Models of certain automorphic function fields

Miyake, Katsuya

doi:10.1007/bf02392033

Cited by 25 publications

(19 citation statements)

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“…One of our main task then is to prove that the isomorphism of V~ to V u, induces an isomorphism ofV~to Vxl for some arithmetic subgroup Fx~ of G1. This can be achieved by studying the isolated fixed points of G o and G1 o" We carry out these considerations in Sections 2 and 3.The same functorial property also holds for the models constructed by Miyake in [3]. In Section 4 we deal with this case briefly.…”

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confidence: 83%

Conjugations of arithmetic automorphic function fields

Shih

1978

Invent Math

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In [1], Doi and Naganuma showed that the conjugation ofa Shimura curve is again a Shimura curve. The present paper deals with the generalization of their result.Consider in general a reductive algebraic group G defined over Q. Let G u be the semi-simple part of G. Assume that G~t modulo a maximal compact subgroup defines a bounded symmetric domain ~, and that a system of canonical models (in the sense of Shimura [11, 2.13]) for the quotients of ~ by the arithmetic subgroups Yx of G exists. Let {Vx, (gx, Jxw(U)} be such a system. Take an arbitrary automorphism z of the complex number field C, and conjugate all the Vx's and Jxw(U)'S by z. Then one expects that {V:~, C~x, Jxw(u) ~} with suitable ~x's forms a system of canonical models for some reductive group G1.In this paper we show that this is the case if G is the type of groups investigated by Shimura in [11]. The corresponding group G1 is defined explicitly in 1.2, and the precise statement of the result is given as Theorem 1.3. In Shimura's construction, for special Yx the model Vx is first realized as a s ubvariety of a moduli-variety Vo for some PEL-type f2. Consider V~ as embedded in V~. It is known that the conjugate variety V~is isomorphic to the moduli-variety V w of another PEL-type f2'. The relations between f2 and f2' are provided by Shimura's work [6]. One of our main task then is to prove that the isomorphism of V~ to V u, induces an isomorphism ofV~to Vxl for some arithmetic subgroup Fx~ of G1. This can be achieved by studying the isolated fixed points of G o and G1 o" We carry out these considerations in Sections 2 and 3.The same functorial property also holds for the models constructed by Miyake in [3]. In Section 4 we deal with this case briefly. We shall not give the proof, because the argument is similar to, and actually simpler than, the one presented in this paper.We assume the reader is familiar with Shimura's work [10] and [11], which will be quoted respectively as [A] and [C] hereafter.

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confidence: 83%

Conjugations of arithmetic automorphic function fields

Shih

1978

Invent Math

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“…(5) What is the structure of each component as a topological or a real analytic manifold? (5) What is the structure of each component as a topological or a real analytic manifold?…”

Section: Introductionmentioning

confidence: 99%

On the real points of an arithmetic quotient of a bounded symmetric domain

Shimura

1975

Math. Ann.

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“…The issue (ii) is treated in [3], [5], [7], and papers of Shimura. See forthcoming papers of the present author for treatment of (i).…”

Section: Paul B Garrett Iii) If P Is In Aw(eqab) Where W Is In X Imentioning

confidence: 99%

Arithmetic Properties of Fourier-Jacobi Expansions of Automorphic Forms in Several Variables

GARRETr¹

1981

American Journal of Mathematics

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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics.In this paper we establish some basic arithmetic results concerning automorphic forms in several variables. Our results are, in part, analogues of the classical facts for elliptic modular forms, and results of Shimura on Hilbert and Siegel modular forms, [9] and [10]. In essence, this involves relating rationality properties of Fourier-Jacobi coefficients to rationality properties in algebraic geometry.Our automorphic forms are C-valued holomorphic functions on certain Cartesian powers Hnd of Siegel upper half-spaces Hn. We consider appropriate arithmetic groups r acting on Hnd such that r\H,d is noncompact. In particular, we reduce our study to that of congruence subgroups r of reductive linear algebraic groups G such that the derived group G' is almost simple over Q. Further, let K be a maximal compact subgroup of G'(R)+, the connected component of the identity in the real points of G'. We only consider G such that G'(R)+/K is isomorphic to Had. Up to an irrelevant central torus, all such groups G are obtained as totally indefinite quaternion unitary groups (see [8] and [15]). Therefore, we are able to describe very explicitly the action of such groups on the appropriate domains.It will become clear that the groups obtained as unitary groups over even-dimensional quaternionic vectorspaces, or from quaternion matrix algebras, have a simple nature: for such groups, one can write Fourier expansions for automorphic forms in the same way that one does for Hilbert of Siegel modular forms. In particular, the Fourier coefficients are numbers.The groups arising from odd-dimensional vectorspaces over a quater-nion division algebra are more complicated: we can only write a Fourier-Jacobi expansion as in [6]. Specifically, we use coordinates zz uX (z, u, T) = Vtu TJ on H d +1 such that z is in H2nd, T is in H1d, u is in C2nd. An automorphic form f will have a Fourier-Jacobi expansion f(z, u, T) = i ax(u, T) exp(trxz), x as x ranges over a suitable lattice of d-tuples of positive semidefinite 2n-by-2n real matrices. Here ax is a theta function in the u variable, and expL(T)ax(uo, T) is an automorphic form on H1d, when uo is a division point of the period lattice of ax, and L is a holomorphic function of T, which depends on uO. For these groups and domains (among others), Shimura describes in [8] a field of functions with special arithmetic properties. Their values at CM-points in the domain generate some abelian extensions of related CMfields. By the results of [2], the quotients r\H d have normal irreducible quasi-projective models. By [8], there exist m...

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Models of certain automorphic function fields

Cited by 25 publications

References 13 publications

Conjugations of arithmetic automorphic function fields

Conjugations of arithmetic automorphic function fields

On the real points of an arithmetic quotient of a bounded symmetric domain

Arithmetic Properties of Fourier-Jacobi Expansions of Automorphic Forms in Several Variables

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