Shimura Variety

The Shimura manifold (sometimes the Simura manifold ) is an analogue of a modular curve in higher dimensions, which arises as a factor of the in the congruent subgroup of the reductive algebraic group defined over Q. The term “Shimura variety” refers to high dimensions, in the case of one-dimensional manifolds they speak of Shimura curves . and are among the best known classes of Shimura varieties.

Special cases of Shimura varieties introduced Goro Shimura in the course of generalizing the theory of (modular curves). Shimura showed that initially analytically determined objects are arithmetic in the sense that they satisfy the models over the number field , the field of reflection of the Shimura variety. In the 1970s, Pierre Deligne created an axiomatic scheme for Shimura’s work. At about the same time, Robert Langlands noted that the Shimura varieties form a natural domain of examples for which the equivalence between the and postulated in the Langlands program can be verified. , realized in the cohomology of the Shimura variety, are more amenable to study than general automorphic forms. In particular, there is a construction that attaches to them.

Definition

Shimura Baseline

Let S = Res _{C / R} G _m be the Weyl restriction of the multiplicative group from complex numbers to real numbers . It is an algebraic group , the group of R- points of which is S ( R ) is C ^* , and the group of C- points is $\mathbf {C} ^{*}\times \mathbf {C} ^{*}$ ${\ displaystyle \ mathbf {C} ^ {*} \ times \ mathbf {C} ^ {*}}$ $\mathbf {C} ^{*}\times \mathbf {C} ^{*}$ . The original Shimura data is a pair ( G , X ) consisting of a reductive algebraic group G defined over the field Q of rational numbers , and G ( R ), the conjugacy class X of homomorphisms h : $S\rightarrow G{\mathbf {R} }$ ${\ displaystyle S \ rightarrow G {\ mathbf {R}}}$ $S\rightarrow G{\mathbf {R} }$ satisfying the following axioms:

For any h from X to g _C , only weights (0,0), (1, −1), (−1,1) can occur, that is, the complexified Lie algebra of G decomposes into a direct sum

{\mathfrak {g}}\otimes \mathbb {C} ={\mathfrak {k}}\oplus {\mathfrak {p}}^{+}\oplus {\mathfrak {p}}^{-},

{\ displaystyle {\ mathfrak {g}} \ otimes \ mathbb {C} = {\ mathfrak {k}} \ oplus {\ mathfrak {p}} ^ {+} \ oplus {\ mathfrak {p}} ^ {- },}

{\mathfrak {g}}\otimes \mathbb {C} ={\mathfrak {k}}\oplus {\mathfrak {p}}^{+}\oplus {\mathfrak {p}}^{-},

where for any z ∈ S h ( z ) acts trivially on the first term of the sum and by

z/{\bar {z}}

{\ displaystyle z / {\ bar {z}}}

z/{\bar {z}}

and

{\bar {z}}/z

{\ displaystyle {\ bar {z}} / z}

{\bar {z}}/z

) to the second and third members respectively.

The conjugate action h ( i ) generates a on the conjugate group of the group G _R.
The adjoint group for G _R does not obey the factor H defined over Q , so the projection of h onto H is trivial.

From these axioms it follows that X has a unique structure of a complex manifold (possibly incoherent), such that for any representation $\rho :G_{\mathbf {R} }\rightarrow GL(V)$ ${\ displaystyle \ rho: G _ {\ mathbf {R}} \ rightarrow GL (V)}$ family $(V\rho \cdot h)$ ${\ displaystyle (V \ rho \ cdot h)}$ is a holomorphic family of Hodge structures . Moreover, it forms a variation of the Hodge structure and X is a finite union of (disjoint) .

Shimura Variety

Let A _ƒ be group Q. For any sufficiently small compact open subgroup K of the group G ( A ), a

Sh_{K}(G,X)=G(\mathbb {Q} )\backslash X\times G(\mathbb {A} _{f})/K

{\ displaystyle Sh_ {K} (G, X) = G (\ mathbb {Q}) \ backslash X \ times G (\ mathbb {A} _ {f}) / K}

is a finite union of locally symmetric manifolds of form $\Gamma \smallsetminus X^{+}$ ${\ displaystyle \ Gamma \ smallsetminus X ^ {+}}$ where superscript plus denotes a connected component . Varieties $Sh_{K}(G,X)$ ${\ displaystyle Sh_ {K} (G, X)}$ are complex algebraic varieties and they form an over all sufficiently small compact open subgroups K. This inverse system

(Sh_{K}(G,X))_{K}

{\ displaystyle (Sh_ {K} (G, X)) _ {K}}

obeys natural right action $G(\mathbf {A} _{f})$ ${\ displaystyle G (\ mathbf {A} _ {f})}$ . It is also called the Shimura variety associated with the original Shimura data ( G , X ) and is denoted by Sh ( G , X ).

History

For special types of Hermitian-symmetric domains and congruent subgroups of Γ, an algebraic variety of the form $\Gamma \smallsetminus X=Sh_{K}(G,X)$ ${\ displaystyle \ Gamma \ smallsetminus X = Sh_ {K} (G, X)}$ and its were introduced in a series of articles by Goro Shimura during the 1960s. Shimura’s approach, later presented in his monographs, was largely phenomenological and pursued the goal of a broad generalization of the formulation of the reciprocity law of the theory of (modular curves). In retrospect, the name "Shimura variety" was introduced by Deligne , who tried to isolate abstract properties that play a role in Shimura's theory. In Deligne’s formulation, the Shimura variety is a domain of parameters of Hodge structures of some type. Then they form a natural generalization of modular curves of higher dimension, which are considered as moduli spaces of elliptic curves with a level structure.

Examples

Let F be a completely real number field and D be a quaternion division algebra over F. The multiplicative group D ^× generates the canonical Shimura variety. Its dimension d is the number of infinite places into which D splits. In particular, if d = 1 (for example, if F = Q and $D\otimes \mathbf {R} \cong \mathrm {M} _{2}(\mathbf {R} )$ ${\ displaystyle D \ otimes \ mathbf {R} \ cong \ mathrm {M} _ {2} (\ mathbf {R})}$ ), fixing a sufficiently small arithmetic subgroup of the group D ^× , we obtain the Shimura curve and the curves arising from this construction are already compact (that is, ).

Some examples of curves with known equations, given by low-genus Hurwitz surfaces :

3
Macbit Surface (genus 7)
(genus 14)

and the Fermat curve of degree 7. ^[1]

Other examples of Shimura varieties include and .

Canonical models and special points

Any Shimura variety can be defined over a canonical number field E called a reflection field . This important result, which belongs to Shimura, shows that Shimura varieties, which a priori are only complex varieties, have an algebraic and, therefore, have an arithmetic value. This forms the starting point in the formulation of the law of reciprocity, in which some arithmetically determined special points play an important role.

The qualitative nature of the Zarissky closure of sets of points on the Shimura variety is described by the Andre-Oort hypothesis . Conditional results can be obtained from this hypothesis, based on the generalized Riemann hypothesis .

Role in the Langlands Program

Shimura varieties play a prominent role in the Langlands program . It follows from the that the Hasse-Weyl zeta function of a modular curve is a product of L-functions associated with explicitly defined modular forms of weight 2. In fact, Goro Shimura introduced his varieties and proved his reciprocity law in process of generalization of this theorem. Etaler, Shimura, Kuga, Sato and Ihara studied the zeta functions of Shimura varieties associated with the GL ₂ group over other numerical fields and their internal forms (that is, the mutiplicative groups of quaternion algebras). Based on their results, Robert Langlands predicted that the Weyl zeta function of any algebraic variety W defined over a number field should be the product of positive and negative powers of automorphic L-functions, that is, should arise from a set of . However, statements of this type can be proved if W is a Shimura variety. According to Langlands:

The statement that all L-functions associated with Shimura varieties, and then with any motive defined by Shimura varieties, can be expressed in terms of automorphic L-functions [of his 1970 paper], is weaker, even very weak, assertions that all motif L-functions are equal to such L-functions. Nevertheless, although it is expected that a more rigorous statement is true, there is no good reason, as far as I know, to expect that all the motif L-functions will be attached to Shimura varieties.

Notes

↑ Elkis , section 4.4 (pp. 94–97) in Levy, 1999 .

Literature

Montserrat Alsina, Pilar Bayer. Quaternion orders, quadratic forms, and Shimura curves. - Providence, RI: American Mathematical Society , 2004. - T. 22. - (CRM Monograph Series). - ISBN 0-8218-3359-6 .
Harmonic Analysis, The Trace Formula and Shimura Varieties / James Arthur, David Ellwood, Robert Kottwitz. - AMS, 2005. - Vol. 4. - (Clay Mathematics Proceedings). - ISBN 978-0-8218-3844-0 .
Pierre Deligne . Travaux de Shimura // Séminaire Bourbaki, 23ème année 1970/71 Exp. No. 389 . - Springer, Berlin, 1971. - T. 244. - p. 123-165. - (Lecture Notes in Math.).
Pierre Deligne . Variétés de Shimura: interprétation modulaire, et de ectories de construction de modèles canonotes // Automorphic forms, representations and L-functions; Proc. Sympos. Pure Math., XXXIII (Corvallis, OR, 1977), Part 2. - Providence, RI: Amer. Math Soc., 1979. - p. 247–289.
Pierre Deligne , James S. Milne, Arthur Ogus, Kuang-yen Shi. Hodge cycles, motives, and Shimura varieties. - Berlin-New York: Springer-Verlag, 1982. - T. 900. - C. ii + 414. - (Lecture Notes in Mathematics). - ISBN 3-540-11174-3 .
The eightfold way / Silvio Levy. - Cambridge University Press , 1999. - V. 35. - (Mathematical Sciences Research Institute Publications). - ISBN 978-0-521-66066-2 .
Encyclopedia of Mathematics / Michiel Hazewinkel. - Springer Science + Business Media BV / Kluwer Academic Publishers, 2001. - ISBN 978-1-55608-010-4 .
Milne J. Shimura varieties and motives // Motives , Proc. Symp. Pure Math, 55: 2 / Jannsen U., Kleiman S .. Serre J.-P .. - Amer. Math Soc, 1994. - p. 447-523.
Milne JS Introduction to Shimura Varieties . - 2004.
Harry Reimann. The semi-simple zeta function of quaternionic Shimura varieties / Dold E., Takens F .. - New York, London, Paris, Tokio: Springer, 1997. - V. 1657. - (Lecture Notes in Mathematics). - ISBN 3-540-62645-X .
Goro Shimura . The Collected Works of Goro Shimura // {{{title}}}. - 2003—2016. - T. 1–5.
Goro Shimura . Introduction to Arithmetic Theory of Automorphic Functions. - Princeton, New Jersey: Princeton University Press, 1994. - Vol. 11. - (Publications of the mathematical society of Japan). - ISBN 0-691-08092-5 .

[1] Elkis , section 4.4 (pp. 94–97) in Levy, 1999 .

Shimura Variety

Definition

Shimura Baseline

Shimura Variety

History

Examples

Canonical models and special points

Role in the Langlands Program

Notes

Literature

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