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# 合同数問題と保型形式

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2018/09/23に開催された、梅崎さん主催の「数学について話す会 https://unaoya.github.io/event.html 」で発表した資料です。

※スライド１枚目でハッシュタグ「#数学について語る会」とありますが、正しくは「#数学について話す会」でした。

http://tsujimotter.info

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### 合同数問題と保型形式

1. 1. @tsujimotter 2018/09/23 # #{n = 2x2 + y2 + 32z2 } = 1 2 #{n = 2x2 + y2 + 8z2 } L(En, 1) = 4 p n a2 n
2. 2. 4 3 5 6
3. 3. 5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47, … 1, 2, 3, 4, 8, 9, 10, 11, 12, 16, 17, 18, 19, 25, 26, 27, 32, 33, 35, 36, 40, 42, 43, 44, …
4. 4. =) n n n = 2x2 + y2 + 8z2 n = 2x2 + y2 + 32z2 (x, y, z) (x, y, z) Thm. Tunnel (1983) ( )BSD (=
5. 5. n X, Y, Z 8 < : X2 + Y 2 = Z2 XY 2 = n n ()
6. 6. X2 + Y 2 = Z2 (X Y )2 = Z2 4n (X + Y )2 = Z2 + 4n (X2 Y 2 )2 = Z4 42 n2 ✓ (X2 Y 2 )Z 8 ◆2 = ✓ Z 2 ◆6 n2 ✓ Z 2 ◆2 ⇥Z2 /26 x := ✓ Z 2 ◆2 y := ✓ (X2 Y 2 )Z 8 ◆2 ±2XY = ±4n y2 = x3 n2 x XY = 2n
7. 7. •  n 　 •  En En : y2 = x3 n2 x () En(Q)
8. 8. BSD E/Q : elliptic curve rank E(Q) > 0 () ords=1 L(E, s) > 0 L(E, s) = 0 Thm. 1977 E/Q rank E(Q) > 0 =) ords=1 L(E, s) > 0
9. 9. n () rank En(Q) > 0 BSD =) ()() L(En, 1) = 0
10. 10. 1.  2.  3.  Waldspurger
11. 11. 1. L(f, s) = 1X n=1 bnn s Mellin Mellin つ Mellin L(En, s) gEn 2 32n2 f(z) = 1X n=1 bnqn
12. 12. 2. k/2 N k–1 N’ Sk/2(˜0(N), ) Shimura ! Mk 1(N0 , 2 ) N’ = N/2 Niwa 1975] 2 32 gE1 = 1X m=1 bmqm 7 ! 3/2 128 f = 1X m=1 amqm
13. 13. 3. Waldspurger ( , 1983) n f = X amqm 2 S3/2(˜0(128)) Shimura(f) = g = X bmqm L(En, 1) = 4 p n a2 n L(En, 1) = 0 an = 0 (Waldspurger, 1980, 1981) k–1 L k/2 q- 2
14. 14. f q = e2⇡iz Rem. 1/2 f(z) = (⇥(z) ⇥(4z))(⇥(32z) 1 2 ⇥(8z))⇥(2z) 2 S3/2(˜0(128) (8z))⇥(2z) 2 S3/2(˜0(128)) ⇥(z) := X n2Z qn2
15. 15. f(z) = (⇥(z) ⇥(4z))(⇥(32z) 1 2 ⇥(8z))⇥(2z) = ⇥(z)⇥(32z)⇥(2z) 1 2 ⇥(z)⇥(8z)⇥(2z) ⇥(4z)⇥(32z)⇥(2z) + 1 2 ⇥(4z)⇥(8z)⇥(2z) = X x,y,z q2x2 +y2 +32z2 1 2 X x,y,z q2x2 +y2 +8z2 X x,y,z q2x2 +4y2 +32z2 + 1 2 X x,y,z q2x2 +4y2 +8z2
16. 16. an = #{n = 2x2 + y2 + 32z2 } 1 2 #{n = 2x2 + y2 + 8z2 } #{n = 2x2 + 4y2 + 32z2 } + 1 2 #{n = 2x2 + 4y2 + 8z2 } n : 0 () #{n = 2x2 + y2 + 32z2 } = 1 2 #{n = 2x2 + y2 + 8z2 } an = 0
17. 17. =) n n n = 2x2 + y2 + 8z2 n = 2x2 + y2 + 32z2 (x, y, z) (x, y, z) Thm. Tunnel (1983) ( )BSD (=
18. 18. N. (2006) –  1 –  3 3.3 –  4 4.4
19. 19. 7
20. 20. 7 ✓ 35 12 ◆2 + ✓ 24 5 ◆2 = 30625 + 82944 3600 = ✓ 337 60 ◆2 1 2 · 35 12 · 24 5 = 7
21. 21. 157
22. 22. =) n n (x, y, z) (x, y, z) Thm. Tunnel (1983) ( )BSD n = 8x2 + 2y2 + 64z2 n = 8x2 + 2y2 + 16z2 =)