1 The problem of the representation of an integer n as the sum of a given number k of integral squares is one of the most celebrated in the theory of numbers... Almost every arithmetician of note since Fermat has contributed to the solution of the problem, and it has its puzzles for us still. G. H. Hardy 1 2 3 4 2 3 4 3 4 S 3, S 4 tetsushi@math.kyoto-u.ac.jp

2 p 2 p = x 2 + y 2 (x, y Z) 1 2 1 2 1 12 2 1 2 2, 3,, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 3, 9, 61, 67, 71, 73, 79, 83, 89, 97,... 7 a, b n a b n a b (mod n) a b n 2.1 ( ). p p p = x 2 + y 2 x, y Z 2 p 1 (mod 4) 2.1 1749 4 12 2.1 p 1. p = x 2 + y 2 ( x, y Z) 2. p 1 (mod 4) 2 (1) (2) 2.2 (2) (1) 2.1 p x 2 + y 2 100 4 1 =1 2 +2 2, 13 = 2 2 +3 2, 17 = 1 2 +4 2, 29 = 2 2 + 2, 37 = 1 2 +6 2, 41 = 4 2 + 2, 3 = 2 2 +7 2, 61 = 2 +6 2, 73 = 3 2 +8 2, 89 = 2 +8 2, 97 = 4 2 +9 2 2.1 p = x 2 + y 2 x, y p 4 3 p =3, 7, 11, 19, 23,... p = x 2 + y 2

3 2.2. x, y x 2 + y 2 3 (mod 4). x 0 (mod 4) x 2 0 (mod 4) x 1 (mod 4) x 2 1 2 1 (mod 4) x 2 (mod 4) x 2 2 2 4 0 (mod 4) x 3 (mod 4) x 2 3 2 9 1 (mod 4) x 2 0 1 y 2 0 1 x 2 + y 2 0 1 2 x 2 + y 2 3 (mod 4) 2010 2 1 2010021 2010021 = 1171 3433, 1171, 3433 1171 4 3 2.2 1171 = x 2 + y 2 3433 4 1 2.1 3433 = x 2 + y 2 3433 = 27 2 + 2 2 2.1 3 [Za] S = {(x, y, z) Z 3 x, y, z > 0, x 2 +4yz = p} f (x +2z, z, y x z) x<y z f :(x, y, z) (2y x, y, x y + z) y z<x<2y (x 2y, x y + z, y) x>2y #S g :(x, y, z) (x, z, y) 2 #S S S f f(f(x, y, z)) = (x, y, z) f f(x, y, z) =(x, y, z) (x, y, z) S p 4k +1 f (1, 1,k) 2.3 (). p r 2 1 (mod p) r p 1 (mod 4). r 2 1 (mod p) r F p 1 4 4 F p p 1 p 1 4 p 1 (mod 4) 2.1 p p 1 (mod 4) 2.3 r 2 1 (mod p) r Z 0 a, b < p (a, b) ( p + 1) 2 α α p +1 > p ( p + 1) 2 > p ( p + 1) 2 p +10 a, b < p (a, b) p +1

4 a rb p p a 1 rb 1 a 2 rb 2 (mod p) (a 1,b 1 ) (a 2,b 2 ) 0 a 1,a 2,b 1,b 2 < px = a 1 a 2, y = b 1 b 2 (x, y) (0, 0) x 2 (a 1 a 2 ) 2 r 2 (b 1 b 2 ) 2 y 2 (mod p) x 2 + y 2 0 (mod p) x, y < p 0 <x 2 + y 2 < 2p x 2 + y 2 0 2p p x 2 + y 2 = p p p 1 (mod 4) p = x 2 + y 2 m mp = x 2 + y 2 (0 <m<p) (x, y, m) : 2.3 a 2 +1 0 (mod p) a (1 a<p) p a a 1 a< p 2 a2 +1< p2 4 +1<p2 a 2 +1=mp (0 <m<p) mp = x 2 + y 2 (0 <m<p) m m 0 m 0 > 1 m 0 p = x 2 + y 2 (x, y) r, s x = x rm 0, y = y sm 0, x m0 2, y m0 2 (x ) 2 +(y ) 2 x 2 + y 2 0 (mod m 0 ) (x ) 2 +(y ) 2 m 0 m 1 = (x ) 2 +(y ) 2 m 0 (x ) 2 +(y ) 2 m2 0 4 + m2 0 4 = m2 0 2 m 1 m0 2 x, y m 0 : x, y m 0 m 0 p m 2 0 0 <m 0 <p (x ) 2 +(y ) 2 0 m 1 0 (xx + yy ) 2 +(xy x y) 2 =(x 2 + y 2 ) ( (x ) 2 +(y ) 2) = m 2 0m 1 p x x (mod m 0 ),y y (mod m 0 ) xx + yy x 2 + y 2 0 (mod m 0 ), xy x y xy xy 0 (mod m 0 ) xx + yy = m 0 α, xy x y = m 0 β α 2 + β 2 = m 1 p m 1 m0 2 m 0 m 0 =1 2.1 3 2 2.1 2.1 Z 1 1 Z[ 1] Z[ 1] A 2 B 2 =(A + B)(A B) A = x, B = y 1 x 2 + y 2 =(x + y 1)(x y 1) 2.1 4 1 p p =(x + y 1)(x y 1) p Z[ 1] p Q( 1) [Ta1], [HW], [Co] p

Q Gal(Q/Q) p Q Q p Gal(Q p /Q p ) Gal(Q/Q) Gal(Q p /Q p ) Gal(Q/Q) Gal(Q p /Q p ) Gal(F p /F p ) I p p Gal(F p /F p ) Frob p : F p x x 1/p F p Frob p Gal(Q/Q) Frob p Gal(Q/Q) p Frob p Gal(Q/Q) Frob p well-defined K/Q K/Q p I p Gal(K/Q) p K/Q Frob p Gal(K/Q) well-defined p Frob p Q Q p Frob p ρ: Gal(Q/Q) GL n (C) Q n p ρ(i p ) ρ p 2 Q( 1)/Q 1 ρ Q( 1)/Q : Gal(Q/Q) Gal(Q( 1)/Q) = {±1} C = GL 1 (C) ρ Q( 1)/Q 2 p ρ Q( 1)/Q (Frob p ) GL 1 (C) well-defined GL 1 (C) 1 p Q( 1)/Q ρ Q( 1)/Q (Frob p )=1 p ρ Q( 1)/Q (Frob p ) GL 1 (C) ρ Q( 1)/Q ρ Q( 1)/Q (Frob p ) p = x 2 + y 2 Q 3.1 (Q ()). 1. n ρ: Gal(Q/Q) GL n (C) N N p ρ p ρ(frob p ) p (mod N) 2. ρ (1) N 3. N (1) ρ Gal(Q(ζ N )/Q) ζ N = exp(2π 1/N ) 1 N 1920 30 3.1 (3) Q(ζ N ) N Q

6 exp(2π 1z) ρ: Gal(Q/Q) GL n (C) ρ(frob p ) n =1 ρ(frob p ) p (mod N) (N ) ρ ρ(frob p ) 2.1 Q( 1) = Q(ζ 4 ) 3.1 ρ Q( 1)/Q (Frob p ) p (mod 4) 2.1 2.1 1. p x, y Z, p= x 2 +2y 2 p 1, 3 (mod 8) 2. 3 p x, y Z, p= x 2 +3y 2 p 1 (mod 3) 3. p x, y Z, p= x 2 +y 2 p 1, 9 (mod 20) 3.1 (1) (3) p Q 2 Q [Ta1], [HW], [Co] 4 p = x 2 + y 2 p p =6x 2 + xy + y 2 (x, y Z) x, y Z, p=6x 2 + xy + y 2 p a 1,...,a r (mod N) a 1,...,a r,n 1000 p =6x 2 + xy + y 2 23, 9, 101, 167, 173, 211, 223, 271, 307, 317, 347, 449, 463, 93, 99, 607, 691, 719, 809, 821, 829, 83, 877, 883, 991, 997 26 23 = 6 2 2 +2 ( 1) + ( 1) 2 9 = 6 2 2 +2 ( 7) + ( 7) 2 =6 2 2 +2 + 2 101 = 6 4 2 +4 ( ) + ( ) 2 =6 4 2 +4 1+1 2 p =6x 2 + xy + y 2

7 4.1. f = (1 q n )(1 q 23n )= a n q n n=1 n=1 a n p 23 x, y Z, p=6n 2 + nm + m 2 a p =2 f 1 23 4.1 p p =6x 2 + xy + y 2 f 4.1 4.1 4.1 Y X f(q) =q (1 q n )(1 q 23n )= a n q n = q q 2 q 3 +q 6 +q 8 q 13 q 16 +q 23 q 24 +q 2 +q 26 +q 27 q 29 q 31 +q 39 n=1 n=1 q 41 q 46 q 47 +q 48 +q 49 q 0 q 4 +q 8 +2q 9 +q 62 +q 64 q 69 q 71 q 73 q 7 q 78 q 81 +q 82 +q 87 +q 93 +q 94 q 98 + 2q 101 q 104 2 q 118 +q 121 +q 123 q 127 q 128 q 131 +q 138 q 139 +q 141 +q 142 +q 146 q 147 +q 10 q 11 +q 162 q 163 + 2q 167 +2q 173 q 174 2 q 177 q 179 +q 184 q 186 q 192 q 193 q 197 +q 200 2 q 202 +q 208 +2q 211 +q 213 +q 216 +q 219 + 2q 223 q 232 q 233 q 239 q 242 q 246 q 248 +q 24 q 27 +q 262 q 269 +2q 271 q 277 +q 278 q 282 +q 289 +q 294 q 299 +q 302 2 q 303 +2q 307 q 311 +q 312 +2q 317 q 32 +q 326 q 328 q 331 2 q 334 2 q 346 +2q 347 q 349 q 31 q 33 +2q 34 +q 38 + q 361 q 363 q 368 q 376 +q 377 +q 381 +q 384 +q 386 +q 392 +q 393 +q 394 q 397 q 400 +q 403 q 409 +q 417 2 q 422 q 426 q 432 q 438 q 439 q 443 2 q 446 +2q 449 +q 43 q 461 +2q 463 +q 464 +q 466 +2 q 472 +q 478 q 487 +q 489 q 491 +q 496 q 499 2 q 01 q 09 +q 12 +q 14 2 q 19 +q 29 +q 33 +q 37 +q 38 q 41 2 q 42 q 47 q 2 +q 4 q 68 +q 7 q 77 q 78 +q 79 q 84 q 87 +q 91 +2q 93 +q 98 +2q 99 q 600 q 601 +2q 606 +2q 607 +q 611 2 q 614 +q 621 +q 622 q 624 +q 62 2 q 633 2 q 634 q 637 q 647 q 648 +q 60 q 63 +q 66 +q 662 q 667 2 q 669 q 673 +q 67 q 683 +2q 691 2 q 694 +q 696 +q 698 +q 699 +q 702 + q 706 q 713 +q 717 +2q 719 q 722 q 72 +q 726 +q 729 q 739 +q 744 +q 72 q 74 q 761 q 762 2 q 767 +q 771 q 77 q 783 q 784 q 786 +q 794 q 806 +q 807 +2q 808 +2q 809 q 811 2 q 813 +q 818 +2q 821 q 823 +2q 829 +q 831 q 832 q 834 q 837 +2q 83 q 87 q 89 q 863 q 867 +2q 877 +q 878 +2q 883 +q 886 q 887 +q 897 2 q 898 +q 899 q 906 2 q 921 +q 922 +q 923 2 q 926 q 929 + q 933 q 943 2 q 944 q 947 +q 949 2 q 91 q 967 +q 968 +q 974 +q 97 q 978 +q 982 +q 984 +2q 991 +q 993 +2q 997 +q 998 + 4.1 p =6x 2 + xy + y 2 6x 2 + xy + y 2 α = x 1+ 23 2 + y (x, y Z) α α = ( x 1+ 23 2 + y )( x 1 23 2 + y ) =6x 2 + xy + y 2 α α K = Q( 23) O K = Z [ 1+ ] 23 2 p p =6x 2 + xy + y 2 O K p = α α K/Q Q K/Q 1 ρ K/Q : Gal(Q/Q) Gal(K/Q) = {±1} C = GL 1 (C) p 23 p K/Q ρ K/Q (Frob p )=1

8 23 K/Q p K/Q (p) O K (p) =Q 1 Q 2 O K K Q(ζ 23 ) p K/Q p (mod 23) p K/Q (p) =Q 1 Q 2 O K K 3 Q 1,Q 2 Q 1,Q 2 K H K H K Gal(H/K) = Cl(K) Q O K Q H Cl(K) K H/K K = Q( 23) H H p 23 x, y Z, p=6x 2 + xy + y 2 (p) =Q 1 Q 2, Q 1,Q 2 p K/Q Q 1,Q 2 H/K p H/Q Gal(H/Q) Frob p =1 K = Q( 23) H X 3 X 1 Gal(H/Q) = S 3 3 p 23 Frob p Gal(H/Q) S 3 2 τ ρ H/Q : Gal(Q/Q) Gal(H/Q) = S 3 τ GL2 (C) Q 2 S 3 x S 3 Tr(τ(x)) = 2 x x, y Z, p=6x 2 + xy + y 2 Tr ρ H/Q (Frob p )=2 2 2 ρ H/Q Tr ρ H/Q (Frob p )=2 p ρ H/Q p (mod N) 4.1 f f 2 ρ f : Gal(Q/Q) GL 2 (C) Tr ρ f (Frob p )=a p ρ f ρ H/Q 4.1 p = x 2 + y 2 p Q( 1) 1 Q( 1) Q( 1) Q( 1)/Q

9 Q p = x 2 +2y 2, p = x 2 +3y 2, p = x 2 +y 2 Q Q p =6x 2 + xy + y 2 Q( 23) H Q Gal(H/Q) = S 3 Q p 1 2 1 2 3 2 f Q( 23) H Q p =6x 2 + xy + y 2 f f f = 1 { q 6n2 +nm+m 2 } q 6n2 +nm+2m 2 2 n,m Z n,m Z 23 2 S 3 2 2 1 f Q( 23) GL(1)/Q( 23) GL(2)/Q f 1 [Se] GL(n) GL n (C).1 ( ). K ρ: Gal(K/K) GL n (C) n GL n (A K ) π L(s + n 1 2,ρ)=L(s, π).1.1 1 2 l K ρ: Gal(Q/Q) GL n (Q l )

10 K l GL n (Q l ) l ι: Q l = C l.2 ( (GL(n) )). K l ι: Q l = C n l ρ: Gal(K/K) GLn (Q l ) GL n (A K ) π L(s + n 1 2,ρ)=L(s, π) ρ π l l p l l π π = vπ v v π v π (isobaric) [Cl], p.84 GL(1) A 0 1 4.1.2 n =1 n 2 l l 2 4 n R red = T K = Q, n=2 ρ l [Sa] K = Q, n =2 mod l n 2 ρ L/K ρ Gal(K/L) ([T], [It2], [It4], [It]) 6 k (k =2, 4, 6, 8) n k r k (n) := # { (x 1,...,x k ) Z k n = x 2 1 + + x 2 } k 2.1 p r 2 (p) 0 p 1 (mod 4)

11 p r k (p) n r k (n) p r k (p) p r k (n) r k (p) p L Fundamenta Nova Theoriae Functionum Ellipticarum 1829 r 2 (n) r 4 (n) r 6 (n) r 8 (n) ϑ(q) = n= 1 2 ( ϑ(q) ) k q n2 ( ) k ϑ(q) =1+ r k (n)q n r k (n) k 2 ( ϑ(q) ) k r k (n) [Gl], [Na], [We] [Na] r 2 (p) 2.1 p 1 (mod 4) p = x 2 + y 2 (x, y Z, x, y 1) x, y x, y 8 (±x, ±y), (±y, ±x) p = x 2 + y 2 r 2 (p) =8 { χ(p) =( 1) (p 1)/2 1 p 1 (mod 4) = 1 p 3 (mod 4) n=1 r 2 (p) =4 ( 1+χ(p) ) = { 8 p 1 (mod 4) 0 p 3 (mod 4) r 4,r 6,r 8 p r 4 (p) = 8(1 + p) r 6 (p) = 16 ( χ(p)+p 2) 4 ( 1+χ(p)p 2) r 8 (p) = 16(1 + p 3 ) r 2 (p), r 4 (p), r 6 (p), r 8 (p) p (mod 4) p 7 r 10 (p) k =2, 4, 6, 8 p (mod??) r 10 (p) p r 10 (p)

12 r 10 (p) 1866 [Na] r 10 (p) = 4 ( 1+χ(p)p 4 ) + 64 ( χ(p)+p 4 ) + 8 (x + y 1) 4 p=x 2 +y 2 p = x 2 + y 2 (x, y) (x, y) (x, y) p=x 2 +y 2(x + y 1) 4 p Q( 1) r 10 (p) p = (±1) 2 0 2 2 10 C = 8064 1 (±2) 2 1 (±1) 2 8 0 2 2 2 10 9 = 360 r 10 () = 8424 χ() = 1 Re (x + y 1) 4 = x 4 + y 4 6x 2 y 2 4( 1+χ() 4 ) + 64 ( χ() + 4 ) + 8 (x + y 1) 4 =x 2 +y 2 = 4 ( 1+ 4 ) + 64 ( 1+ 4 ) + 32 (x 4 + y 4 6x 2 y 2 ) =x 2 +y 2, x,y 0 = 4268 + 32 2 (24 +1 4 6 2 2 1 2 ) 4268 448 = = 8424 r 10 (p) Q( 1) p r 10 (p) r 10 (p) 8 r 12 (p) p (mod??) p K/Q K r 12 (p) r 12 (p) r 10 (p) r 12 (p) r k (p) k 2 ( ϑ(q) ) k k 2 f 1,...,f r α 1,...,α r C f i 2 l ( ) k r ϑ(q) = α i f i i=1 ρ fi : Gal(Q/Q) GL 2 (Q l ) 2 l ρ fi p r k (p) = r α i Tr ρ fi (Frob p ) i=1

13 r k (p) r k (p) α i 2 l ρ fi Tr ρ fi (Frob p ) 2 k k =2, 4, 6, 8 ( ϑ(q) ) k f f 2 l ρ f Tr ρ f (Frob p ) p (mod??) p χ(p) =( 1) (p 1)/2 Q( 1)/Q 1 p n Q l ( n) r 10 (p) ( ϑ(q) ) 10 Q( 1) r 10 (p) (x + y 1) 4 p=x 2 +y 2 f l ρ f Q ρ f 2 Gal(Q/Q( 1)) Gal(Q/Q) Q( 1) ρ f Gal ( Q/Q( 1) ) 1 f 4.1 r 12 (p) ( ϑ(q) ) 12 6 Γ 0 (4) 6 b n g = q (1 q 2n ) 12 = b n q n n=1 n=1 r 12 (p) = 8(1 + p ) + 32 b p g b p g 2 l ρ g : Gal(Q/Q) GL 2 (Q l ) r 12 (p) r 12 (p) = 8(1 + p ) + 32 Tr ρ g (Frob p ) [BLGHT] Tr ρ g (Frob p ) [BGG] 8.1 ( g ). 0 α<β π N p C(N,α,β) lim N cos β Tr ρ g(frob p ) 2p /2 cos α C(N,α,β) (N p ) = 2 π β α sin 2 θ dθ

14, 1974 1 Tr ρ g(frob p ) 2p /2 1 p Tr ρ g(frob p ) [ 1, 1] 2p /2 1963 sin 2 θ p (mod??) K/Q r 12 (p) r 12 (p) : k r k (n) k k r k (n) k ( ϑ(q) ) k rk (n) k SL 2 (A) k =3 n n >4 24h( n) n 3 (mod 8) r 3 (n) = 12h( 4n) n 1, 2,, 6 (mod 8) 0 n 7 (mod 8) h( d) d L [Ko] [O] x 2 + y 2 + 10z 2 [OS] x 2 + y 2 + 10z 2 3, 7, 21, 31, 33, 43, 67, 79, 87, 133, 217, 219, 223, 23, 307, 391, 679, 2719 [BLGHT] Barnet-Lamb, T., Geraghty, D., Harris, M., Taylor, R., A family of Calabi-Yau varieties and potential automorphy II, preprint, to appear P.R.I.M.S. http://www.math.harvard.edu/ rtaylor/ [BGG] Barnet-Lamb, T., Gee, T., Geraghty, D., The Sato-Tate conjecture for Hilbert modular forms, preprint (http://arxiv.org/abs/0912.104) [Cl] Clozel, L., Motifs et formes automorphes: applications du principe de fonctorialité, Automorphic forms, Shimura varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), 77 19. [Co] Cox, D. A., Primes of the form x 2 + ny 2. Fermat, class field theory and complex multiplication, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1989. [Gl] Glaisher, J. W. L., On the numbers of representations of a number as a sum of 2r squares, where 2r does not exceed eighteen, Proc. London Math. Soc. (2) (1907), 479 490.

1 [HW] Hardy, G. H., Wright, E. M., An Introduction to the Theory of Numbers, th ed., Oxford University Press, Oxford, 1979G. H., E. M., I,II,,, 2001 [Ko] Koblitz, N., Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics, 97. Springer-Verlag, New York, 1993. ( : N. (, ),,, 2006 ) [Na] Nathanson, M. B., Elementary Methods in Number Theory, Graduate Texts in Mathematics, 19. Springer-Verlag, New York, 2000. [Neu] Neukirch, J., Algebraic number theory, Grundlehren der Mathematischen Wissenschaften, 322. Springer-Verlag, Berlin, 1999. : J. ( (), ()),,, 2003 ) [O] Ono, K., Ramanujan, taxicabs, birthdates, ZIP codes, and twists, Amer. Math. Monthly 104 (1997), no. 10, 912 917. [OS] Ono, K., Soundararajan, K., Ramanujan s ternary quadratic form. Invent. math. 130 (1997), no. 3, 41 44. [Sa],,, 2009. [Se] Serre, J-P., Modular forms of weight one and Galois representations, Algebraic number fields: L- functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 197), pp. 193 268. Academic Press, London, 1977. [We] Weisstein, E. W., Sum of Squares Function, From MathWorld A Wolfram Web Resource. http://mathworld.wolfram.com/sumofsquaresfunction.html [Za] Zagier, D, A one-sentence proof that every prime p equiv (mod 4) is a sum of two squares, Amer Math Monthly 97 (1990). [It2] [] 2009 9 34 3,. [It3] 2011 1 64 67,. [It4] 2011 1, :, 2011 1, 40 4,. [It] 20 199, 2011 4,. [Ta1] 2,, 1971. [T] : 2008.