2011 3(13)
511.2
1
. .
. . . , . ,
E-mail: grechnik@mccme.ru
. , , . , .
: , , , .

, [1, 2], [3] [4] . , . , , , , .
. , . , ; , . , , 3, [3]. , , .
, . , , . . 1 .
. , . 2 3. . 4 , . 5, 6 7 .
11-01-12098.
1.
1.1. , D Z ,
D < 0 D = 0 (mod 4), D = 1 (mod 4).
K = Q(v D); d. d < 0

d = 1 (mod 4) d , (

dd d = 0 (mod 4), , - = 1 (mod 4). (
, D = f2d, f N. O = Z (d + v^d)/2

K OD = Z (D + vD)/2 O f. M , ; , = .
Od K, Od- ; Od , Od ( ). () Od () , , { K : } = OD. OD [5, 7], I (Od ). , , . . aOD K*, I(OD), P(OD). , ; ~ b b. . P(Od) , - Hd = I (Od )/P(Od ). [5, 7]. Od K , ; HD.
Ax2 + Bxy + Cy2, A, B, C Z. (A, B, C). B2 4AC, , (x,y) 1. , A > 0, , gcd(A, B, C) = 1. D, . () At2++C = 0,
H = {z C : Im z > 0}, . . =---+ ^ K H. -
2A
, |B| ^ A ^ C, B < 0, |B| < A < C. [5, Theorem 2.8].
Hd - , h, [5, Theorem 7.7]: = (A, B, C) h() = h(A, B,C) Od, (1, )z ( Od), .
: , |B| ^ A ^ \/|D|/3 A, B C. Hd .
j H [6, 46] j- C [5, 10], , , H j () = j ((1, )z). Od , C. K*; , ^'() . , Ax2 + Bxy + Cy2 , ~ (1, )z, ^'() = j ().
n Z [n], P nP. , [1] . ( ), , . n Z [n] [7, Example III.4.1] ( [7, Theorem III.4.8]); Z- n^ = [n] ^, n Z, ^ . {[n] : n Z} , Z [7, Proposition III.4.2].
C {[n] : n Z}, [7, Corollary III.9.4 Exercise 3.18b]. , . |Hd | Od [7, Proposition C.11.1], , j- j (), Od . . K L = L(D) = K(j()), OD (ring class field) [5, Theorem 11.1]. L/K Hd [5, 9]. , , b , ^'() j(-1) [5, Corollary 11.37]. : ^'() = j () j [5, 10], ~ OD [5, (7.6)], ,
j() = -1).
h
HD[j](x) = (x j(^)), h = |HD| ї
i= 1
. L, Gal(L/K) , Q. , j(^ [5, Theorem 11.1], HD [j](x) Z[x].
p , n , q = pn, E Fq. E Fq- Fq. Fq-. End(E) , Q [7, Corollary III.9.4 Theorem V.3.1]. E , . End(E) = Od D, (1); End(E) = ([1],a)Z, E. [8, 13.14], ї C : L', E', , ' E', B' OL>, p ( B'nZ = pZ), E' B', E, ' . {[n] : n Z}, ' {[n] : n Z}, End(E') = Z E' . [8, 13.12], End(E') End(E). ' , End(E') = End(E) = OD.
End(E) Fr : (x,y) - (xq,yq) Fr, Fr Fr = [q] [7, Theorem III.6.2 Proposition 2.11] [|E(Fq)|] = [| Ker([1] Fr)|] = ([1] Fr) ([1] Fr) ( , Fr Fq , [7, Theorem III.4.10, Corollary III.5.5, Theorem III.6.2]). OD = End(E), Fr, , , Fr, ; = q (1)(1) = = |E(Fq)|. , R , ; R, R Od = Z, , Fr = [], Fr = Fr [7, Theorem III.6.2], .
OD, u,v Z, , = (u + vvD)/2. = (u vvD)/2 q = = (u2 Dv2)/4, . . 4q = u2 + |D|v2. E 1 + = q +1 u; [7, Exercise 5.10] , (q, u) = 1 .
: E Fq, D, L' E' , ,
E' Od ;
E', E;
E q + 1 u, u Z , v Z
4q = u2 + |D|v2.
, :
1) = , , , V, , (1) :
+1 , .
.2.
4q = u2 + IDI^2; gcd(u,q) = ^
(4)
()
2)
3)
[]().
[]() ; , , , -
. - " -, .
// . , -. , , //, + 1 .
, Z, p , , p
p , p =2 = 0,1 (mod 4)
1, a = 1 (mod 8),
-1, a = 5 (mod 8),
, a = (mod 4).
N .
(4) (5) . , .
1. 6 < 0 (2) (3), = 6/2. = , V 4 :
1)
u2 + IDIv2, gcd(q,u) = 1.
2)
3)
p tf; _ _ pO = pp, p = p u + v\f~D
O;
2
u + vv^D
(9
D;
O = pn

u + v^fD
. 4q = u2 + |D|v2 p f u , p f D p f ; , p u2 Dv2 = = 0 (mod p), D = (uv-1) (mod p). , D = df2.
p f D.
(, [9, 13.1.3 13.1.4]).
2
4q = u2 + ID I v2. D , u , OD
Z
(D + vD)/2
= Z
(vD)/2
(u + vvD)/2 = u/2 + vvD/2 OD. D , u = v (mod 2), OD

Z
(D + vD)/2
Z
(1 + vD)/2 (u + vVD)/2 = (u v)/2 + v(1 + vD)/2 Od.
2
2
u v / Dd u___v / /D
, ----- ----O ---- ----O = qO = pnO = pnpn. p f u,
_ u ± v^J~D
pp = pO f----------O. -
O.
.
q .
q , D, (). D (); , D. D (). [0], x2 DIy2 = m, m = 4q. , D. , , q ± x .
q , , , D, (), u,!, q (4) q 1 ± , . [] [2] , , , ( ) q = (u211DIv2)/4 q 11 ± u. , p q 1 ± u ( [2]), , D = 5 (mod 8), v ; [2] u = 210u0 1, = 210v0 105, u0,v0 , ( ) u 106 104 = 210 106. , 210 = 105 2 = 2 3 5 . (u21 ID I!2)/4 q ± u 2, 3, 5, . [] , . [2].
Hd [j](x). h = = IHd I , ,... , Th, j (),... , j (Th) . Hd [j](x). , 0,, , , .
M , C OM , z OM, RC(z) z C ( , Rc Om ). Rc Om[x] .
, Rpz(Hd[j](x)) Fq, j-.
pO = pp. B Ol , p. B Z = p Z = pZ,
RpZ(Hd[j](x)) = rb(Hd[j](x)) = (x - RB(j(ai))) (7)
i=l
( []()) ^/. , .
1.
Ol/ Fq.
. C ^ , C OL, -
c O, , [, ] ( K , K , ), ([, Lemma .9]) Gal(/K), a (a) = aNorm(c) (mod C) a Ol. Gal(/K) = Hd , c [, Corollary .2] ^^ . b = c^1 ... ckk O Q , , ( / ) = ( ) ... ( ) . ( )
V b V ci VcW \ J
O, , , Gal(/K). .
Pk,z(/) [, 9] O, aO, a O, a = a (mod /O) a Z, gcd(a, /) = 1. [, 9], OD K, ,
O, , /O (, , PK,Z(/) : a = a (mod /O), gcd(aO, /O) = = gcd(aO, /O) = gcd(a, /)O = O, aO /O);
PK,Z(/).
= (u Vv/D)/2; , pn = O, pn = O. pn Pk,z(/) ( = (u V\/D)/2 = = (u /Vd)/2 (mod /O), = (u VV/)/2 = (u /!d)/2 (mod /O), Pk,z(/), (4)), , .
( /K \ n
, ( pJ = Id. , n-
Norm(p) = p, pn = q, Ol/B , , Ol/B Fq.
[7, Proposition ..], , j Fq j C , Fq C, j -, j :
, 0 , j = 0 j = 128:
2 3 j
y2 = x3 3cx 2c, c =------;
128 j
, 0 , j = 0: y2 = x3 1;
, 0 , j = 128: y2 = x3 x;
Fq 2 j F*: 2 xy = x3 j-1;
Fq 3 j F*: y2 = x3 x2 j-1.
j = 0 2 3 [7, Exercise .7, Theorem 4.], .
, // Od (, , ). .
(7) E" , j- RB(j(a)), Od . j () Ol, [13, 4.3] L/ L, ;, L;, j-, j (), B Ol, B, ( ).
j- / , / Od . j- j- ( j- , ), B OL = B, (j()) = RB(j()), . . j- / B j- 11. , j- [7, Proposition III.1.4], // /. 1 [8, 13.12] , // End^//) = End^/) = OD.
, //, Fq Od. , // q +1 u, 4q = u2 + |D|v2 gcd(q,u) = 1, u,v u, V. = (u + vvD)/2 OD O. 1 nO = p, nO = p. nO. , nO = nO nO = O. nO^ Od q f Od , nOD = nO OD [5, Proposition 7.20]. OD = nO OD. , nOD = nOD, nOD = OD, . . OD , .
(., , [9, 13.1.5]), OD {±1} D { 3, 4}, {±1, ±(, ±) D = 3, Z3 = e2ni/3 = = (1 + /3)/2, {±1, ±i} D = 4.
D { 3, 4}, = ± = ±, u = ±u. , |"^)| = q + 1 ±u. |E//(Fq)| = q +1 u, // , // . p = 2, y2 = f (x), f 3 1, y2 = c3f (x/c), Fq, ( [14]). p = 2, y2 + xy = x3 + a2x2 + , y2+xy = x3 + (2+y)x2 + a6, TrFq/F27 = 1, [14].
D = 3, H-3[j](x) = x, j = 0. (6)
( 3 | = 1, p = 1 (mod 3), , p > 3. y2 = x3 + b, p
b = 0, j-, [7, Proposition A.1.1], Fq- ( j-).
Fq 2 S2(b) = x(x3 + b). , y2 = x3 + b
xeFq
q + 1 + S2(b). p = 1 (mod 3), q = 1 (mod 3); , -
[15] ( q = p, , ), k, l Z, , F* S2(1) = 2k, S2(2) = k ± 3l, S2(c-2) = k ^ 3l q = k2 + 3l2. , S2(b) = x(t)s2(bt3) t F*.
// y2 = x3 + 1; u = S2(1) = = 2k, q = k2 + 3l2, l = ±S2(c )-6S2(c", Fq; | |2 = q = k ± l\/3. -1 k l^/3, l = (S2(c2) S2(c-2))/6.
, - O-3; u = 2 Re - . , u :
u = 2 Re = ±2 Re = ±2k. q +1 ± 2k , y2 = x3 + 1 y2 = x3 + g3, g Fq, ;
u = 2 Re = ±2 Re ((3) = ±(k + 3l). q + 1 ± (k + 3l)
, y2 = x3 + 2 y2 = x3 + 23 ;
u = 2 Re = ±2 Re ((|) = ±(k 3l). q +1 ± (k 3l)
, y2 = x3 + -2 y2 = x3 + -23 .
D = 4, H_4[j](x) = x 1728,
f4\
j = 1728. (6) =1, p = 1 (mod 4),
p
- p > 3. y2 = x3 + bx, b = 0, j -, 1728 [7, Proposition A.1.1], Fq- ( j -).
S1(b) = ^2 x(x)x(x2 + b), , , -
xeFq
Fq 2. , y2 = x3 + bx q + 1 + S1(b). p = 1 (mod 4), q = 1 (mod 4); , [16], k,l Z, , k , S1 (1) = 2k, S1(b) = ±2l b S1(b) = x(t)S1(bt2) t F*.
// y2 = x3 + x; u = S1(1) = = 2k, q = k2 + l2. ||2 = q, = k ± li.
u, ±2 Re = = ±2k ±2 Re () = ±2l. u = ±2k, y2 = x3 + x y2 = x3 + g2x, g Fq, . u = ±2l, y2 = x3 + gx y2 = x3 + g3x .
1.3.
Hd[j] |D|. , H_40[j](x) = x2 425692800x + 9103145472000. , L, .
z H, q = e2niz. [6]
(9)
N . [17] N- ( 61),..., (^,^, ,
|^(^, *, ) : 1 ^ ^ ,} -
gcd(A^ N) = 1; Bj = Bj (mod 2N).
, (Aj, Bj, Cj) Bj = D (mod 2), , Bj = Bj (mod 2) , j.
, (, ), N-. [17, Proposition 3]. ( N .)
1) gcd(Aj, N) = 1 .
, gcd(Aj, N0l) = 1 , gcd(A^ N0) = 1, l N, N0.
l Aj, Aj + N0B^ + NQ Cj, l2Aj + lN0Bj + +NgCi, Aj,Bj,Cj l (Aj, Bj, Cj) .
l { Aj, gcd(Aj, N0l) = 1 .
11 Aj + N0Bj + Nq Cj, Ajx2 + Bjxy + Cjy2 x = x/, y = N0x/ + y/ (, 1); x/2, Aj + N0 Bj + Nq Cj, (Aj, Bj, Cj).
l {l2Aj + lN0Bj + NqC, a, b Z, , al bN0 = 1, x = lx/ + by/, y = N0x/ + ay/ (, 1); x/2, l2Aj + lN0Bj + Nq Cj, (Aj, Bj, Cj).
2) Bj = B1 (mod 2N) . x = x/ + ay/, y = y/ (Aj,Bj,Cj) (Aj, Bj + 2aAj, Cj + aBj + a2Aj); gcd(Aj, N) = 1, a = A_1(B1 Bj)/2 mod N .
2 (Theorem 1 [17]). H
;
(, , ), 2 | , 32 | ,
2 4 = = 4, N. $() :
(mod 8),
f(a)3, m = 3 (mod 8), \3
;)M , m = 5 (mod 8),
3
m = (mod 8),
3
m = 2 (mod 4),
3
m = 4 (mod 8).
g(a) G OL.
a1 = a,... , ah 16-, g(a^) Q.
3 (Theorem 2 [17]). a G H
, 3 { , 1 = ,..., , 3-, 72(^) Q. , 72(^) .
[18] 1,2
4 (Theorems 3.2, 3.3, Corollary 3.1 [18]). D (1), N = p1p2, 1 p2 , :
(,61), , gcd(A1,N) = 1 N | 1. 1 . 12 (1) . 2 (1) 2 (*), * ^. (*) .
1 2, (*) (. . -^2(*) ).
1 2
(A,B,C), 3 t A, 3 | B
B2 4AC = D.
s
24
gcd(24, (p1 1)(p2 1))'
. p, 26} p|f p1 = p2 = p.
= 1 p1,p2 t f p1 = p2;
(") = 1 I ^ = 2 = =2;
= 1, 2 | f 4 (mod 32) 1 = 2 = 2,
1 ,2 (^).
. 2, 3, 4 1(/) . [17, 18] .
1 [17]. 9 , 2, 3, 4 ( 1,2), , N- , , ,
: ^ 1(/) .
9()()-1) = 9 ().
9 = , .
1 , 1(/) . , () , N - .
. 9, ^ , , . , ^ ,
^() = ^()-1.
,
9()() = 9 (). (10)
mp1 p2 , p1 2 ; , D p1 , p2 .
2 3, , 2 3 \ D, 3 { A, 3 | B, 2 g(a) . , m = 3 (mod 8). (9),
(f(a)3)3 72(a)
f(a) = (f(a) ) 8Y2(a). (11)
(f(a)3)8 - 16 V J
f(a)3 G L Y2(a) G L, f(a) G L. 1 , Gal(L/K) f(a)3 f(a')3 y2(a) y2(a;), a' ; (11) , f(a) f(a'). , f(a) , , f(a)3. , .
9 a* = {a1,... , ah} , 2-4.
HD[9, a*](x) = (x - 9(ai)).
i=1
9 , 2 3, [9, *] . 9 = , 1,
, Hd [9, a*] Gal(L/K) , , K[x]; 4 , Hd [9, a*] .
, H_40[y2, a*](x) = x2 780x + 20880, H-40[g, a*](x) = x2 x 1, g(a) =
AQ f1(a)2, H_4o[m5 ,7, a*](x) = x2 x 1, H_4o[mn ,13, a*](x) = x2 ±2x + 1. ( a*, a* .) , mp1p2 (a^) , Hd [mpip2, a*] .
Hd[9, a*], Hd[j], , 9(a^) x 9(a^) . 9(a^) G Ol, 9(a^) OL/B Fq ( 1), HD[9,a*](x) = 0 Fq. 9(a^) Fq j- , RB(j(a)). y2 f 2 (9). mp1 p2 . 1 ,p2 (x,y) G Z[x,y], , 1 ,2(mp1 ,2(z), j(z)) = 0 [18]; z = a^ B ( B Z = pZ, 1,2 ), Fq RB(j(a)), RB(j(a)). , , , q +1 m. , , , , q + 1 m , , , q +1 m , . , , Od , ; [18].
2. Q
Hd - I (Od ) Od P (Od ).
, OD f, a + fOD = OD. [5, Lemma 7.18], gcd(Norm(a), f) = 1, . , I(Od), I(Od , f). P(Od), aO_D gcd(Norm(a), f) = 1, P(OD, f). I(OD, f) I(OD) I(OD, f )/P(OD, f) = HD [5, Proposition 7.19].
O f , gcd(Norm(a), f) = 1 [5, Lemma 7.18]. O, , I(O, f). , P_K,z(f) O, aO, a G O, a = a (mod f O) a G Z, , gcd(a, f) = 1. [5, Proposition 7.20] a M aO
1 : I(OD, f) M I(O, f), . [5, Proposition 7.22], I(OD, f)/P(OD, f) = I(O, f)/PK,z(f). , 2 : HD M /(O,f)/PK,Z(f).
I(O, f) M Gal(L/K), ^ ^ , I(O, f)/PK,Z(f) M Gal(L/K), 2, [5, 9], , 1.
:
I (Od ) I (Od ,f) ----I (O,f)
I I I
HD I(Od,f)/P(Od,f) I(O, f)/PK,Z(f) Gal(L/K), (12)
2
-, .
5. (A, B, C) , , gcd(A, D) = 1. q | D , :
|q| , q = 1 (mod 4);
q G {4, ±8}, D/q = 0 (mod 4) D/q = 1 (mod 4).

1) -q G L;
2)
3)
(a, (B + VD)/2^ g I(Od, f), Norm(a) = A;
/ L/K \ . . (q \ / L/K \ ^ ^
( ) (/?) = /?, ( ------- ) ,
-
1, , . 21.
. [19, Theorem 2.2.23 (2.2.8)].
[5, Theorem 7.7] Od . |Od/|; , 0,... , A 1, Norm(a) = A. gcd(A, f) = 1, a f.
. 1() = p1 ... ps, pj O ( ). A = Norm(a) = Norm(p1) ... Norm(ps) , , p | 1()
p
Q
Norm(p)
13)
^fq , ±^/q.
p , p | 1(), p Z = pZ, . B OL p. gcd(A, D) = 1 q | D, 2q G B , ,
q = q (mod B).
(LK)(q)=qNorm<,)
Norm(p)-1 :_ , 4
q 2 yfq (mod B).
n
pO (. . p = pO), Norm(p) = 2 (13) q.
_ Norm(p)-1 , 1 +1 ' 1 ' ' N
, q 2 = (qp-1) 2 = 1 (mod ), (13)
y/q B , , ^/q, (13) .
pO , Norm(p) = (13) ( ^ ) q.
\pj
, q ^ = q = (mod ), (13)
I ^ ) q B, , ( ^ ) q, (13) pp .
p | 1 (a), p Z = 2Z, B OL p. 2 | A, , 2 { D q . B2 4AC = D, D = B2 = 1 (mod 8). , d = 1 (mod 8) 2O [9, 13.1.4], Norm(p) = 2. , (13) -
(q\ ^ (i3) (LlK\ f1 + q\
V2/ q. (13) I p II ------ I .
(1 ± ^/q) /2, B.
2
(mod B).
= 1, q = 1 (mod 8), (q 1)/4 , , B; = 1, q = 5 (mod 8), (q 1)/4 , , 1 = 1
'1 ,2
B.
(13)
2. d < 0 (2) (3). d d = q1 ... q*, q* , q* = (1)q, q > 0 , q* G {4, ±8}, q = 2.
. , .
(2) d d = q1 ... qt, q^ ; q* = ±q^, d = ±q1 ... q*; , d = 1 (mod 4) q* = 1 (mod 4) i, .
(3) d/4 d/4 = q1 ... qt-1, d/4 = 2q1 ... qt-1, q^ . d/4 , , , d/4 = ±q1 ... q*-1, (3) d/4 = 1 (mod 4), , 4 q* = 4. , d/4 , d/4 = ±2q1 ... q*-1, , q* = ±8 , .
, q* 2 5 , , K (VH,... , /qf) L. K (/,... , /qf) K ( d, f) (genus field) K. K (/,... , /1?) = KG.
3.
, K = Q KG = K (q1, . . . , qtr).
Kg.
q* 2, :
1) |qi| ;
2) q* = ±8;
3) q* = 4.
Kg; d = q1 ... q*, d G Q(!,... , qtF) , KG = Q(qf,... , qf).
3. M , p G Z , pZ M. c G M , pc2 G OM. c G OM.
. , c G OM. cOM , q11 ... q^, qj s1 < 0. q1 pOm 1 pZ, q1 c2Om 2, , q1 pc2Om , , pc2 G Om .
6. q1,..., qr , |qj| / = 1 (mod 4). = (1 + /qj)/2 = (1 v^/i)/2.
1) 11 ... ^, (sj) {0,1}r, Q(^/qT,... , yfoT);
2) (511 a1-s1... <5^a1-Sr, (sj) {0,1}r, 0(^/1 ... , /<?).
. . r = 0 . , Mj = = Q(/1,..., qi) i = 1,..., 1.
4. G Z , /1,... , /-1. pZ Mr-1.
. , 1 ^ i ^ 1 Mj-1, pOMi-1, Mj = Mj-1^y/qj').
p Mj-1, , p Z = pZ. Mj-1 Mj aj; , (1, aj) Om4/Om*- 1. Mj Mj-1 (1, aj). [20, III.8 III.14] p , p
det ^ = (aj <5j)2 = /j. qj, .
4 = |/r|. OMr-1 . , ^/qT G Mr-1, (qrOMr-1) =
= (^/-)2. , (1,) -. + , -1. + 6(1 ) + . , = 2 + = 2 + + 2 (1 )/4 , , -1. , 2 4 = 2 -1. 3 , -1. , 2 -1, 2 + -1, 22 = 2(2 + ) 2 -1. 4 3 = 2, _ 1. , + , -1, , -1; , , (1,) -1 , 1 ... . , (1 , ) = ((1 )/2, (1 + ^/)/2)
-1.
7. /1,... ,/-1 , 6, = ±8. = \//4. 0(^51,..., ):
1) 11 ... ^, (^) {0,1};
2) !1 1-51 ... --1 1-1 ^, (*) {0,1}.
. = 0(^/,..., /-). 4 =2 6. 2Z , , , ^/ (1, ^/) (^/) .
+ , . + . , 2 2 +
22
, , . , (2)2 4(2 + 22) = ±2(2)2 , 3 , 2 . 2(2 + 22) ± (2)2 = 22 , 3, . , 2 (2 + 22) = ±22 , 3 . , (1, ) (^/) , 6 .
8. ,... , -1 , 6, = 4. = //4 = .
0(^,..., V):
1) 11 ... ^, (^) {0,1};
2) 151 ... --1 1-1 ^, (^) {0, 1}.
. = 0(^/,..., /1). 2 = (1 + )2 , 2Z , . 4 6, 2Z . , .
+ , . + . , 2 2+2 , , . , 2(2 + 2) + 2 2 = 2( + )2 4 3, = 2, 6 + . 2 ( + ) = , ( + )( ) = = 22 , 22 , 22 ; 4 3 6,
, . , (1,*) (^/) , 6 .
, [ : ] = 2*, Q(... ,
\Zqj-i, \/1+1,...) 1 ^ ^ . , 1(/) -
, :
*; = V ^^
^ = 41 ... 4 1(/0) ^ {0,1}*; ^ , , ^ . 2* = | 1(/0)| 1(/0), , , .
Z- . (, , ) (, , Z-). * . *, 0 ^ < , , 1 > 0, ... , > 0, +1 < 0, ... , * < 0.
, +1... *. , , 1(( )/0) - 1(/0) , . = 0, , 1(( )/0)
= 1,...,(_1 = 1,...,*_1,0 = 11 . . . ^-11 (15)
{0,1}*-1, .
/ , ... /*, , <5^.
9. 1,... , * , 2, , , * > 0 1 ^ ^ * < 0 < ^ , 0 1 ; = 0(^1,... , /*"); (15).
1)
&!,..,*_! = ^1,...,5_1 (1 ,...,* ) =
^ ( ( *?1-*) + ( 5
=1 /\ \=+1 / \=+1 /
(^) {0,1}*-1, ;
2)
,...,*-! = *1,...,5_1 (11 ,..., ) =
\ / / *-1 \ / *-1 \5 ^,1-5 \ ( ( " / , 1-5 \ ^ / I ~1-5
( ^)5 ^ ) ( ^)5 ^ ( ^( ^^ ) <5*
=1 / \ \=+1 / \=+1 /
(*) {0,1}*-1, Z- 0(3 ;
3) , V {0,1}*-1
(-1)+---, ( ; ^ 4 = "
^{0,1}-1 |0, .
. . 2 6, (1,..., *), *....
0 , . , * 1,...,, *...,8-8+...-8(. , Z- * , (^) *...,5, * 1,...,^,1-+1,...,1-4 . 6 , {* 1,...,4-1 , + * 1,...,,1-+1,...,1-4-1,1} . ; , ...,4-1, .
,
. {* 1,...,4-1, - * 1,...,,1-+1,...,1-4-1,1} ^-. ' , ± 1,
.
, . , ,
, ^...-1
,1-^^ V ( <5^1-(,)) * +( 1-(,44),4
=1 ) \ \=+1 ) \=+1 )
, (,;, ,-1) = ( (-1)"'-*"-'^^
*(( (-1)"','"'1-(,'"' * -( (-1)"'1-(,'"'),'" *) .
\ \=+1 ) \=+1 ) )
,(...,(-1 ),(^...^^ 1), (+6)(-^), , +6-^-. : 8 = 1, = 0 = ]. ,
^ ( 1),' ( 1)^ (,)+(,/)1-(, ') + 1 - (, ^)
,'=0
(-1)^' ^'+^'^ (+ -) - 5 (' + ')'+^'^ (2 - 2) = //,
^ (- 1), (- 1)^ 1-(,'')+(,'^') (, ) + 1-(,^) =
,=0
( 1)^ (1-+^1+-^ 1+-^1-+^) = (2 2) =
(-1) (, - ^ = +^ (, - +^,1\/<?
/'- + '^ +-^ ~+-^^ -+ _ / 2 ~,2>
^ = +^ (,
^ , , (-1).
,
(-1),1+...+,-1 , (1,...,,-1 1,...,^,-1)
,{}
\ / *-1 *-1
/^] ( * /^" + ** +Vi\/q;
=1 / \ =+1 =+1
-** (-+^,1/^) - 52 (-^ /7)
=+1 =+1
1 ... * , . , 1 ... * < 0 ,
-1
(-1) = (1) = 1;
=+1
(-1)^1+...+^-!(>...>1 ^,...-) =
0,1}-1
(* *) ( = /∞? "^/0?
, 1 \ 1
11 "^ V = V 11 "^,
=1 / \=+1 / =1
.
10. , , * , 9, 1 = 8. * , * > 0 1 ^ ^ * < 0 < ^ , 1 1 ; = (/0! , , /?); (15).
1)

&,..,*- = ...)4-1 (1 ,,0*) = ^ 1 - I
181 ~8* ~1-5
\=2
( ( ?1- * + ( 1-5* * ,
\ \=+1 / \=+1 / /
() {0,1}*-1, ;
2)
;...-1 = :...)4_1 (1 ,...,?**) = ^21-51 ( (*)5*1-5* I
\=2
^ ()541-^ * (" 1-54(^ *
V \=+1 / \=+1 /
() {0,1}*-1, Z- 0(3 ;
3) V, {0,1}*-1
(1)^>+...+--1 (1.....,-1 ;.,_1) = /5, =
^{0,1}-1 [0, .
9. 9
(1 ((1/2)1 ((1)^1 V2)1-"1 = ^21+1-"1 (1 + (1)1+"1) = 2^2^1,1.
1 =
11. 1,... , **_1 , 9, ** {4, 8}. ; , * > 0 1 ^ ^ * < 0 < ^ , 0 2 ; = 0(^/,... , /*); (15).

&...- = *...-(∞, , V?) = ( 1 ^
(( ,1 ,1 *-1*4 1 + ( 1 ,?1 *-1(*)5( 1
V \=+1 / \=+1 /
() {0,1}*-1, ;
2)
;, ,,-1 = ;, (/;,..., /*) = ( (-&<)
*-2 \ / *-2 \
;^,1-,4 1 ^,1-,-1 ,1-,^_^,^,. 1-- 1
I I ( ), ^ I *-1* 41 I ^( ), ) *-1( *)
\ \=+1 / \=+1
() {0,1}*-1, Z- 0(3 ;
3) , V {0,1}*-1
( 1)^1+...+^-1 (>...>-1 *...,^4-)
/^,
= V,
^' \ / V '/1>--->'/ 1 4^1, ,^4 1/ I
{0,1}4-1 .
. . 2 8, (1,..., *), , 1 .
, . , , 1,...,,4 (1),, 1,...,,,1-,+1,...,1-,4-1,,,. , , ,4 Z , () , 1,...,,, , 1)...,,)1-,+1 ,...,1,, 1,,, (1),. 8 , {, 1,...,-2,, + (1),, 1,...,,,1-,+1 ,...,1--2,1,} . ' ,
,...,-2,,,, .
.
. 9
( 1)1+...+-1 (1,...,,-1 ^,...,^,-1) =
{0,1}4-1
\ / _ *-2
^ /* ,-1+-",- /*-1 ^ /* +
=1 / \ =+1
( ^4 )
=+1
+(1-1+ї-1 (\/0?1) ,,\/*)
=+1 /
... * < 0, *, ,
*-2

=+1
, , (1) = (1)* 2 = 1.
( 1)^1+...+^-1 (>...>1 ^.^-) =
0,1}4-1
$/0 ) (1 + (1)^-1+-1 )1+ 1 1 = %--1 / ) 2*.
I V ^ "44-1^4-1 111 "^
=1 / \=1
.
12. ,... ,*_1 , * = 4 * = 8.
-1
1) ...,4-1 = ...,4-1 (1,..., *) = (5|-5, () -
=1
{0,1}*-1, ;
2) *1,...,5-1 = *1,...,54-1 (91* ,...,*) = ^( ^ 1-"^ ,
{0,1}*-1, Z- 0(3 ;
3) , V {0,1}*-1
(1)^1 + ...+^4-1 (1,..,4-1 *1 ,...,^4-1 )
/^,
= V,
^' \ / V '/1>--->'/-1 ^1,...,^4_ 1/ I
{0,1}4-1 .
. , = (^/*), . , = , = .
6, , 9.
= 1 ,...,4-1 ^ {0,1}*-1. {} , .
0. , , + = 2 , = 2 . , , , , 2, 2
. , 2, = ^ 2 = ^ .
, , , {0,1}*-1. , . , , , /* , * < 0. , , , , .
.
0 [] [0, *] ./2*-1, [0, *]. . . 4.
(. 5).
. 5. . 6.
. 7 , , .
4. [0, *](x)
a G . (A, B, C) , h(A, B, C) = gcd(A, D) = 1; , h , , [5, Lemma 2.25, Lemma 2.3], (A, B, C) gcd(A, D) = 1. ^ ^ {±1}*
:
((A)...( a
,
( q * \
I(,/)/,(/), 5 , ( 1
(, , ) gcd(A, ) = 1.
13. ^ {(^...,^) {±1}* : = 1} . ^

. (^ = { : () = ( <^)}
,..., ^).
. , ^ , . 3 5 , .
, ^ 1. , , 5, (, , ).
q*\ 1 fL/K
(\/q?)
) \^1(),
, , 2
( )( )= (
^ , ( / ^ , 1(/), .
V 1())
, ( <^) . <^() = (1, , 1), . 2 5.
(^
1() , () (12), , .
,
(,, ,,) {±1} : = , (^, , *) ("^
1((^/) = 1(/)/(<^) = /^ = 1<^,
[(^ : ] = | <^| ^ 2*-1. . 3 , [(/*, , /*) : ] = = 2*. [ (/1,..., /*) : ] = 2*-1. [ (/,..., /*) : ] ^ ^ [(^ : ] = | <^| ^ 2*-1, | <^| = 2*-1
(/1,..., /* ) = ( ^.
[9, *]
[9, *]() = ( - 0()), (16)
^(;,;,;))=(1,...,1)
{(, , )} -, 9 (. . 2-4, 9), (,,).
, [9, * ] 1(/). (10) , ^ , , ( <^) [9, *](); , ^. 2-4 9() , [9, *] . , [9, *] ^ . [9, *] , .
, 1993 . [4], [9, *] : , ^; 1(^/), , , , . , 9() - [9, *]. , .
[9, *]; , 9() , <^(^()) = (1,..., 1). 13 , 2*-1 , -.
5. [9, *]
9-12 , [9, *] - + ), ,^ ^, , * .
2 V /

. -
^ .
J ± , . .
' 1
2 ( +
, Hd[j,a ;] ;, D [j] = D [j, ;].
[21].
Hd [j] ^0 {0,1}*
#D)W [j](x)= (x - j(1T)
:^(^(;,;,;))=^0
, , (A^B^C) , ^
(Aj,Bj,Cj).
Hd[j] = HD,(i,...,i) [j]. , Hd[j], ^0 D>^0[j] OKG[]. , Gal(L/Q) , b Hd , 1 HD,^0[j]∞" = HD,^0^(6)-i[j]. KG Gal(L/Q), Gal(KG/Q) ^0 = ^0() ( ), , HD>^1 [j] = D,^1^0(r)[j] {±1}*.
14. Hd,^0 [j]
exp ^C5h + c,N ^ln2 N + 4y ln N + + ln N ^ ^ ^
^ exp (c,N ln2 N + c2N ln N + c3N + c, ln N + c4) = T0,
N = V|D|/3; y = 0,577... ; , = 3 = 5,441...; 2 = 18,587...; 3 = 17,442...; 4 = 11,594...; 5 = 3,011...; = 2,566...
T0 = exp O (VlDln2 |D|)
0.
. [21, Section 4].
(17) , (Aj, Bj, Cj) ( SL2 ^^ , j ).
(A, B, C) j ^(B + //D)/(2A)^. j
{z G H : |z| ^ 1, | Re z| ^ 1/2}. , Imz ^ /3/2 |q| = |e2niz| ^ e-^. ,
1
j(z) = - + 744 + ' cmqm,
~,m
,_ ?
q m=1
4^^
, [22], || ^ ^--------. -
23/4
,
^ 744 + V ----------^ = , = 2114,566
^ 2m3/4 1
m=1 v
j ((- + VD)/(2A^ ^ 1/|q| + , ^ /, &2 = 1 + k,e = 10,163...
, {(,,') : 1 ^ i ^ deg HD[j]} , (17), |1/| = ^/^/. HD,^0 [j]
deg H Dj| , h/2t-1
Cdeg HD j| 6 < <2k2)h/2-1 e'^iSi/A-.
=+1 | =1
, HD,^0 [j] h h/2t1 _ h 1
2*3, ln(2k2)+ AT ^ h ln(2k2)+ nVlDl AT.
=1 =1
, [21, Theorem 1.2], j. [23, Proposition 3].
, .
[23] , , Hd[j]
A,
(A,B,C)
. , j, , , , degj /degfl , j , ((), j(z)) = 0.
[23] Hd [j]
lnTo ~ /|D| max (1S)
{±} ' A
(A,B,C):^(h(A,B,C))=e
j. degj /deg^ .
z = + b^ I /2 H[j], b^,^ Z.
V m m /
, Gal(KG/Q) Hd,^1 [j] , {0,1}*
.
, [24]; , , -, [24, Chapter 6A]. , . , , , , .
[25]. [25] ( ), M ([25] , M = 1) ([25] M/Q ).
15. M R , , M/Q m. WT,...,Wm ї*,...,ї Q- M M : Gal(M/Q) ^ R ( ), , 1 ^ l,l' ^ m
1, l = l', , l = l'.
()(ї/ї/*)
Oal(M/Q)

i = 2,
C = |() (Wi)|
Oal(M/Q)
=/d
m
C,

Oal(M/Q) =/d
() ((W,) WiiW!
!,... , ,
,ї* = z ^ ,
i=1
( ,ї*
,=1

Z m-1
Gal(M/Q)^ = Id.

|^ ^ ^(^)^^ ;
|1| > , (^) = 0 = 2,... ,
,_ W
, Wi

I ( |1|- \ m-1 |1 |M(/d)Wi| )
. l = ,
m
m
/ ^ /^ () (W/W/*)
^=1 GGal(M/Q)
()(W/) //(W/*)
Oal(M/Q) \//=1
M(Id)W/Z+ ( ) (W/ ) (Z).
Oal(M/Q) =/d
f19)
I = 1:
1 = (1^)^1^ + ( ) (^ (%). (20)
1(/) =/
(%),

|1 (1^)1%| ^ ^ .
% -1
.
|1| > . 1^(1^)^!% ^ |1| . , (1^) = 0
^ |1| ,
^ |(1)1|. (21)
(19) (20), /^,
(%):

1
1
^
% -1
|1|:

1
1
1
|1|%
(21) .
[25] ( -

) ^* . -
=1
, . .
= ,
15 = . = [ : ] = 2*-1, ^ 2,
1(/) {0,1}*-1, (15).
, , {0,1}*-1. , - , * * ^ {0,1}*-1, : 1(/>) ^ , :
1) <..,0 = 1;
2) {*} ;
3) , ' {0,1}*-1,
^ / * I 1, = ', , ,
(*)* ( ' ) = \ , ' (22)
*{0,1}4-1 I 0 = .
-. , 15,
^1+1+2+22+...+24-2
, 1
W-
1+,1+2,2+22,++2 2,1
= ,
, , (3 . 9-12 , . , / , -/! ... .
1.

,
,
,...,<

,1,...,,(1
,1.....,*1
,...,
(23)
(,1,...,,41) = (-1),1+...+,-
,1,...,,(1 (,..., ,...,) [
1-3 . 2.
,1,...,,(1
,1,.^1
,...,
,1.....,*1
,...,
(24)
(,...,1) = (-1),1+...+,1
,1,...,,(1 (,...,) ,...,) [
1-3 .
Wi, Wi* , 15 ^ . ,, ,

15 -
1+1+2 +22^+...+2-24-1
.
{0,1}*-1, = (0,... , 0). = (? )1 ... (^4-1 (4*)+1 --1. , = /^/2, = (1 + ^?)/2. ∞.
. , 0,1,2,... 0,1,2,... : 0 = X, = |_], +1 = 1/( ), (. . ) , . , -
2 -1
1 2
,01,02
-1 = 1,0-1 = 0,
:
^-1
= , = 1,+1 = +1 + -1,^+1 = +1^ + - (, [26, 9 12]), ^ 0
<
1
+2 ;
(25)
(26)
; [27, 11.10], .
2. , , , , 5 = 2 > 0 . 2 + 2 + = 0 5.
= ( + \/5)/ 5. 5 = = ( + /5)/, , . = |_] 1 = = (5 2)/ , :
1 1 1, ^ 1 ,
5 = + 1, ^ 0, (27)
2 1 ( 1), ^ 1.
, ^ 0
( 1)
1 . . . = -5 ~ 1 + 21^1 + ^1 = (1).
1 ^1
( + /5)/, , , , ( + /5)/ > 1
1 < ( /5)/ < 0. ( + /5)/ , 0 <\/5: < < /5 + . , .
.
, {0,1}* 1, = 0. , , , , , ,. , 2, = , , ,. ().
5 , 5, ,0 = 1, ,0 = 2, ,1 = (5 1)/2. (27) , , ; ; = (, 1)/2 = ,/2 . 2 + 2 + , , , 2, 2 , (). , 2 2(,1 ,1)(,1 ,1) = (1), =
= (1,-
5 , 5/4, , = 0, , = 1, ,1 = 5/4. , = , , = ,. 2 + 2 + , , , 2, 1 , (). , 2 (, 1 ,1)(,1 , 1) = (1), = (1),.

_ + , , =----------,
,
(,1 ^,1)(,1 ^,1) ( 1) ,. (28)
2 ^, , , = [,], , ^,. ;,
^, = (,1 . . . ,) = ( 1)(,1 ^,1) ∞. (29)
(28) , ^ 0
^,^(^,) = (-1) , (30)
, , ((-1)∞(^,)) = ,, ,. 2 - = ,
(29) , .
. ,*

( 1)+1^ (,+1) = ,+1/,+1 = ,+1,+1 = 9 + ,+1
(-1)^ (,) , , , .
, , ,, ,
= (^,
, \ ?
= +1 = . {,} - 9, , ,
9 =
,
,
/ 9 + _ 1 ^ *1^ * ^ ? ,? + , *
(?{ ] ~~ = (?"" + ] = ?
.
. $, 9, , , N > 0. -
,, , |,...,| ^ N0 ,/,..., ,/,..., {0,1}*-1.
2*-1 ,, , (,) (, 5), {0,1}*-1, = (0,... , 0). : 3 (. ) , = ^,
(,) = ( , > -1), (, ) = (^, ,^,-1)^ ,, = ((-1) ^(^, )).
, =
.
1) . : ,..., := 1; := 0; := 0; (,) := (1, [^/4]); (^,5) := (1,9) {0,1}*-1, = (0,... , 0).
2) . | | < N0, .
3) , , = ,
,=(,...,)
4) = [(9 + )/].
5) (,) := ( ,).
6) = , := 4 {/4},
(,/) := ( ( ),). ( ,
, .)
7) , = 1 ,^ + , I / . ( ,
, ^ .) , := ,.
[28].
16. 0(1 N0) . 0 ^ < -\/^ 9; 0 < < /$; ^ = ,, ^ 1;
^ = (0,..., 0).
1
Z m-1
. ', yA .
5. A G {0,1}t-1, = 0; n ^ 1 .
0 ^ ') < VA - gA, 0 < yA,n < Va, -1 < ^a(Xa ,) < 0.
. $. $ . = 1/(gA |_gA_l)- , > 1. , () = 1/(1 gA |_gA_|) , gA > 1, 1 <
< ^a(Xa;1 ) < 0. , 2 gA, , $, . .
0 < /^xA,n < yA,n < /^+xA,n n ^ 1. XA n = (xA,n 1)/2 yAn = (,)/2, .
$ . , = 1/(gA |_gA_l) > 1. , 0() = 1/(gA + LgAj) , gA > 1, 1 < () < 0. , 2 gA, , $/4, . . 0 < \/2 , <
< , < V^A/2 + XA,n n ^ 1. x'A,n = XA,n yA , = ,, .
XA,n = (gA + xA n)/yA n, . n ^ 1
XA, < V^A. (31)
, = 0 3 . Z = ((1) cta(za , )) ^ 1 -
=0
5 za , = 1/(, 1 ... ,
).
6.
,
\/|.
^.11, )z'
. 1, . , , , 3, . .
zAm = zun . nA = nA + 1 n' = nu = , = (0,..., 0).
'=(0 ,... ,0) '
, ,/ > 1, zA , n < zA , . :
1) ZA ^ ! zu . min zu n = ! zu , ,
, '=(0,...,0) ' '=(0,...,) ' ^ '=(0,...,0) '
zu zu
=(0,...,0) ', =(0,...,0)
! Zu ! Z,
'=(0,...,0) ' " '=(0,...,0) '
S ---- S
2) zA,nA < ! Z',M. ! zM,n^ = zA,nA; (31),
zu '=(0,...,0) M,n^ Za
! zM,/ Za
'=(0,...,0) '
S = ,+1 < v^A S x/idT-
.
, () . , ^ ().
/()... (91-1)-1 )-+>. --^ =
= ((1)"1 \/1... ((1-1 yq;-7),-l ^+1--1 =
4-1
2 /
= (1)4=1 ( 1*)1... (1)-1 (* )+1...-1,
t-1
' | ^(^) AjU* = 0 (rnod 2) -
i=1
.
zM,n = . (6) = (0,... , 0) -'=(0,...,0)

-= S zA^ S . (29), (30) 5 ,
V|d|
1 S ZA,^ | (za,^)| < VA S VTdT. , 1 S \ ^,)| S TdT. -
m1 I 71
S |d|
( |d| \ m1
Z = |^ ^,)| S ) ,
A=0
_ ........... - 1
=0 \ J Z m-1
t1
= 0. * = 0 (rnod 2),
i=1
S {0,1}t1, n/2 , .
VT
| |Z', I S + m = |d|m" S
21 . 2|E Ai'i V / Zm-1

^=0
, 0(1 N0) . 15, |0,...,0| ^ |(/^)0,...,0|^ /|^|, (/^)0,...,0 = 0 .
(30) ^ = ^ (=0 ^,) , (29), (25) (26)
=0 ^, '
1 -(-1)
. | | . -1 =0
Z ^ ( |, 1 Q,-1 gA ^ Q, ^ 22 = 2
A=0 A=0 A=0
] . , 0(1 N0)
=0
X ^
+ ^[
|(/^),...,|
, ,
|0,...,0| ^ N0, .
7.
^ ,
^ &. . 5
( 6**



*

(32)
0 , . . 6 /,..., /0 0 , -
N0.
16, . ( , ^ Wi, -





'
|
,
16 1/( 1) .)
*, * , (23) 6* (24) 6*. , 1-3 - :
2'. , {*} .
6*; 6* .
Q = 2*-1 . , = ; 0,...,0 = 1 = = (0,... , 0), . 2'
*
(33)
? ^. ( = , ? -
? . = .) > 0 ?? , . . 7,
,

6?& 7
< . ,...,
6?

| |
|,..., 1
6? *?)
,...,

| | |,...,|
(34)
= ?? ! (22) ,
?
' = ^ ()(^*,) = ()(*,) ( ^
\
'' = () ' . (35)
' ' / V /
= 0 ; (34) . = 0 16. : ^ = ( ? ? | = ?(), (32)
, ,

'
^ 0 |()| . , (35), (34) 16
' - (/^)^7
0,...,1
|{^| -^ + \{)|()|, (36)
|,...,| = Z1
, , Z = ^.

Z-1. ,..., = 1, (21) , N ,
Z > 4 |()()^/!^|- ) , (37)
=
-1
(36) 1/4.
N , Z. ,
_ < 1 \,...,\ (38)
4 |(^)|Z. ( )
(36) 1/4, , (36) 1/2. ' ^,
'
, (/^^
7
,...,
?,
= ( ? ) ?.
? V /
7. ( ? I .
' ?,{0,1}4-1
. , . , ,
**? = ∞. (40)
*
. () = = * (*2), (2)'1'2 = * (*), (X)>2 = "" 3 (*), 4 *(). (22) 1 32 = , . , 1, 2 3 . , (33) *, 41 = 1. X = -141, X = 4()-1 = -13-1432. , 3 4, , 2 = 42. ^, ? = *.
. (40) {0,1}*-1:
** = ∞, ( ) ( = ∞.
* *
, , . 16. 7.
, {*} , .
.
1) = , ,>, 1 . 1.2. , , + 1 .
2) , 0 (18), 0, , (37), (21). . 6.
3) (38). [] (16) .
4) *, (39) (36) . *. ( , , 2.)
5) , , ; , . - 11 -, .
6) " , , , . 1.2.
, j , . 1.3. 0 2, . 5, 3 Hd[0,*], 5 j- , . 1.3.
, , .

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n
Chxn-hyh = P // Giorn. Mat. Batt. 1908. No. 46. P. 33-90.
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