case, the main problem would...

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“…Contributions from conjugacy classes of regular elliptic elements. To compute the total contribution from conjugacy classes of regular elliptic elements, it is unnecessary to classify their conjugacy classes explicitly [12]. The total contribution can be written as T.yM(y) ■ I(y) where y ranges over all representatives of con- Here a and ft are constants independent ofk.…”

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confidence: 99%