 BostConnes, ,   
 


 BostConnes   -  ,     ,     -  .





 BostConnes, ,   



 



 ,2026



ISBN978-5-0069-4564-7

     Ridero


 BostConnes   ,   

 BostConnes   -  ,    ,     -  .










 BostConnes  C*-   $ A_K = C^* (\mathbb {Q} /\mathbb {Z}) \rtimes \mathbb {N} $,      $ \sigma_t $,     .


    $ \mathbb {Q} $    $ \sigma_t (u_n) = n^ {it} u_n $,  $ u_n $ ,   $ \mathbb {Q} /\mathbb {Z} $.


       $ \zeta (s) = \sum_ {n=1} ^\infty n^ {-s} $.







KMS-  


KMS_?- (Kubo-Martin-Schwinger)       ?.  ?> 1  KMS_?-;  0<? ? 1  KMS_?-  ,  -   .


  -    T=0,    .










,     - .


         Drinfeld-    .




 

Bost J.-B., Connes A. Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking // Selecta Math. (N.S.). 1995. Vol. 1, 3. P. 411457. ArXiv: math/9507002


.

Connes A., Marcolli M., Ramachandran K. Bost-Connes systems for Shimura varieties // Selecta Math. (N.S.). 2008. Vol. 14. P. 179229. ArXiv: math/0603773


.

Ha E., Paugam F. Bost-Connes systems for Shimura varieties. II.The complex case // Adv. Math. 2010. Vol. 224, 4. P. 16211655


.



?



 KMS-  BostConnes

KMS- (Kubo-Martin-Schwinger)  BostConnes       $ \beta> 0$,    $ \beta = 1$.







 KMS-


 $ \phi $ C*- $ A$   $ \sigma_t $  KMS_\beta-,    $ x, y \inA$    $ F_ {x,y} (z) $  $ 0<\Im (z) <\beta $,   , :













KMS- Bost-Connes

  $ (A_\mathbb {Q}, \sigma_t) $,  $ A_\mathbb {Q} = C^* (\mathbb {Q} /\mathbb {Z}) \rtimes \mathbb {N} $ $ \sigma_t (u_n) = n^ {it} u_n $ ($ u_n $   $ 1/n \mathbb {Z} /\mathbb {Z} $),  KMS- :




 $ \beta> 1$:  KMS_\beta-頖  (  ).

 $ 0<\beta \leq 1$:  KMS_\beta-  $ \mathrm {Gal} (\mathbb {Q} ^ {ab} /\mathbb {Q}) $,  -.

  $ \phi_u$ ( $ u\in\widehat {\mathbb {Z}} ^\times $) :








 $ H $  $ e^ {-\beta H} e_n = n^ {-\beta} e_n $, $ Z_\beta = \zeta (\beta) $, $ \pi_u$ .







 


  $ Z_\beta = \mathrm {Tr} (e^ {-\beta H}) = \sum_ {n=1} ^\infty n^ {-\beta} = \zeta (\beta) $,  - .


  $ T=0$ ($ \beta = \infty $) KMS_\infty-   KMS_\beta  $ \beta \to\infty $.






?



     ? (?)  KMS-

  $ Z(\beta) $  BostConnes      ,    ? (?).







  


C*- $ A_\mathbb {Q} $   $ u_n $ (n ? 1),  $ u_n u_m = \delta_ {n,m} u_n $, $ \sum_n u_n = 1$,   $ s_ {m,n} $ $ s_ {m,n} s_ {n,k} ^* = \delta_ {n,k} s_ {m,k} $, $ s_ {m,n} ^* s_ {m,n} = u_n $.


  $ \sigma_t (u_n) = n^ {it} u_n $, $ \sigma_t (s_ {m,n}) = (m/n) ^ {it} s_ {m,n} $.


  $ H $   $ {u_n} $  $ H u_n = (\log n) u_n $,   $ e^ {-\beta H} u_n = n^ {-\beta} u_n $.







  


  KMS- $ Z(\beta) = \mathrm {Tr} (e^ {-\beta H}) $,  䠖    :








    $ {\mathrm {id} \otimes u_n} $   .










 ?> 1  ;  ?=1     Gal ($ \mathbb {Q} ^ {ab} /\mathbb {Q} $).


     ?_K (?)  .






?



      BostConnes

     BostConnes    ,     O,   ,    ?_K (?).







 K-


 -  K = ? (?d)   O= ? + ?? (? ? ?)    GK   1- K- (?, ?),  ? ? ? O- K? ? K, ?: K/O? K?/?.


 C*- AK = C* (GK)      K-,    C*.


  ?_t (f) (L, L) = |L/L|^ {it} f (L, L),  |L/L|  .







 


  K- L = (?, ?)  (GK) _L   J ? Ovia J^ {-1} L = (?, ?) ? = {x ? ? | xJ ? ?}.


   n (J) = covol (?) / covol (?),  H ?? ()  H ?_J = (log n (J)) ?_J.


  e^ {-?H} ?_J = n (J) ^ {-?} ?_J.







 


 Tr (e^ {-?H}) = ?_ {J  O} n (J) ^ {-?} = ?_K (?),  ?_K (?) = ?_ {J} n (J) ^ {-?}  ,        ?_J.


 KMS_?-  L: ?_ {?,L} (f) = ?_K (?) ^ {-1} ?_J f (J^ {-1} L, J^ {-1} L) n (J) ^ {-?}, ?> 1.







 


  Connes-Marcolli-Ramachandran (2005): KMS states and complex multiplication,        -.


     Drinfeld-  ?_ {k,?} (?).






?



     

      C*-     - ,   BostConnes  GL  CM-    .





















1.   

 C*- , 堖  ,   (,  ).


























 ( )  KMS- ( )    ;   -     




























2. KMS-  

KMS-

 C*- 堖 KMS,      






































 KMS-堖   -KMS .


























 

   ,   KMS- 























  BC ()













 GL-













  CM-  - $$

Z_K (\beta) =\sum_ {J\ \ O} N (J) ^ {-\beta} =\zeta_K (\beta). [file:56]








3. Q- BC/GL 








1- Q- BC-

Q- : , 堖 ,. [file:56]























:,    . [file:56]













  1- Q-     








    BostConnes. [file:56]

  , . [file:56]













2- Q- GL-








2- Q- : ,. [file:56]























  2- Q- up toscale    -

$$

\mathrm {Sh}


\pm) =GL_2(\mathbb {Q}) \backslash (M_2(\mathbb {A} _f) \times H^\pm).




    















$$

(f_1*f_2) (g,\alpha, z) =\sum_ {s\in\Gamma\backslash GL_2^+ (\mathbb {Q}),\ s\alpha\inM_2(\widehat {\mathbb {Z}})}

f_1(gs^ {-1},s\alpha, s (z)),f_2(s,\alpha, z). [file:56]

 :








    

$$

Z(\beta) =\sum_ {m\in\Gamma\backslash M_2


 {-\beta} =\zeta (\beta) \zeta (\beta-1).




4. K- CM- ( )

 K-

 - ,   ,.




1- K-: , 


























  -;


















  -.
















: .
















 1- K-  








$$

\rho\in\widehat {O},\ s\inA_K


\ast,\quad (\rho, s) \sim (x


 \ast. [file:56]

  up toscale  

$$

A_ {K,f}


\ast \cong \widehat {O} /K^\ast.







 







   K- (up toscale),    $ \mathbb {C} ^\ast$:








$$

G_K=\widetilde {R} _K/\mathbb {C} ^\ast. [file:56]

 CM-:








 ,      








$$

X\simeq \widehat {O} \times_ {\widehat {O}


 \ast/K^\ast).




    :

$$

|L/L|=\frac {\mathrm {covol} (\Lambda)} {\mathrm {covol} (\Lambda)},\quad

\sigma_t (f) (L,L) =|L/L|^ {it} f (L,L). [file:56]

  : CM- -    








$$

C_0(A_ {K,f} /O^\ast) \rtimes (K


\ast).







, 







蠖 invertible K-,  K-,  ,    























J^ {-1} L= (\Lambda,\varphi),\quad J= {x\inO\mid x\Lambda\subset \Lambda}.




  













  

$$

Z_K (\beta) =\mathrm {Tr} (e^ {-\beta H}) =\sum_ {J} N (J) ^ {-\beta} =\zeta_K (\beta). [file:56]

KMS- CM- (5.1)

  : [file:56]








   KMS-.













  KMS-   K- ( ):























   ,   :













  KMS-  ;  - . [file:56]























5.   

    ; ࠖ , ࠖ  . [file:56]

 ,    -

































 KMS-;    








        . [file:56]








 蠖  K-,


































$$

\alpha\circ\varphi_ {\infty,L}

=\varphi_ {\infty, L} \circ\theta^ {-1} (\alpha) \ \ A_ {K,\mathbb {Q}}.




      (Hilberts 12- )   :   KMS- ,  -    .





















       (,    5.1  ,, 


), 堖      .

?



     

      ; 堖 ,    ,     .






1.  (Section1)

:       C*- (KMS-)  ,    BostConnes (BC) GL-.











 :

BC-: C*-   ,      (KroneckerWeber).
















GL-:  BC 2- -  ; KMS-     .


























  : CM-  - ,    ,   K-.
















 :  KMS- CM-  ,    -   .
















  :

BostConnes, Hecke algebras, type III factors and phase transitions Selecta Math. 1(1995).




ConnesMarcolli, From physics tonumber theory via noncommutative geometry. Part I(Q-lattices, GL-system).











Connes, Trace formula innoncommutative geometry and the zeros ofthe Riemann zeta function.







, 


, 


  /    (Shimura, Stevenhagen, Weil).






2.      (Section2)

2.1.  KMS-

		 - :











		  C*- ().








		 1-   ( ).








		:, ,.





















		KMS- :













		    .
















		KMS:   KMS .





















2.2.   

:    KMS-,  .





















 :   ,     KMS.





















 ,  ,  .   ?     .


























 :













2.3.    (Hilbert12)

  :

, , -,  .































   :








   :

   , :
















  ,  .
















:








  ;


















.
















   :




























 2:

BratteliRobinson, Operator algebras and quantum statistical mechanics.




, Haag, HaagHugenholtzWinnink  KMS-.




, 


    Hilbert 12.






3. Q-, BC GL   - (Section3)








3.1. Q- BC-(GL)








 3.1. Q- : , 堖 ,.


























: .
















:








,.
















 :,   


















BC:








   :








 , -.
















   :

$$

H\epsilon_k=\log k,\epsilon_k,\quad Z(\beta) =\sum_ {k\ge1} k^ {-\beta} =\zeta (\beta). [file:56]

KMS- BC-: .    ( ,   ,  ). [file:56]




























 :




, 


 BostConnes. [file:56]

,  -  KMS-. [file:56]

3.2. GL-








2- Q-:,,. [file:56]


















 








$$

\Gamma\backslash (M_2(\widehat {\mathbb {Z}}) \times\mathbb {H}), \quad \Gamma=SL_2(\mathbb {Z}).




       -













$$

\mathrm {Sh}


\pm, GL_2) =GL_2(\mathbb {Q}) \backslash (M_2(A_f) \times H^\pm). [file:56]

 : 


















 ,  (3.31). [file:56]








 :








.  :








$$

Z(\beta) =\zeta (\beta),\zeta (\beta-1).




KMS- GL-:  KMS ;   KMS  




  .


   .

   ,     (https://www.litres.ru/pages/biblio_book/?art=73516368)  .

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