I. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ) modular symbol., notation. H = { z = x

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1 I. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ) modular symbol., notation. H = z = x iy C y > 0, cusp H = H Q., Γ = PSL 2 (Z), G Γ [Γ : G] = e <., G H, X G = G\H modular symbol. α, β H G 2,, β = M(α) ( M G)., α β H smooth X G, closed path, H 1 (X G, Z). modular symbol α, β G,, α, β., H 1 (X G, Z) modular symbol. ( ) M Γ, H M(0) M( ) smooth (M) = M(0), M( )., M : M(0), M(1), M( ) : (M), (MT S), (M(T S) 2 ) Date: October 24, Key words and phrases. modular forms, elliptic curves, L-functions. 1

2 2.,, S = Γ. ( ) 0 1, T = 1 0 ( ) Remark 1.1. (M) G, index, X G,, G modular symbol., M G, (M) G (MT S) G (M(T S) 2 ) G =0 (M) G (MS) G =0.,, (M M) G = (M) G ( M G), Γ/G, X G closed path. (M 1 ) G,..., (M e ) G (, e = [Γ : G]) modular symbol. C(G), (M 1 ) G,..., (M e ) G symbol Q e. modular symbol B(G) (M) G (MT S) G (M(T S) 2 ) G (M) G (MS) G C(G)., C 0 (G) G-cusp [α] G ([α] G = [β] G β = M(α), M G) Q. modular symbol δ : C(G) C 0 (G) δ((m) G ) = [M( )] G [M(0)] G, Z(G) = Ker (δ)., modular symbol, H(G) = Z(G)/B(G). Proposition 1.2. modular symbol, H(G) H 1 (X G, Q). Remark 1.3. G Γ, cusp α, β modular symbol α, β, Manin Drinfeld, Q-,, H 1 (X G, Q)., 0, H 1 (X G, Q).

3 I. (CREMONA ) M-symbol.,., M-symbol (M Manin ),., G = Γ 0 (N), H(N) = H(Γ 0 (N)), X 0 (N) = X Γ0 (N) M-symbol. gcd(c, d, N) = 1 (c, d) Z 2, (c 1, d 1 ) (c 2, d 2 ) c 1 d 2 c 2 d 1 (mod N)., (c, d) (c : d), M-symbol., M-symbol P 1 (N) = P 1 (Z/NZ). (c 1, d 1 ) (c 2, d 2 ),, N M-symbol., M-symbol modular symbol,. M-symbol v.s. modular symbol Proposition 1.4. P 1 (N) [Γ : Γ 0 (N)] (M) : M [Γ : Γ 0 (N)]., ( ) a b (c : d) M = (M) = b/d, a/c c d., a, b Z ad bc = 1. ( [C, Proposition 2.2.1],.) modular symbol, M-symbol,.,,. M-symbol M-symbol (c : d) (c d : c) (d : c d) (c : d) ( d : c) δ : (c : d) [a/c] [b/d],.

4 4 Example 1.5. M-symbol, 11 modular X 0 (11) H 1 (X 0 (11), Q) H(11). I. M-symbol (c : 1) mod 11 (1 : 0) 12 (c) ( ). II. B(G) a). (c : d) ( d : c) = 0 (0) ( ) = 0, (1) ( 1) = 0, (2) (5) = 0, ( 2) ( 5) = 0, (3) ( 4) = 0, ( 3) (4) = 0., (2 : 1) ( 1 : 2) = 0,, ( 1) (mod 11), ( 1 : 2) = (5 : 1), (2) (5) = 0. b). (c : d) (c d : c) (d : c d) = 0 (0) ( ) ( 1) =0, (1) ( 2) (5) = 0, (2) (4) ( 4) =0, (3) ( 5) ( 3) = 0., (1 : 1) (2 : 1) (1 : 2) = 0,, 2 1 ( 2) ( 1) (mod 11) ( 2) (mod 11), (2 : 1) = ( 2 : 1) (1 : 2) = (5 : 1), (1) ( 2) (5) = 0., A = (2), B = (3), C = (0), (0) = C, ( ) = C, (1) = ( 1) = 0, (2) = ( 2) = A, (3) = B, ( 3) = A B, (4) = B A, ( 4) = B, (5) = ( 5) = A., M-symbol A, B, C. III. Z(G) [a/b] = [0] if b 0 (mod 11), [a/b] = [ ] if b 0 (mod 11), Γ 0 (11)-cusp [0] [ ]. δ((m)) = [1/m] [0] =0 δ((0)) = [ ] [0] 0 m 0 (mod 11), A, B Ker (δ) C = (0) Ker (δ). H 1 (X 0 (11), Q) H(11) A, B.

5 I. (CREMONA ) 5 ( ) Example 1.5. I. II. III., III. II.,,,. 2. Hecke, Hecke, modular form f, a p0 p 0,, Heilbronn.,, a p0, p a p Hecke.,, N N, N p, Hecke T p. modular symbol T p ( α, β ) [ ( ) p 0 ( ) ]α, 1 r β = pα, pβ p r mod p r mod p α r p, β r. p, modular symbol H(N). modular form, Γ 0 (N) 2 cusp form, ( ) C a b S 2 (N)., 2 2 M = c d (ad bc > 0) cusp form f(z) S 2 (N) ad bc (f M)(z) = (cz d) f(az b 2 cz d ) [ ( ) p 0., Hecke T p = 0 1 r mod p form f(z) (f T p )(z) = pf(pz) 1 p, S 2 (N). p 1 f( z r p ) r=0 ( ) ] 1 r cusp 0 p

6 6 Hecke γ H 1 (X 0 (N), Q) f S 2 (N), 2πif(z)dz γ, f γ.,, α, β, f M = Mα, Mβ, f,, Hecke T p α, β, f T p = T p α, T p β, f, Hecke T p H(N) S 2 (N). Fricke involution W q N q, H(N) S 2 (N) Fricke involution W q,., L-,. Hecke T p Fircke involution W q Q Hecke, T., W N = Π q W q,, z 1/Nz,., α N q α N q α1 N, x, y, z, w Z q α xw (N/q α )yz = 1., ( ) q W q = α x y Nz q α w H(N) S 2 (N), Wq 2 Γ 0 (N) involution., W q x, y, z, w Z. Example 2.1. (Example 1.5..) modular X 0 (11) H 1 (X 0 (11), Q) H(11) A, B., M-symbol A = (2 : 1) Hecke T p (p 11) Fricke involution W 11. I. M-symbol modular symbol M-symbol A = (2 : 1) modular symbol Proposition 1.4. ( 1 1 ), ( ) 1 0 A = (2 : 1) M = (M) = 0/1, 1/ II. modular symbol )., Hecke T p. p = 2,, T 2 (A) = T 2 ( 0, ) = 0, 1 0, 4 2, 3 4

7 I. (CREMONA ) 7., modular symbol M-symbol (1 : 1) (4 : 1) (1 : 2) ( 4 : 1) = 0 (B A) ( A) ( B) = 2A T 2 (A) = 2A. Hecke, S 2 (11) rational newform f(z) = a n e 2πinz a 2 = 2, (, f A A, f = 0, ). )., Fricke involution W 11. W 11, ( ) 0 1 W 11 = (x = w = 0, y = 1, z = 1 ) M-symbol modular symbol,, )., W 11 (A) modular symbol 0, 2 1 ( ) 0 1 0, 1 2 W 11 (A) = =, ,, M-symbol, 0 0, 1 5 1, = (1 : 0) ( 5 : 1) (11 : 5) = ( ) ( 5) (0) = A,, Fricke involution W 11 (A) = A., L- ( 5.1),. Hecke, modular form a p0 p 0,, modular symbol, Heilbronn., M-symbol, T p W q, modular symbol.,, M-symbol Heilbronn.,,., modular symbol, T p, p 1 ( ) ( ) p 0 1 r, (r mod p) p

8 8. M-symbol ([C, Proposition ])., p, M 2 (Z) R p (Heilbronn ), T p M-symbol T p ((c : d)) = M R p (c : d)m., R p p, R p, Hecke. Proposition ( 2.2. p ) N., R p x y y x M 2 (Z) (xx yy = p). (1) x > y > 0, x > y > 0, yy > 0; or (2) y = 0, y < x /2; or (3) y = 0, y < x/2. Heilbronn (1 ) ( ) ( ) ( ) R 2 =,,, (1 ) ( ) ( ) ( ) ( ) ( ) R 3 =,,,,, (1 ) ( ) ( ) ( ) ( ) ( ) ( ) R 5 =,,,,,,, ( ) ( ) ( ) ( ) ( ) ,,,, Example 2.3. (Example 2.1. ) modular X 0 (11) H 1 (X 0 (11), Q) H(11) A, B,, M-symbol A = (2 : 1) modular symbol, T 2 (A) = 2A, Heilbronn R 2, M-symbol

9 I. (CREMONA ) 9. T 2 (A) = (2 : 1)R 2 ( ) ( ) ( ) ( ) (2 : 1) (2 : 1) (2 : 1) (2 : 1) =(2 : 2) (4 : 1) (4 : 3) (3 : 2) =(1) (4) (5) ( 4) =0 (B A) ( A) ( B) = 2A., T 3 (A) = (2 : 1)R 3 ( ) ( ) ( ) (2 : 1) (2 : 1) (2 : 1) ( ) ( ) ( ) (2 : 1) (2 : 1) (2 : 1) =(2 : 3) (6 : 3) (3 : 3) (6 : 1) (6 : 1) ( 1 : 3) =(8 : 1) (2 : 1) (1 : 1) (6 : 1) (5 : 1) (4 : 1) =(A B) (A) (0) ( A) ( A) (B A) = A. Hecke ( ) modular form a p0 p 0,. I. M-symbol modular symbol,, Hecke.,. II. Heilbronn, M-symbol Hecke., Heilbronn. 3. modular form, modular form., modular form 2, N rational newform modular. f rational newform, Λ f Λ f = α, β, f α, β H, α β mod Γ 0 (N), 2 ( C). E f = C/Λ f, f modular.

10 10 E f Q. L(E f, s) = L(f, s). E f N 3.2. Hecke. rational newform f(z) = n 1 a nq n (q = e 2πiz ), a 1 = 1.,. p N p, f T p = a p f. q N q, f W q = ϵ q f (ϵ q = ±1) ϵ q if q 2 N a q = 0 if q 2 N., a p n1 = a p a p r δ N (p)pa p r 1 (r 1),., 1 if p N δ N (p) = 0 if p N., n m, a mn = a m a n. 4. H (N) S 2 (N) R,.,, H (N) S 2 (N) R. z H, involution z z = z., modular symbol, H 1 (X 0 (N), R) R-linear involution., H 1 (X 0 (N), R) = H 1 (X 0 (N), R) H 1 (X 0 (N), R)., H ± 1 (X 0 (N), R), ±1. Remark 4.1. A R, H ± 1 (X 0 (N), A) = H ± 1 (X 0 (N), R) H 1 (X 0 (N), A)., H ± (N), H ± 1 (X 0 (N), Q) H(N).

11 modular form I. (CREMONA ) 11 f S 2 (N), f (z) = f(z ), S 2 (N) R-linear involution. S 2 (N) R, S 2 (N). (1) f(z) = a n q n f (z) = a n q n (q = e 2πiz ) (2) γ, f = γ, f for all f, γ., (2), f S 2 (N) R, γ, f R γ H 1 (X 0 (N), R), γ, f ir γ H 1 (X 0 (N), R). Remark 4.2., M-symbol. z z = z, H (N),., M-symbol (c : d) = ( c : d),. [C, 2.5]. 5. modular form a p0 (p 0 : ), L-, p, a p, modular form L-. rational newform f(z) = n 1 a nq n, L- L(f, s) = a n n=1 n s (R(s) > 2/3)., Euler : L(f, s) = p N (1 a pp s p 1 2s ) 1 p N (1 a pp s ) 1 i Mellin : L(f, s) = (2π) s Γ(s) 1 ( iz) s f(z) dz z. Mellin, L(f, s)., L- Γ(s) Λ(f, s) = N s/2 (2π) s Γ(s)L(f, s) = 0 0 f(iy/ N)y s 1 dy, s s 2,. Fricke involution W N, f W N = ϵ N f (ϵ N = ±1), W N z 1/Nz, f( 1/(Nz)) = ϵ N Nz 2 f(z)., z = iy/ N f(i/y N) = ϵ N y 2 f(iy/ N),, Λ(f, 2 s) = ϵ N Λ(f, s).,, ϵ N = 1, L(f, 1) = 0.

12 L-. rational newform f, f Ω(f) L(f, 1)/Ω(f), E f BSD., Mellin L- s = 1 L(f, 1) = 2πi i 0 f(z)dz = 0,, f,, f ( )., L(f, 1)/Ω(f) a p p N p, Hecke T p modular symbol 0,, (2 ), T p ( 0, ) = 0, p 1 p 1 k/p, = (1 p) 0, k/p, 0., k=0 (1 p T p ) 0, p 1 = 0, k/p k=0, T p f = a p f, f k=0 ( ) (1 p a p ) 0, p 1, f = 0, k/p, f, a p., Ω(f). Ω 0 (f), f., f E f E f (R), Ω(f) 2Ω 0 (f) E f (R) Ω(f) = Ω 0 (f) E f (R). 6.1., Ω(f) f ω 1, ω 2, p 1 k=0 0, k/p, f, Ω 0 (f), Ω 0 (f)., ( ) , n(p, f) (, ) L(f, 1) n(p, f) ( ) = Ω(f) 2(1 p a p ).,, a p a p < 2 p non-zero. ( ),. k=0

13 I. (CREMONA ) ( ). ( ). ). p 0, a p0 n(p 0, f), BSD. ). p 0, a p0 n(p 0, f), ( ) L-, p n(p, f) 2(1 p a p ) = n(p 0, f) 2(1 p a p0 ), n(p, f), a p.,, a p0 (p 0 : ), p, a p, modular form ( ),. ( ) p 0, a p0 n(p 0, f). L(f, 1) 0, ( ), n(p 0, f) 0, p a p = 1 p n(p, f)(1 p 0 a p0 ) n(p 0, f).,, a p., n(p, f), p 1 k=0 0, k/p, f Ω 0 (f), p 1 k=0 0, k/p H (N) (4 ). Example 5.1. (Example 2.3. ) modular X 0 (11) H 1 (X 0 (11), Q) H(11) A, B,, M-symbol A = (2 : 1), T 2 (A) = 2A,, a 2 = 2., A modular symbol 0, 1/2., A = A, H (11) A. (1 2 a 2 )L(f, 1) = 0, 1/2, f = A, f = Ω(f) (Example 6.1. Type 1 ). ( ) L(f, 1) Ω(f)., ( ). ).,, a 2, L(f, 1) 0., E f (Q) Mordell-Weil rank 0. = 1 5

14 14 ). ( ), a p. p 1 n(p, f) 0, k/p = A 2 k=0 n(p, f)., H (11) A? n(p, f), 2. p = 3 0, 1/3 0, 2/3 = 0, 1/3 0, 1/2 1/2, 2/3 M-symbol, A = (3 : 1) (2 : 1) (3 : 2) = (3) (2) (7) = B A ( B) = A, a 3 = n(3, f) = 1. 2 p = 5 0, 1/5 0, 2/5 0, 3/5 0, 4/5 = A, a 5 = n(5, f) = 1. 2 p = 7 0, 1/7 0, 2/7 0, 3/7 0, 4/7 0, 5/7 0, 6/7 = 2A, a 7 = n(7, f) = 2. 2, a 13 = 4., 3.2., n = 16 a n, a 1 = 1, a 2 = 2, a 3 = 1, a 4 = 2, a 5 = 1, a 6 = 2, a 7 = 2, a 8 = 0, a 9 = 2, a 10 = 2, a 11 = 1, a 12 = 2, a 13 = 4, a 14 = 4, a 15 = 1, a 16 = 4., Λ f E f. L(f, 1) = 0, E f N = 37,. ( ), n(p 0, f) = 0, a p.,. α = n/d Q (gcd(d, N) = 1) (1 p T p ) α, = α, pα p 1 α k α, p k=0

15 I. (CREMONA ) 15,, integral modular symbol (, H 1 (X 0 (N), Z) ) p 1 α k 0, pα 0, (p 1) 0, α p k=0. modular form f,, n(α, p, f) ( ) R α,, f Ω(f) = n(α, p, f) 2(1 p a p ). ( ), L(f, 1) 0,. ( ), L(f, 1) 0. L(f, 1) = 0,,., a p0 (p 0 : ). I. A i, f = Ω(f) H (N) M-symbol A i. II. p 0 1 k=0 0, k/p0 M-symbol Ai, n(p 0, f). III. p, ( ) a p = 1 p n(p, f)(1 p 0 a p0 ) n(p 0, f), a p., n(p, f) II.. 6. Λ f,, 2 Λ f ( C) γ 1, γ 2,..., γ 2g H 1 (X 0 (N), Z) Z,, H(N) Q,. (1). rational newform f, Hecke Fricke involution, ( ) v, v. (2). γ ± H ± (N) ( ), v γ = v γ = 1.

16 16 (3). R x, y,. x = γ, f, y = i γ, f. Type 1: v v (mod 2) ( ) = ω 1 = 2x, ω 2 = x yi. Type 2: v v (mod 2) ( ) = ω 1 = x, ω 2 = yi. Example 6.1. (Example 5.1. ) M-symbol A = (2 : 1), B = (3 : 1), modular X 0 (11) H 1 (X 0 (11), Q) H(11) A, B. M-symbol, (c) ( c), A = A, ( B = ) A B. 1 1, A B,, 0 1 v ± v = (2, 1), v = (0, 1)., Λ f Type x y., x y, direct method indirect method, Λ f direct method (L- ). γ, f = (v γ)x (v γ)yi, γ, v γ v γ non-zero, γ, f, x y. I f (α, β) = β α 2πif(z)dz, I f (α) = I f (α, ), I f (α, M(α)) = I f (α) I f (M(α)), α. f α, M(α) P f (M). P f (M),. P f (M), lemma. Lemma 6.2. f = a n e 2πinz (z = x iy H) 2 cusp form, z 0 = x 0 iy 0 H, a n I f (z 0 ) = 2πif(z)dz = n e2πinx 0 e 2πny 0. z 0 n=1

17 I. (CREMONA ) 17,. I f (z 0 ), e 2πny 0, a n,. Tingley lemma, e 2πny 0, y 0,., M Tingley ( ) α α = d i cn, M(α) = a i cn. ( ) a b,, M = Γ cn d 0 (N).. Proposition 6.3., P f (M) =I f ( d i cn = n=1 ) I f( a i cn ) a n n e 2πn/cN (e 2πina/cN e 2πind/cN ). x y (direct method) γ = α, M(α), v γ v γ non-zero, P f (M) = γ, f x = R(P f(m)), y = Im (P f(m)) v γ v γ, x y., P f (M) e ny indirect method (L- ). L(f, 1)/Ω(f),,, L(f, 1), Ω(f),., L(f, 1) = 0, 2 χ L(f χ, 1). L(f, 1) 0 rational newform f, L(f, 1) 0 Ω(f)., L-,, a n, L(f, 1).

18 18 rational newform f, ϵ N = ±1 Fricke involution W N.,, L(f, 1) L(f, 1) = i 0 2πif(z)dz =I f (, 0) =I f (, i/ N) I f (i/ N, 0) =I f (, i/ N) ϵ N I f (i/ N, ) =(ϵ N 1)I f (i/ N)., L(f, 1) 0, L(f, 1) = 2I f (i/ N)., lemma 6.2. Proposition 6.4. f = n=1 a ne 2πnz, f W N = f, L(f, 1) L(f, 1) = 2 n=1 a n N n e 2πn/. L(f, 1), L(f, 1) = a n /n,., Proposition, e 2π/ N,,. Remark 6.5., Fricke involution, I f (i/ N, 0) ϵ N I f (i/ N, ), e 2π/ N ( ). Example 6.6. (Example 6.1. ) modular X 0 (11) H 1 (X 0 (11), Q) H(11) A, B,, Example 5.1., A 0, 1/2, ( ) L(f, 1) Ω(f) = 1 5., Example 6.1., Λ f Type 1, (ω 1, ω 2 ) = (2x, x iy) ω 1 = Ω(f) = 5L(f, 1)

19 I. (CREMONA ) 19., Example 5.1. n = 16 a n Proposition 6.4., L(f, 1) L(f, 1) (0.15) 1 2 (0.15)2 1 3 (0.15)3 2 4 (0.15)4 1 5 (0.15)5 2 6 (0.15)6 2 7 (0.15)7 0 8 (0.15)8 2 9 (0.15) (0.15) (0.15) (0.15) (0.15) (0.15) (0.15) (0.15)16 = (, e 2π/ ),.,,, ω 1 = Ω(f) , L(f, 1) = 0,, 2 χ, L(f, 1) variation L(f χ, 1). l N, χ l 2., χ( ) = ( /l) ( ). (f χ) = χ(n)a n e 2πinz S 2 (Nl 2 ) n=1,, L- variation i L(f χ, s) = (2π) s Γ(s) 1 ( iz) s (f χ)(z) dz z. Proposition 6.4. variation,,. Proposition 6.7. χ( N) = ϵ N, L(f χ, 1) χ(n)a n L(f χ, 1) = 2 e 2πn/l N. n n=1, L(f χ, 1), L(f, 1)/Ω(f) = n(p, f)/2(1 p a p ) variation., γ l = l 1 k=0 χ(k) 0, k/l, f γ l, f P (l, f)., P (l, f) = χ( 1)l L(f χ, 1). -, (γ l ) = χ( 1)γ l, χ( 1) = ±1. 0

20 20 ). x (χ( 1) = 1 ), χ( 1) = 1, (γ l ) = γ l, γ l H (N)., P (l, f) Ω 0 (f),, m (l, f)x (m (l, f) )., m (l, f) non-zero, x, x = l ). y (χ( 1) = 1 ) L(f χ, 1) m (l, f) = P (l, f) m (l, f)., χ( 1) = 1, (γ l ) = γ l, γ l H (N)., m (l, f), P (l, f) = m (l, f)yi., m (l, f) non-zero, y, y = l L(f χ, 1) m (l, f) = P (l, f) im (l, f). Remark 6.8. f N perfect square, Murty-Murty, m (l, f) m (l, f) non-zero l., N perfect square,, 0., N = 49,, m (l, f) = 0, y ([C, Appedix, Example 4: N=49] )., N perfect square, direct method. Example 6.9. (Example 6.6. ) modular X 0 (11) H 1 (X 0 (11), Q) H(11) A, B,, Example 6.6., H (11) A, ω 1., y (i.e. ω 2 ), l 3 (mod 4) l, ). y. l, l = 3, γ 3 = 0, , = (3) ( 3) = A 2B 0 3, m (3, f) 0, ).. m (3, f), γ 3 H (11) = H(11)/H (11), H (11). H (11) A,, B 2 m (3, f),. γ 3 t ( 1, 2), v = (0, 1) m (3, f),, m (3, f) = 2., )., y = 1 3 2i P (3, f) = L(f 3, 1). 2

21 I. (CREMONA ) 21 Proposition 6.7. n = 16 a n, L(f 3, 1) L(f 3, 1) (0.53) (0.53) (0.53) (0.53) (0.53) (0.53) (0.53) (0.53) (0.53) (0.53) (0.53) (0.53) (0.53) (0.53) (0.53) (0.53)16 = (, e 2π/ ), y Λ f Type 1, ω 2 = x yi,. ω i x y (indirect method) L(f, 1)/Ω(f) 5, L(f, 1) 0, L(f, 1) a n,,, Ω(f)., l, L(f l, 1),. Λ f ( ) I. -, ±1 v ±., Λ f Type. Type 1 = Type 2 = ω 1 = 2x, ω 2 = x yi Λ f ω 1 = x, ω 2 = yi Λ f II. x y L- direct method, L- indirect method., e nk,,. 7. E f, 6 Λ f, E f = C/Λ f.

22 22 c 4 c 6 ( Z), ω 1 /ω 2 ω 2 /ω 1, τ.,, τ R(τ) 1/2 τ 1,, τ., q = e 2πiτ, c 4 (= 12g 2 ) c 6 (= 216g 3 ) c 4 = ( 2π ω 2 ) 4 ( n=1 n 3 q n 1 q n ) ( 2π ) 6, c 6 = ω 2 ( n=1 n 5 q n 1 q n, q < 0.005,. E f Q, c 4 c 6.,, Edixhoven,., c 4 c 6,,., c 4 c 6 N (1) c 3 4 c 2 6 = 1728., ( ) N, (2) 5 p N p c 4 p c 6 p 2 N, (3) c 6 9 (mod 27), (4) c 6 1 (mod 4), or c 4 0 (mod 16) c 6 0, 8 (mod 32) ).,. E f [a 1, a 2, a 3, a 4, a 6 ], E f : y 2 a 1 xy a 3 y = x 3 a 2 x 2 a 4 x a 6., c 4, c 6 b 2 = c 6 mod 12 5,..., 6 ; b 4 = (b 2 2 c 4 )/24; b 6 = ( b b 2 b 4 c 6 )/216; a 1 = b 2 mod 2 0, 1 ; a 3 = b 6 mod 2 0, 1 ; a 2 = (b 2 a 1 )/4; a 4 = (b 4 a 1 a 3 )/2; a 6 = (b 6 a 3 )/4, c 4 c 6 E f., 11 rational newform f E f.

23 I. (CREMONA ) 23 Example 7.1. Example 6.9. ) Λ f = ω 1, ω 2, ω ω i. c 4, c 6 c , c (n = 16 a n ), c 4 = 496, c 6 = , n = 100 a n c , c ,, 11 y 2 y = x 3 x 2 10x 20, E f. E f ( ) ω 1 ω 2 c 4 c 6. Edixhoven c 4 c 6,,,, N. 1. M-sybmol,. 2. a p0 (p 0 ). 3. L-, p a p. 4. L-,. 5. c 4 c 6 ( ),. References [C] Cremona, J.E.: Algorithms for modular elliptic curves. Second edition. Cambridge University Press, Cambridge, vi376 pp. Department of Mathematics, Hokkaido University, Sapporo , Japan address: morita@math.sci.hokudai.ac.jp

SAMA- SUKU-RU Contents p-adic families of Eisenstein series (modular form) Hecke Eisenstein Eisenstein p T

SAMA- SUKU-RU Contents p-adic families of Eisenstein series (modular form) Hecke Eisenstein Eisenstein p T SAMA- SUKU-RU Contents 1. 1 2. 7.1. p-adic families of Eisenstein series 3 2.1. modular form Hecke 3 2.2. Eisenstein 5 2.3. Eisenstein p 7 3. 7.2. The projection to the ordinary part 9 3.1. The ordinary