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Automorphism Groups of Shimura Varieties

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25 Documenta Math Automorphism Groups of Shimura Varieties Haruzo Hida Received April 14 2003 Revised February 5 2006 Communicated by Don Blasius Abstract In this paper we determine the scheme automorphism group of the reduction modulo p of the integral model of the connected Shimura variety of prime to p level for reductive groups of type A and C The result is very close to the characteristic 0 version studied by Shimura Deligne and Milne Shih 2000 Mathematics Subject Classification 11G15 11G18 11G25 Keywords and Phrases Shimura variety reciprocity law There are two aspects of the Artin reciprocity law One is representation theoretic for example Homcont Gal Qab Q C Homcont A Q C via the identity of L functions Another geometric one is Gal Qab Q GL1 A Q They are equivalent by duality and the first is generalized by Langlands in non abelian setting Geometric reciprocity in non abelian setting would be via Tannakian duality so it involves Shimura varieties Iwasawa theory is built upon the geometric reciprocity law The cyclotomic field Q p is the maximal p ramified extension of Q fixed by b p A Q R removing the p inertia toric factor Z We then try to Z p study arithmetically constructed modules X out of Q p Qab The main idea is to regard X as a module over over the Iwasawa algebra which is a completed Hecke algebra relative to are used to determine X GL1 A b p Q GL1 Z and ring theoretic techniques Documenta Mathematica 11 2006 25 56 26 Haruzo Hida If one wants to get something similar in a non abelian situation we really need a scheme whose automorphism group has an identification with G A Z Q for a reductive algebraic group G If G GL 2 Q the tower V Qab of modular curves has Aut V Q identified with GL2 A Z Q as Shimura proved The decomposition group of p is given by B Qp SL2 A p 1 for a Borel subgroup B and I have been studying various arithmetically constructed 2 A modules over the Hecke algebra of GL GL b p U Z Q relative to the unipotent Z 2 p subgroup U Zp B Zp removing the toric factor from the decomposition group Such study has yielded a p adic deformation theory of automorphic forms see PAF Chapter 1 and 8 and it would be therefore important to study the decomposition group at p of a given Shimura variety which is basically the automorphism group of the mod p Shimura variety Iwasawa theoretic applications if any are the author s motivation for the investigation done in this paper However the study of the automorphism group of a given Shimura variety has its own intrinsic importance As is clear from the construction of Shimura varieties done by Shimura Sh and Deligne D1 2 4 7 their description of the automorphism group of Shimura varieties of characteristic 0 is deeply related to the geometric reciprocity laws generalizing classical ones coming from class field theory and is almost equivalent to the existence of the canonical models defined over a canonical algebraic number field Except for the modulo p modular curves and Shimura curves studied by Y Ihara the author is not aware of a single determination of the automorphism group of the integral model of a Shimura variety and of its reduction modulo p although Shimura indicated and emphasized at the end of his introduction of the part I of Sh a good possibility of having a canonical system of automorphic varieties over finite fields described by the adelic groups such as the ones studied in this paper We shall determine the automorphism group of mod p Shimura varieties of PEL type coming from symplectic and unitary groups 1 Statement of the theorem Let B be a central simple algebra over a field M with a positive involution thus TrB Q xx 0 for all 0 6 x B Let F be the subfield of M fixed by Thus F is a totally real field and either M F or M is a CM quadratic extension of F We write O resp R for the integer ring of F resp M We fix an algebraic closure F of the prime field Fp of characteristic p 0 Fix a proper subset of rational places including and p Let F be the subset of totally positive elements in F and O denotes the localization of O at disregarding the infinite place in and O is the completion of O at again disregarding the infinite place We write O F O We have an Documenta Mathematica 11 2006 25 56 Automorphism Groups of Shimura Varieties 27 exact sequence 1 B M Autalg B Out B 1 and by a theorem of Skolem Noether Out B Aut M Here b B acts on B by x 7 bxb 1 Since B is central simple any simple B module N is isomorphic each other Take one such simple B module Then EndB N is a division algebra D Taking a base of N over D and identifying N D r we have B EndD N Mr D for the opposite algebra D of D Letting Autalg D act on b Mr D entry by entry we have Autalg D Autalg B and Out D Out B under this isomorphism Let OB be a maximal order of B Let L be a projective OB module with a non degenerate F linear alternating form h i LQ LQ F for LA L Z A such that hbx yi hx b yi for all b B Identifying LQ with a product of copies of the column vector space D r on which Mr D acts via matrix multiplication we can let Autalg D act component wise on LQ so that bv b v for all Autalg D Let C be the opposite algebra of C EndB LQ Then C is a central simple algebra and is isomorphic to Ms D and hence Out C Out D Out B We write CA C Q A BA B Q A and FA F Q A The algebra C has involution given by hcx yi hx c yi for c C and this involution of C extends to an involution again denoted by of EndQ LQ given by TrF Q hgx yi TrF Q hx g yi for g EndQ LQ The involution resp induces the involution 1 resp 1 on CA resp on BA which we write as resp simply Define an algebraic group G Q by G A g CA g gg FA for Q algebras A 1 1 e of G by the following subgroup of the opposite group and an extension G AutA LA of the A linear automorphism group AutA LA e G A g Aut A LA gCA g 1 CA and g gg FA 1 2 Since C EndB LQ we have B EndC LQ and from this we find that gBg 1 B gCg 1 C for g AutQ LQ and if this holds for g then gg F gx g 1 gxg 1 for all …


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