2014 F/ E 1 The arithmetic of elliptic curves from a viewpoint of computation 1 Shun ichi Yokoyama / JST CREST,.
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- としみ さかわ
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1 2014 F/ E 1 The arithmetic of elliptic curves from a viewpoint of computation 1 Shun ichi Yokoyama / JST CREST,. K Q, C, F p.,, f = 0.,,., K 3 E : y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 (a i K) elliptic curve. 3.,.,.,,.,.,., Sage 2. Sage 10,, Python., Sage SageMath Cloud Pari/GP s-yokoyama@math.kyushu-u.ac.jp 2, Windows Sage. Sage SageMath Cloud., Sage Pari/GP. 5.
2 2, 2014 F 4 E. 1, pdf...,...,,.,,,. 2.,.,. 3.39, x, y = min {x + y, 100} 0 x 40.,..,..,.,,.,. 5.,.,..,.,...,.,,. TEX Word,, A4 or A4..,,.,., s-yokoyama@math.kyushu-u.ac.jp.,..,.
3 2014 F/ E 3 1, rational point.,. f(x, y) x, y 2, f(x, y) = 0 (x, y) f = 0., x, y R (x, y) f(x, y) = 0. x, y Q, (x, y). x, y Z, (x, y) f = 3x y. f = 0 (x, y), y = 3x. x y,. (t, 3t) t Q. (n, 3n) n Z f = x 2 + y 2 1. f = 0 (x, y), x 2 + y 2 = 1., 4 (±1, 0), (0, ±1)..., ( 1, 0) a, ( ) 1 a a 2, 2a 1 + a 2. a Q 1 a2 1+a Q 2a 2 1+a Q,. a Q 2,. ( ) 1 a 1.4., 2 2a 1+a, 2 1+a f = x 2 + y 2 3. f = 0 (x, y), x 2 + y 2 = 3.,... x 2 + y 2 = 3. m, n, r Z gcd(m, n, r) = 1 1, (x, y) = ( m r, n r ). m 2 + n 2 = 3r r 2 0 mod 3. m 2 0 or 1, n 2 0 or 1 m 2 + n 2 0 m 2 0 n 2 0. m, n 3, r 3. gcd(m, n, r) = ,. x 2 + y 2 = c c N 2, c. 1.6 (5 ). x 2 + y 2 = c, x 2 + y 2 = r 2 c r N.. r Q >0., c square-free. c = p 1 p 2 p k p i..
4 x 2 + y 2 = c c = p 1 p 2 p k, 1 i k p i = 2 or p i 1 mod (6 ). x 2 + y 2 = c c 3, 1. c = 1, f = 0 (f = ax 3 + bx 2 y + cxy 2 + dy 3 + ex 2 + gxy + hy 2 + ix + jy + k) a, b, c, d 1 0, f = x 3 y 2 2. f = 0 (x, y), y 2 = x ( 129 (3, 5), 100, 383 ) ( ) , , , ( , ), f = x 3 y f = 0 (x, y), y 2 = x ,. ( 1, 0), (0, ±1), (2, ±3),.,.,.,, (C, 6 ). 1.9 y 2 = x 3 2, ( ) , , Sage Pari/GP 3., (C),. Word TEX,,. 3, OK
5 2014 F/ E 5 2, 3.,. K., K = Q. K, Q number field., 3 E : y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 (a i K) 4. K 3. K 2, (x, y) ( ) x, y a1x a3 2 Ẽ : y 2 = 4x 3 + b 2 x 2 + 2b 4 x + b 6. K 2 3 (x, y) ( x 3b 2 36, E : y 2 = x 3 27c 4 x 54c 6 y 108). K = Q 0., E : y 2 = x 3 + ax + b. b i, c j a k. b 8,. b 2 = a a 2, b 4 = 2a 4 + a 1 a 3, b 6 = a a 6, b 8 = a 2 1a 6 + 4a 2 a 6 a 1 a 3 a 4 + a 2 a 2 3 a 2 4, c 4 = b b 4,. c 6 = b b 2 b 4 216b (E) = b 2 2b 8 8b b b 2 b 4 b 6 E discriminant (E) 0, E K elliptic curve. K = Q, (E) 0. (E) = , 5.,,. 2.5.,.. 4, a 5 a 6,.,. 5, x2 a 2 + y2 b 2 = 1 a, b R.
6 6 2.6 (C ). K = C, C/Λ Λ 2,. 2.7 (5 ). E : y 2 = x 3 + ax + b (E ). 2. y 2 = x 3, cusp. y 2 = x 3 + x 2, node. (E) = 0, (E) = 0., c 4 = 0., c (12 ). E E, 2, 1.., Sage. Q Sage. sage: E=EllipticCurve([1,2,3,4,5]); E E y 2 + xy + 3y = x 3 + 2x 2 + 4x + 5. Sage 5 a 1, a 2, a 3, a 4, a 6, Q. ;,. E, Elliptic Curve defined by y^2 + xy + 3y = x^3 + 2x^2 + 4x + 5 over Rational Field. E : y 2 = x 3 + ax + b sage: E=EllipticCurve([a,b]); E a,b. Cremona s index,., b i,. Sage discriminant. sage: E.discriminant(), E. Sage..,., c 4. Sage sage: E.c4(). b i.. sage: E.plot(), png. pdf 6. 6, eps, pdf., eps.
7 2014 F/ E (C,, 9 ). 3,. 1. y 2 = x 3 3x y 2 = x 3 + x 3. y 2 = x 3 x, 1., 1,., E : y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 P = (x 1, y 1 ), Q = (x 2, y 2 ), P = (x 1, y 1 a 1 x 1 a 3 ), P + Q = (x 3, (λ + a 1 )x 3 ν a 3 ), x 3 = λ 2 + a 1 λ a 2 x 1 x 2. λ, ν x 1 x 2, λ = y 2 y 1 x 2 x 1, ν = y 1x 2 y 2 x 1 x 2 x 1, x 1 = x 2, λ = 3x a 2 x 1 + a 4 a 1 y 1 2y 1 + a 1 x 1 + a 3, ν = x3 1 + a 4 x 1 + 2a 6 a 3 y 1 2y 1 + a 1 x 1 + a 3., np = P + P + + P n. O (6 )., E : y 2 = x 3 + ax + b, 2.,. 3, P, Q, R P + Q + R = O. P + ( P ) = O , E {O} O.,, E. O E K, E E(K) Mordell-Weil group (Mordell-Weil). E(K). E(K) Z r G. G. G E(K) torsion part, E(K) tors , r E, rank. E(K),. Z r 3, E(K) tors. K = Q.
8 (Mazur 7, 1977). E(Q) tors. m Z Z/mZ, 1 m 12, m 11, Z/2Z Z/2mZ, 1 m 4. K,. K = Q( 5) E : y 2 + xy + y = x 3 + x 2 3x + 1 E(K) tors Z/15Z, Mazur., E(K) tors, (C, 8 ). E : y 2 = x 3 + 2x + 3, E(Q) Z Z/2Z. E E P = ( 1, 0) 2 i.e. 2P = O. 2. E Q = (3, 6), n N nq O. Mazur P, Q E(Q). n 2., 2 E 1, E 2,., E 1 E 2 ϕ : E 1 E 2 1. P = (x, y) E 1, ϕ(p ) E 2 x, y. 2. P, Q E 1, ϕ(p + Q) = ϕ(p ) + ϕ(q). ϕ. ϕ E 1 isogeny., E 1 E 2.,. up to isogeny ϕ, ϕ. E 1 E 2, E 1 E 2, E 1 E 2. K, j(e) = c3 4 (E) E j(e ) = 1728 (4a)3 (E ) E E j j-invariant K K. E 1 E 2 K j(e 1 ) = j(e 2 ). 7 15, Ogg 1975.
9 2014 F/ E (6 ). j 0 0, E : y 2 + xy = x 3 36 j x 1 j ( ) 3 j 0 (E) =, j(e) = j 0 j Tate.. E(Q). E y 2 + y = x 3 + x 2 2x 8. sage: E=EllipticCurve([0,1,1,-2,0]) sage: E.gens() [(-1:1:1), (0:0:1)] 2, 9. ( 1, 1) (0, 0),. E 2. 2,. Q,. K,., E 2. Q 3 yes, sage: E=EllipticCurve([0,0,0,-82,0]) sage: E.gens(). 4 sage: E=EllipticCurve([0,-1,0,-24649, ]) sage: E.gens()., n N, n Q 1., Q.. Elkies n = 28. E E : y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 a 1 = 1 a 2 = 1 a 3 = 1 a 4 = a 6 = Cremona s index 389a. 389 conductor,, a index. 2 b, c,. 9, 3 1, 1 x, 2 y.
10 10 28 P 1 = ( , ) P 2 = ( , ) P 3 = ( , ) P 4 = ( , ) P 5 = ( , ) P 6 = ( , ) P 7 = ( , ) P 8 = ( , ) P 9 = ( , ) P 10 = ( , ) P 11 = ( , ) P 12 = ( , ) P 13 = ( , ) P 14 = ( , ) P 15 = ( , ) P 16 = ( , ) P 17 = ( , ) P 18 = ( , ) P 19 = ( , ) P 20 = ( , ) P 21 = ( , ) P 22 = ( , ) P 23 = ( , ) P 24 = ( , ) P 25 = ( , ) P 26 = ( , ) P 27 = ( , ) P 28 = ( , ) 28, (C, ). Elkies E 28. n = 28, n 29, ( ) , K, (Kida). E : y 2 = x 3 x K m = Q( p 1, p 2,, p m ). p i p i 5 or 7 mod 8, m E. j. Sage sage: E=EllipticCurve([0,0,1,-1,0]) sage: E.j_invariant() / (C, 6 ). E : y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 j.,. Sage. sage: var( a,b ); E=EllipticCurve([a,b])
11 2014 F/ E 11, E(Q) tors. Schmitt,., E : y 2 = x 3 + a 2 x 2 + a 4 x + a 6 a 1 = a 3 = (Lutz-Nagell). E a 2, a 4, a 6., P = (x 1, y 1 ) E(Q) tors x 1, y , P = (x 1, y 1 ) E(Q) tors y 1 = 0 or y = 27a a 3 2a 6 + 4a 3 4 a 2 2a a 2 a 4 a 6, = 16 0., E : y 2 = x 3 43x E(Q) tors Z/7Z (C, 12 )., E(Q) tors O {(3, ±8), ( 5, ±16), (11, ±32)} 6. 3., (3, 8) E(Q) tors., 2. (3, 8), E(Q) tors Z/7Z.,. j., K = Q. E, mod p F p. 4. E (x, y) F p F p, discrete.., 2.3 (E) F p {0} E F p E Q, E : y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 (a i Z). mod p E p : y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 (a i F p ), E p reduction at p. E p i.e. (E p ) 0., p (E) (E p ) = 0.,,.
12 E p F p i.e. (E p ) 0, E p has good reduction at p. E p, F p i.e. (E p ) = 0, E p has bad reduction at p , p / good prime/bad prime. (E),., E p, c 4 0., E p. E p, E p has multiplicative (semistable) reduction at p. E p, E p has additive (unstable) reduction at p..,, E Q. N(E) = p : prime p fp(e) E conductor. E E N(E) = N(E ). f p (E) E p f p (E) = 0, E p f p (E) = 1, E p f p (E) = 2 + δ p. δ p depth of wild ramification 0, p 2, 3 δ p = p = 2, 3, f p (E) f 2 (E) 5, f 3 (E) 3., 1) (E), B E, 2) p B E, c 4 f p (E), 3) p fp(e), 3. Sage. sage: E.conductor() 2.38 (C,, 5 ). Q E : y 2 + xy + 3y = x 3 + 2x 2 + 4x + 5, 3 N(E). Sage conductor, 3., N(E) 1. N(E) = 1, p E Q E N(E) = 1, E everywhere good reduction.,.
13 2014 F/ E Q E, B E. E B E everywhere good reduction outside B E. p p., Q, 1., Tate (Tate). Q (C, 16 ). 2.41,. E : y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 (a i Z) 1. (E) = c3 4 c E Q. (E) = ±1. 3. c 2 6 = c 3 4 ± (c 4, c 6 ), (c 4, c 6 ) a i Z (c 4, c 6 ). E y 2 = x 3 27c 4 x 54c 6. K Q, mod p 5 mod p., K, O K p, mod p., (1) K. Q, K (20 ). K Q. O K, p. E K. 1. E p reduction at p. 2. E N(E). depth of wild ramification (C, 20 ). K = Q( 6), ε = E 1 : y 2 = x 3 4(2 + 6)x 2 + 4εx, E 2 : y 2 + 6xy y = x 3 (2 + 6)x E 1 c 4, c 6, (E 1 ), j(e 1 ). 2. E 2 c 4, c 6, (E 2 ), j(e 2 ). 3. E 1, E E 1 E 2, E 2 global minimal model.
14 (8 ). Q( 3 28) E : y 2 + a 1 xy + a 3 y = x 3 (a 2 = a 4 = a 6 = 0). (E) = a 3 3(a a 3 ), E Q( 3 28) (a 1, a 3 ) (C, )., 2 11.,., K Q( ±m) m > 0: Q( 3 m) m > 0:. K, K.,.., K = Q( m), 1 < m < 100 m = 26, 51, 79, 86, 87, 91..,. PEX/UNDET 1,. PNEX/UNDET, K = Q( m), 1 < m < 100 m = 38, 53, 61, K = Q( 3 m), 1 < m < 100 m = 23..,., K = Q( 3 46)., Non-existence of elliptic curves with everywhere good reduction over some real quadratic fields, Journal of Math-for-Industry 3 (2011), pp On elliptic curves with everywhere good reduction over certain number fields, American Journal of Computational Mathematics 2, No.4 (2012), pp y xy y = x ( )x 2 + C C C 3 x + C C C C 1 = , C 2 = , C 3 = , C 4 = , C 5 = , C 6 =
15 2014 F/ E 15 3,., E(K)., 2 E(K) Z r.,., E, E K, ϕ : E E.. 0 E (K)/ϕ(E(K)) Sel ϕ (E/K) III(E/K)[ϕ] 0 Sel ϕ (E/K) Sel(E/K) E Selmer group. III(E/K) - Tate-Shafarevich group 13. 2, K Galois Gal(K/K) 1 Galois, - Weil-Châtelet group.,. m, 0 E(K)/mE(K) Sel m (E/K) III(E/K)[m] 0. III(E/K)[m] = {t III(E/K) mt = 0} m-torsion. E(K)/mE(K),.,. m = 2, E(K)/2E(K) Sel 2 (E/K), E Sel 2 (E/K). 2-descent. descent m 2 m-descent, 2-descent. III(E/K)[2], E(K)/2E(K)., III(E/K)[2] E(K)/2E(K) Sel 2 (E/K)., III(E/K)[2]., III(E/K).. III(E/K)[2],, Sel 2 (E/K) E(K)/2E(K),., III(E/K)[2] E(K)/2E(K). E(K)/2E(K). E(K). -descent. height., E(K)/2E(K),,. 13 : sha.,. TEX, I 3.
16 E(K). Sage Simon s 2-descent. Sage 14, Simon Pari/GP 15 Sage., K = Q( 7) E : y 2 = x 3 + x + 7 E(K). Sage sage: K.<a>=NumberField(x^2+7) sage: E=EllipticCurve(K,[0,0,0,1,a]) sage: E.simon_two_descent(verbose=1),. a K/Q, a= 7. verbose, courbe elliptique : Y^2 = x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7) points triviaux sur la courbe = [[1, 1, 0], [Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7), 1]] #S(E/K)[2] = 2 #E(K)/2E(K) = 2 #III(E/K)[2] = 1 rang(e/k) = 1 listpointsmwr = [[Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7), 1]] (1, 1, [(1/2*a + 3/2 : -a - 2 : 1)]). #S(E/K)[2] Sel 2 (E/K), 2. 2 #III(E/K)[2] III(E/K)[2], E(K)/2E(K) 2, torsion part 16, E 1. -descent, E(K) ( a+3 2, a 2). E(K).,. E(K) tors = {0}, torsion part K = Q( 59) E : y 2 = x E(K),. courbe elliptique : Y^2 = x^3 + Mod(y, y^2-59)...omitted... #S(E/K)[2] = 4 #E(K)/2E(K) = 4 #III(E/K)[2] = 1 rang(e/k) = 1 [1, 2, [...points...]] 14. Sage Simon s 2-descent, Pari/GP Sage. Pari/GP. Denis Simon ell.gp. Sage, developer-trac,. 15,. 16,.
17 2014 F/ E 17 points ( 12, 0) ( , ). [1, 2]. E 1 2,, 1., Sel 2 (E/K) F 2 -. III(E/K)[2] E(K)/2E(K) Sel 2 (E/K). 1, E(K) Z 2 or E(K) Z E(K) tors with #(E(K) tors /2E(K) tors ) = 2., ( 12, 0) 2, E(K) torsion part. E(K) Z Z/2Z, ( , ) E(K). K = Q( 3 53) E : y 2 = x E(K),. courbe elliptique : Y^2 = x^3 + Mod(y, y^3-53)...omitted... #S(E/K)[2] = 8 #E(K)/2E(K) >= 2 #III(E/K)[2] <= 4 0 <= rang(e/k) <= 2 [0, 3, [...points...]] points, 2 ( 12, 0). III(E/K)[2] 4. III(E/K)[2] = {0}, E(K)/2E(K) 8 E(K)/2E(K). 2 ( 12, 0) E(K) Z 3. E 2. 2, E(K)/2E(K) 2.,. III(E/K)[2],. 3.7 (C, 8 ). Q K, 1, ell.gp E(K). Sage. Pari/GP, gp calculator gp.exe ell.gp. ell.txt. ell.gp. Pari/GP, ell.gp. 2. gp calculator gp: \r ell.gp. gp: bnf=bnfinit(f) gp: ell=ellinit([a1,a2,a3,a4,a6]) f y. K = Q( 43) y^2-43.
18 gp: bnfellrank(bnf,ell). Affichage des calculs DEBUGLEVEL ell= (6 ). K E E(K),. Sel 2 (E/K) 4, III(E/K)[2] 2, 2 2., E(K). 3.9 (20 ). Sage, 2-descent..,. sage: K = CyclotomicField(43).subfields(3)[0][0] sage: E = EllipticCurve(K, 37 ) sage: E.simon_two_descent() # long time (4s on sage.math, 2013) Traceback (most recent call last):... RuntimeError: *** at top-level: ans=bnfellrank(k,[0,0,1, *** ^ *** in function bnfellrank:...eqtheta,rnfeq,bbnf];rang= *** bnfell2descent_gen(b *** ^ *** in function bnfell2descent_gen:...riv,r=nfsqrt(nf,norm(zc)) *** [1];if(DEBUGLEVEL_el *** ^ *** array index (1) out of allowed range [none]. An error occurred while running Simon s 2-descent program, E(K).., , n. 3 n, n congruent number :4:5.
19 2014 F/ E 19,, , 1. a, b, c Q >0 a 2 + b 2 = c ab = a b. α = c2 4, β = (a2 b 2 )c 8 β 2 = α 3 α. a, b, c Q >0 α, β Q, a b β 0., Q y 2 = x 3 x (α, β) α > 0, β 0., (0, 0), (1, 0), ( 1, 0), O 4., n, Q y 2 = x 3 n 2 x n, 3 (α, β) β 0. a = α2 n 2, b = 2nα β β, c = α2 + n 2 β n = 1 y 2 = x 3 x 0, (C, 8 ). n, 3 a, b, c. a, b, c (C, 5 ) a, b , c. 3., n, n..,., n. 17, a, b, c,. 18,.
20 n.. A n = # { (x, y, z) : x, y, z Z n = 2x 2 + y z 2} B n = # { (x, y, z) : x, y, z Z n = 2x 2 + y 2 + 8z 2} C n = # { (x, y, z) : x, y, z Z n = 8x 2 + 2y z 2} D n = # { (x, y, z) : x, y, z Z n = 8x 2 + 2y z 2}. A. n. B. n 2A n = B n. n 2C n = D n., Tunnell (Tunnell, 1983) A B. BSD B A (C,, 8 ). 2014, 3.16., 3.17 BSD.,.,. - Birch and Swinnerton-Dyer conjecture. 2 Birch Swinnerton-Dyer, BSD., (BSD ). E Q, L(s, E) E Hasse-Weil L. L(s, E) C, L(s, E) s = 1 E r Hasse-Weil L, (Kolyvagin, Gross-Zagier). r 1 BSD. Hasse-Weil L. E(F p ) := { (x, y) : x, y F p y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 } {O}. E(F p ) (y 2 + a 1 xy + a 3 y) (x 3 + a 2 x 2 + a 4 x + a 6 ) p (x, y) O, E p E p. E(F p ) L(s, E) := p (E) E Hasse-Weil L. a p (E) := p + 1 #E(F p ) 1 1 a p (E)p s + p 1 2s p (E) 1 1 a p (E)p s
21 2014 F/ E 21 L(s, E) s = 1 E. BSD E., L(s, E) s = 1, E analytic rank. Sage, sage: E.analytic_rank(), BSD., (BSD ). E Q, L(s, E) E Hasse-Weil L. L(s, E) s = 1 Taylor L(s, E) = c ran (s 1) r an +. E r an, c ran = Ω E Reg(E) #III(E/Q) p c p #E(Q) 2 tors. Ω E E Archimedean period, Reg(E) E regulator, c p Tamagawa number. BSD, - III(E/Q) 3.3 K = Q., BSD a p (E). a p (E) Hasse (Hasse, 1933). p (E) p a p (E) 2 p,., h = {z C Im(z) > 0} f : h C, ( ( ) ) γ : z az + b a b z h, γ = SL 2 (Z), c 0 (mod N) cz + d c d f(γ(z)) = ε(d)(cz + d) k f(z) (k Z 2 ) Q k modular form. N f level, ε : (Z/NZ) C character, nebentypus. 19 Mordell-Weil p p #E(F p) p p. #E(F p). n, #E(F p ) = n E Deuring. 20.
22 ,., modular form, automorphic form. f(z), q = e 2πiz. q Fourier f(q) = a n (f)q n (q = e 2πiz ) n 0. a p (E) a p (f),, ( - ). Q E modular., N E, 2 N f, p a p (E) = a p (f). N , BSD 3.19 L(s, E) C,. -, E 1995 Wiles, Fermat n 3 x n + y n = z n 0 x, y, z,., (Breuil-Conrad-Diamond-Taylor, 2001) E : y 2 = x 3 x, 2, 32 f(q) = q 2q 5 3q 9 + 6q q 17 q 25 10q p = 17 #E(F 17 ) = 16 a 17 (E) = = 2, a 17 (f). sage: E=EllipticCurve([0,0,0,-1,0]) sage: E17=EllipticCurve(GF(17),[-1,0]) sage: E.ap(17) sage: E17.cardinality() sage: E17.points() (C, 8 ). Q E : y 2 + y = x 3 x 2, f(q) = q (1 q n ) 2 (1 q 11n ) 2 n 1, -, 100 p.
23 2014 F/ E 23,. Q,. Q, BSD.,, (20 )., abelian variery (20 ). a p (f) eigenform Fourier. 1. Hecke Hecke operator. 2. f (20 )., -.,.. Hilbert modular forms Siegel modular forms Bianchi modular forms 3.35 (20 ). - Langlands correspondence (30 ). BSD,.. Ω(E) E Archimedean period Reg(E) E regulator c p Tamagawa number (30 ) , - L(s, E) C (40 ). BSD p p-adic version.,. 1. p Q p. 2. p L. 3., p BSD. 4. p BSD. 21 manifold..
24 24 2, , (5., 20 ).,.., E Q E 1, E 2 N(E 1 ) N(E 2 ), E 1 E E E(R) := {(x, y) E : x, y R} {O}. E(R). 3. #III(E/Q)[2] = 1, #Sel 2 (E/Q) = 16 E Q. 5. E : y 2 = x 3 + x + 2 p 1 < p 2 #E(F p1 ) < #E(F p2 ). N(E) = 56 = p 1, p 2 2, (C, 20 ). 1 15, A C. 13. Q E 1 : y 2 + y = x 3 + x 2 8x b 2 = 1, b 4 = 2, b 6 = 3, b 8 = 4, c 4 = 5, c 6 = 6, (E 1 ) = 7, N(E 1 ) = 8. E 1 B E1 = { 9, 10 }. Q E 2 : y 2 = x 3 49x. E 2 11 E 2 (Q) tors E 2 (Q)/2E 2 (Q) 14, E A E 1 E 2. B E 1 E 2 p = 23. #E 1 (F 23 ), #E 2 (F 23 ). C E 1 E 2,. Sage., a. Sage, OS Sage Reference Manual a, Sage..
25 2014 F/ E 25 4,., 6.. E = E : y 2 = x 3 + ax + b. F p, F q q = p m p. p.., F p F q P E mp = O m E m m-division point, E[m] m p. E F q, E m E[m] Z/mZ Z/mZ. p m m = p r m p m E[m] Z/m Z Z/m Z or Z/mZ Z/m Z. E/F q m m 2.. Q, F q E(F q ), E(F q ) Z/m 1 Z Z/m 2 Z (m 1 m 2, m 1 q 1). E(F q ) 2. E(F q ) r E(F q ) Z/m i Z (m j m j+1 ) i=1. E m 1 m r r 2.., E F q. q + 1 #E(F q ) 0 mod p E supersingular F 5 E : y 2 = x E(F 5 ) = {(0, 1), (0, 4), (2, 2), (2, 3), (4, 0), O} #E(F 5 ) = (C,, 5 ). F 7 E : y 2 = x 3 + x.
26 26., E F p. P E. Q = dp 0 d < ord(p ). ord(p ) P. P, Q 2 d Elliptic Curve Discrete Logarithm Problem, ECDLP F 7 E : y 2 = x 3 + 2x + 3 P = (3, 6) Q = dp = (3, 7), d d = 14. 2P, 3P, P F E : y 2 = x 3 +2x+3 P = (3, 6) Q = dp = ( , ), d d = P , (C, 6 ). F 103 E : y 2 = x 3 + 2x + 3 P = (2, 85) Q = dp = (11, 29), d. 2, d., d,.,.., 22 RSA RSA cryptosystem. 3 Rivest-Shamir-Adleman M, p, q n = pq. 2. gcd(e, φ(n)) = 1 e N. φ(n) Euler, n n. φ(n) = (p 1)(q 1). 3. de 1 mod φ(n) d Z., (n, e) (p, q, d).. 0 M c M e mod n,. c M c d mod n., n M < n. p, q. ASCII. 22,,.,.
27 2014 F/ E A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Sage, HELLOWORLD RSA. M M = sage: p = (2^31) - 1; p sage: is_prime(p) True sage: q = (2^61) - 1; q sage: is_prime(q) True sage: n = p*q ; n e. sage: e = ZZ.random_element(euler_phi(n)) sage: while gcd(e, euler_phi(n))!= 1:... e = ZZ.random_element(euler_phi(n))... sage: e # random sage: e < n True Python, for indent. 2 Enter. d. sage: bezout = xgcd(e, euler_phi(n)); bezout (1, , ) sage: d = Integer(mod(bezout[1], euler_phi(n))) ; d sage: mod(d*e, euler_phi(n)) 1 (n, e) (p, q, d), n = e = p =
28 28 q = d = sage: mod(m^e, n) RuntimeError Traceback (most recent call last) /home/mvngu/<ipython console> in <module>() /home/mvngu/usr/bin/sage-3.1.4/local/lib/python2.5/ site-packages/sage/rings/integer.so in sage.rings.integer.integer. pow (sage/rings/integer.c:9650)() RuntimeError: exponent must be at most ,., mod. sage: c = power_mod(m, e, n); c power mod 23 def power_mod(a, b, n): d = 1 for i in list(integer.binary(b)): d = mod(d * d, n) if Integer(i) == 1: d = mod(d * a, n) return Integer(d) c,. sage: power_mod(c, d, n) sage: m HELLOWORLD H E L L O W O R L D (C, 15 ). SHUNICHIYOKOYAMA RSA ,. 23 SageMath Cloud,. Sage ver.5.
29 2014 F/ E P, Q = dp d. RSA.. P d = Q, P, Q d. d = log P Q. P, Q R d,, P, Q log P Q,.,., ECDLP Elliptic Curve Cryptography ECC. RSA,. 2 bit. 9 = = RSA k=0 a k2 k, a k = 0, 1, 160., (C, 15 / cf ). n RSA number field sieving method., T 1 (n) = C 1 e n1/3 (log n) 2/3 C 1., m ρ rho method, T 2 (m) = C 2 2 m/2 C 2. T T = T 1 (n) = CT 2 (m) C, T n, m RSA 160 ECC T 1 (1024) = CT 2 (160)., 4096 RSA ECC (6 ). 4.16, > 0, 1,,.., p, (Z/pZ) g. p, g. A,B, 0 p 2,. r A, r B.
30 30 1. A X A g r A mod p, B. 2. B X B g r B mod p, A. 3. A B K A X r A B mod p. 4. B A K B X r B A mod p. K A = K B g r Ar B,. Diffie-Hellman key exchange. X A, X B K A = K B., F p p, F p E, E P. E, P. A,B, 0 ord(p ) 1,. r A, r B. 1. A X A = r A P, B. 2. B X B = r B P, A. 3. A B K A = r A X B. 4. B A K B = r B X A. K A = K B = r A r B P,. Elliptic Curve Diffie-Hellman key exchange, ECDH. X A, X B K A = K B (C, 10 ). F E : y 2 = x 3 + x + 6 E P = (2, 4). ord(p ) = r A, r B 10000, ECDH. X A, X B, K A, K B K A = K B X A, X B, P K A = K B Elliptic Curve Diffie-Hellman Problem ECDHP. ECDHP ECDLP. X A, P r A ECDLP K A = r A X B (6 ). ECDHP ECDLP.. ECDH, p, F p E, E P. E, P. A B M E., 0 ord(p ) 1,. r A, r B.
31 2014 F/ E B P B = r B P. r B, P B. 2. A R = r A P, B X A = r A P B. C = M + X A, (C, R) B. 3. B (C, R) r B X B = r B R. M = C X B. M = M 24,. Elliptic Curve ElGamal Encryption., (C, R),. X A, X B, M E (C, R), M E (C, R )., M + M E (C + C, R + R ) (6 ) ,.,,, RSA.,., Fully-holomorphic Encryption. IBM Craig Gentry, 2009., (20 )., (20 ).,,. digital signature. ECDSA (20 ). ECDLP,. 2P, 3P, 4P, Q = dp. brute force method
32 32 Baby-step Giant-step ρ 2. ρ, (20 ). ρ (20 ). ECDLP. Menezes- -Vanstone., (20 ). ECDLP pairing., F q bilinear map. 1. F q 2 G e : E(F q ) E(F q ) G. 2., ID., (20 ) Bos-Kaihara-Kleinjung-Lenstra-Montgomery, 112 ECDLP., PlayStation 3., ECDLP, Certicom,. Certicom Challenge.
33 2014 F/ E 33 5,,. 1 15, A G Pari/GP A D. Q E : y 2 + xy = x x b 2 = 1, b 4 = 2, b 6 = 3, b 8 = 4, c 4 = 5, c 6 = 6 (E) = 7. factor(n) (E), E B E = { 8, 9, 10, 11 }. E p B E, N(E) = 12. E(Q) 13, P = 14. a p (E) p p 20 { 15 }. A N(E). B #E(F 11 ). C Q = (4344, ). D np + Q E n N. SageMath Cloud. E E BSD, F E. G E p = 11. SageMath Cloud,., A G. SageMath Cloud. Sage, Sage Quick Reference Card 26,. Elementary Number Theory. 26.
34 SageMath Cloud Sage, 2,3 Reference Card, SageMath Cloud. β,,. GitHub ,. SageMath Cloud..,. 27. create an account,., I agree to the SageMathCloud Terms,., sign in.,.. 27.,.,.
35 2014 F/ E 35. ID. Projects,. SageMath Cloud, New Project.,. Description. Create Project,.
36 36 Create or Import a File, Worksheet, Terminal or Directory.. Sage. Sage L A TEX. L A TEX Sage, Sage L A TEX pdf.,,.,.
37 2014 F/ E 37, Sage. Sage. Shift+Enter., Help. Project Control,.
38 38, SageMath Cloud,, GitHub. Sage GitHub. 5.2 Pari/GP Sage Windows OS, Pari/GP. Pari/GP Q.. gp: E=ellinit([1,2,3,4,5]) E y 2 + xy + 3y = x 3 + 2x 2 + 4x + 5., [1, 2, 3, 4, 5, 9, 11, 29, 35, -183, -3429, , /10351, [.... a i i = 1, 2, 3, 4, 6, b i i = 2, 4, 6, 8, c i i = 4, 6, (E), j(e)., gp: E.disc. E.a2, E.j. E torsion part E(Q) tors 28. gp: elltors(e),.. Pari/GP, y 2 = x 3 43x E(Q) free part, elldata.
39 2014 F/ E 39 gp: E=ellinit([0,0,0,-43,166]) gp: elltors(e) [7, [7], [[3, 8]]]., 7 C 7 Z/7Z. y 2 + xy = x x gp: E=ellinit([1,0,0,-1070,7812]) gp: elltors(e) [16, [8, 2], [[34, 88], [-36, 18]]]. 2 16, , E(Q) tors Z/2Z Z/8Z. [[34, 88], [-36, 18]]. E. y 2 + xy = x 3 + 2x 2 + 4x + 5 P, Q E P = ( 1, 1), Q = (1, 4). 2 P + Q = ( 5 4, ) 15 8 E. gp: E=ellinit([1,2,0,4,5]) gp: P=[-1,-1] gp: Q=[1,-4] gp: elladd(e,p,q) [-5/4, 15/8] np ellpow(e,p,n). gp: ellorder(e,p) 0. P,,. a p (E) = p + 1 #E(F p ). a k (E) k N. gp: ellap(e,p) gp: ellak(e,k) gp: ellan(e,n) a p (E), a k (E), a k (E) 1 k n,. SageMath Cloud. Pari/GP,., A D.,. E G,.
40 40,,.., 1. J.H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, GTM 106 (1986). Expanded 2nd Edition (2009). 2. J.H. Silverman and J. Tate, Rational Points on Elliptic Curves, Springer-Verlag, UTM (1992). 3. W. Stein, Elementary Number Theory: Primes, Congruences, and Secrets, Springer-Verlag, UTM (2009) , ,,,, (1995), (2012) Sage, Sage., pdf., J.H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer-Verlag, GTM 151 (1994).. 4., / (2003, 2012)..,.. J.W.S. N.,.,.,,, 5.,, (2008) ,. 29,.,.
41 2014 F/ E 41,.,.,. liao files/taguchi.pdf calc.pdf Sage Tate BSD Jerome Dimabayao, Cid Reyes.,,,.
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(Florian Sprung) p 2 p * 9 3 p ζ Mazur Wiles 4 5 6 2 3 5 2006 http://www.icm2006.org/video/ eighth session [ ] Coates [Coates] 2 735 Euler n n 2 = p p 2 p 2 = π2 6 859 Riemann ζ(s) = n n s = p p s s ζ(s)
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数論 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/008142 このサンプルページの内容は, 第 2 版 1 刷発行当時のものです. Daniel DUVERNEY: THÉORIE DES NOMBRES c Dunod, Paris, 1998, This book is published
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(. x y y x f y = f(x y x y = y(x y x y dx = d dx y(x = y (x = f (x y = y(x x ( (differential equation ( + y 2 dx + xy = 0 dx = xy + y 2 2 2 x y 2 F (x, y = xy + y 2 y = y(x x x xy(x = F (x, y(x + y(x 2
42 3 u = (37) MeV/c 2 (3.4) [1] u amu m p m n [1] m H [2] m p = (4) MeV/c 2 = (13) u m n = (4) MeV/c 2 =
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3 3.1 3.1.1 kg m s J = kg m 2 s 2 MeV MeV [1] 1MeV=1 6 ev = 1.62 176 462 (63) 1 13 J (3.1) [1] 1MeV/c 2 =1.782 661 731 (7) 1 3 kg (3.2) c =1 MeV (atomic mass unit) 12 C u = 1 12 M(12 C) (3.3) 41 42 3 u
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(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0
1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45
x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y)
x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy
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