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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。
合同数問題という初等的な問題が保型形式によって解決する面白さについて、私の知っている限りでお話しました。

※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。

発表者:tsujimotter
http://tsujimotter.info

Published in: Education
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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 =)

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