߹ಉࣜͱزԿֶ ക࡚௚໵ 2021 ೥ 4 ݄ 24 ೔ˏຑ෍ֶԂڭཆ૯߹ 1/36

ࣗݾ঺հ • ക࡚௚໵ • 2017 ೥ 3 ݄౦ژେֶ਺ཧՊֶݚڀՊमྃɺത࢜ʢ਺ཧՊֶʣ ɺ ઐ໳͸਺࿦زԿ • 2017 ೥ 4 ݄͔Β 2020 ೥ 3 ݄·Ͱຑ෍Ͱඇৗۈߨࢣ • ݱࡏ͸גࣜձࣾ͢͏͕͘ͿΜ͔Ͱେਓ޲͚ʹ਺ֶ΍౷ܭֶΛ ڭ͑Δ • झຯͰ਺ֶΛֶͿ 2/36

ࠓ೔ͷ࿩ 1. ਺࿦زԿͷߟ͑ํ 2. ߹ಉࣜʢزԿֶʣ 3. อܕܗࣜʢදݱ࿦ɺௐ࿨ղੳʣ ϓϩάϥϜͰܭࢉ͠ͳ͕Βݱ৅Λ؍࡯͍ͨ͠ɻίʔυ͸ͪ͜Βɻ 3/36

਺࿦زԿͷߟ͑ํ

਺࿦زԿ • ਺࿦ɿ੔਺ͷੑ࣭Λௐ΂Δɻ • زԿɿਤܗͷੑ࣭Λௐ΂ΔɻதֶߴߍͰ͸ϢʔΫϦουۭؒ ಺ͷਤܗΛௐ΂ͨɻݱ୅తͳزԿֶͰ͸ΑΓҰൠͷۭؒ΍ͦ ͷதͷਤܗΛௐ΂Δɻ • ਺࿦زԿɿ੔਺ͷੑ࣭ΛزԿΛ࢖ͬͯௐ΂Δɻ·ͨ͸ͦͷٯɻ 4/36

ૉ਺ ૉ਺Λख͕͔Γʹͯ͠੔਺ͷੑ࣭Λௐ΂Δɻ ྫ 24 ͷ໿਺ͷݸ਺ͱ૯࿨ΛٻΊΔɻ 24 ΛૉҼ਺෼ղ͢Δͱ 24 = 23 × 31 ͳͷͰɺ ໿਺ͷݸ਺͸ 4 × 2 = 8 Ͱ૯࿨͸ (1 + 2 + 4 + 8)(1 + 3) = 60 ͱΘ͔Δɻ ૉҼ਺෼ղͷҰҙੑ͕伴ɻ 5/36

ۭؒͱؔ਺ ۭؒ ؔ਺ ԁ x2 + y2 = 1 ճస पظؔ਺ f (θ) ࡾ֯ؔ਺ cos θ, sin θ Ճ๏ఆཧ ੔਺શମ ֻ͚ࢉ ਺ྻ anʢ਺࿦తؔ਺ʣ ৐๏తؔ਺ ৐๏తؔ਺ n, m ͕ޓ͍ʹૉͳͱ͖ anm = anam ͱͳΔɻૉ਺ p Ͱͷ஋ ap ͕ ॏཁɻ 6/36

θʔλؔ਺ ਺ྻ an ͷ฼ؔ਺ L(s) ͔Β an ΍੔਺ʹ͍ͭͯͷ৘ใΛҾ͖ग़͢ɻ σΟϦΫϨڃ਺ L(s) = ∞ ∑ n=1 an ns = a1 1s + a2 2s + a3 3s + · · · શͯͷ n ʹ͍ͭͯ an = 1 ͱ͢ΔͱɺϦʔϚϯͷθʔλؔ਺ ζ(s) = ∞ ∑ n=1 1 ns = 1 1s + 1 2s + 1 3s + · · · 7/36

ΦΠϥʔੵ ૉҼ਺෼ղͷҰҙੑΛ࢖ͬͯ ζ(s) ΛҼ਺෼ղ͢Δɻ ζ(s) = 1 1s + 1 2s + 1 3s + 1 4s + 1 5s + 1 6s + 1 7s + 1 8s + 1 9s + 1 10s + · · · = 1 1s + 1 2s + 1 3s + ( 1 2s )2 + 1 5s + 1 2s 1 3s + 1 7s + ( 1 2s )3 + ( 1 3s ) + 1 2s 1 5s + 1 11s + ( 1 2s )2 1 3s + · · · = (1 + 1 3s + 1 5s + · · · ) + 1 2s (1 + 1 3s + 1 5s + · · · ) + ( 1 2s )2(1 + 1 3s + 1 5s + · · · ) + ( 1 2s )3(1 + 1 3s + 1 5s + · · · ) + · · · = (1 + 1 2s + ( 1 2s )2 + ( 1 2s )3 + · · · )(1 + 1 3s + 1 5s + · · · ) 8/36

ΦΠϥʔੵ = (1 + 1 2s + ( 1 2s )2 + · · · ) × (1 + 1 3s + ( 1 3s )2 + · · · ) × (1 + 1 5s + ( 1 5s )2 + · · · ) × · · · = 1 1 − (1/2s) × 1 1 − (1/3s) × 1 1 − (1/5s) × · · · ͜ΕΛ༻͍ͯૉ਺͕ແݶʹ͋Δ͜ͱΛࣔ͢͜ͱ͕Ͱ͖Δɻ ΦΠϥʔੵදࣔ ζ(s) = 1 1s + 1 2s + 1 3s + · · · = ∏ p 1 1 − (1/ps) 9/36

σΟϦΫϨͷࢉज़ڃ਺ఆཧ ࢉज़ڃ਺ఆཧ ޓ͍ʹૉͳ੔਺ n, k ʹ͍ͭͯ n Ͱׂͬͯ k ༨ Δૉ਺͸ແݶʹ͋Δɻ n = 4 ͷ৔߹ a1 = 1, a2 = 0, a3 = −1, a4 = 0, a5 = 1, a6 = 0, . . . ͱ͍͏਺ྻ an =        1 n ≡ 1 mod 4 −1 n ≡ 3 mod 4 0 n ≡ 0, 2 mod 4 ৐๏తؔ਺ɺͭ·Γ anm = anam ͱͳΔɻ 10/36

σΟϦΫϨͷࢉज़ڃ਺ఆཧ ͜ͷ an ͷσΟϦΫϨڃ਺΋ΦΠϥʔੵදࣔΛ࣋ͭɻ L(s) = ∞ ∑ n=1 an ns = 1 1s − 1 3s + 1 5s − 1 7s + · · · = 1 1 + (1/3s) × 1 1 − (1/5s) × 1 1 + (1/7s) × · · · = ∏ p≡1 mod 4 1 1 − (1/ps) × ∏ p≡3 mod 4 1 1 + (1/ps) ͜ͷؔ਺ͷੑ࣭͔Βɺ4 Ͱׂͬͯ 1 ༨Δૉ਺΋ 4 Ͱׂͬͯ 3 ༨Δ ૉ਺΋ͲͪΒ΋ແݶʹ͋Δ͜ͱΛࣔ͢͜ͱ͕Ͱ͖Δɻ 11/36

਺ྻͷ࡞Γํ ਺ྻ an ͷ࡞Γํͱͯ͠ɺࠓ೔͸ೋͭͷํ๏Λ঺հ͢Δɻ 1. ߹ಉࣜΛ༻͍Δํ๏ʢزԿֶʣ 2. อܕܗࣜΛ༻͍Δํ๏ʢௐ࿨ղੳɺදݱ࿦ʣ 12/36

߹ಉࣜ

x2 + 1 = 0 ͷղͷݸ਺ ໰୊ x2 + 1 = 0 mod p ͷ 0 ͔Β p − 1 ·Ͱͷ੔਺ղͷݸ਺ ap Λٻ ΊΑɻ ࣮ݧ͢Δɻίʔυ͸ͪ͜Βɻ ap = 0 p = 3, 7, 11, 19, 23, 31, 43, . . . ap = 1 p = 2 ap = 2 p = 5, 13, 17, 29 37, 41, 53, . . . p Λ 4 Ͱׂͬͨ༨Γ p mod 4 ʹ஫໨ɻx2 + 1 = 0 ͱ 4 ͷؔ܎ɻ 1. ൑ผࣜ D = b2 − 4ac = −4 Ͱ͋Δɻ 2. i4 = 1 Ͱ͋Δɻ ap − 1 ͸ઌ΄Ͳͷ 1, 0, −1, 0, 1, 0, −1, 0, . . . ͱ͍͏਺ྻͱಉ͡ɻ 13/36

x2 − 2 = 0 ͷղͷݸ਺ ໰୊ x2 − 2 = 0 mod p ͷ 0 ͔Β p − 1 ·Ͱͷ੔਺ղͷݸ਺ ap Λٻ ΊΑɻ ࣮ݧ͢Δɻίʔυ͸ͪ͜Βɻ ap = 0 p = 3, 5, 11, 13, 19, 29, 37, . . . ap = 1 p = 2 ap = 2 p = 7, 17, 23, 31 41, 47, 71 . . . p Λ 8 Ͱׂͬͨ༨Γ p mod 8 ʹ஫໨ɻx2 − 2 = 0 ͱ 8 ͷؔ܎ɻ 1. ൑ผࣜ D = b2 − 4ac = 8 Ͱ͋Δɻ 2. ( 1 √ 2 + 1 √ 2 i)8 = (cos 45◦ + i sin 45◦)8 = 1 Ͱ͋Δɻ 14/36

ೋ࣍ํఔࣜͷղͷݸ਺ ೋ࣍ํఔࣜ͸ฏํ׬੒Ͱ x2 − a = 0 ͷܗʹͰ͖Δɻ mod p Ͱͷ ղͷݸ਺ ap ͸ 0, 1, 2 ͷ͍ͣΕ͔ʹͳΔɻ͋Δ N ͕ଘࡏͯ͠ɺ ap = 0, 1, 2 ͷ͍ͣΕʹͳΔ͔ p mod N ͰΘ͔Δɻ 1. ൑ผࣜ D = b2 − 4ac ͱ N ͷؔ܎ɻ 2. N ৐͢Δͱ 1 ʹͳΔෳૉ਺ͱ √ a ͷؔ܎ɻ 15/36

ϑΣϧϚʔͷখఆཧ ϑΣϧϚʔͷখఆཧ p Λૉ਺ͱ͠ɺa Λ p ͱޓ͍ʹૉͳ੔਺ͱ͢ Δͱ ap−1 ≡ 1 mod p ͭ·Γɺap−1 Λ p Ͱׂͬͨ͋·Γ͸ 1 ʹͳΔɻ 16/36

ݪ࢝ࠜఆཧ ݪ࢝ࠜఆཧ ૉ਺ p ʹର͠ɺ࣍ͷ৚݅Λຬͨ͢ 1 Ҏ্ p ະຬͷ͋Δ੔਺ a ͕ ଘࡏ͢Δɻa1, a2, . . . , ap−1 mod p ͕ 1 ͔Β p − 1 ͷશମͱҰக ͢Δɻ • mod 5 Ͱ 21 = 2, 22 = 4, 23 = 3, 24 = 1 Ͱ͋Δɻ • mod 7 Ͱ 31 = 3, 32 = 2, 33 = 6, 34 = 4, 35 = 5, 36 = 1 Ͱ ͋Δɻ • mod 11 Ͱ 21 = 2, 22 = 4, 23 = 8, 24 = 5, 25 = 10, 26 = 9, 27 = 7, 28 = 3, 29 = 6, 210 = 1 Ͱ͋Δɻ 17/36

ࡾ࣍ํఔࣜͷղͷݸ਺ ࡾ࣍ํఔࣜͰ΋ಉ༷ͳݱ৅͕ى͜Δ৔߹͕͋Δɻ mod p Ͱͷղ ͷݸ਺ ap = 0, 1, 2, 3 ͷ͍ͣΕ͔ɻ ྫ x3 + x2 − 2x − 1 = 0 x3 − 3x + 1 = 0 x3 + x2 − 4x + 1 = 0 x3 + x2 − 6x − 7 = 0 18/36

x3 + x2 − 2x − 1 = 0 ͷղͷݸ਺ ໰୊ x3 + x2 − 2x − 1 = 0 mod p ͷ 0 ͔Β p − 1 ·Ͱͷ੔਺ղͷݸ਺ ap ΛٻΊΑɻ ࣮ݧ͢Δɻίʔυ͸ͪ͜Βɻ ap = 0 p = 2, 3, 5, 11, 17, 19, 23, . . . ap = 1 p = 7ʢ3 ॏղʣ ap = 3 p = 13, 29, 41, 43 71, 83, 97 . . . p Λ 7 Ͱׂͬͨ༨Γ p mod 7 ʹ஫໨ɻx3 + x2 − 2x − 1 = 0 ͱ 7 ͷؔ܎ɻt6 + t5 + t4 + t3 + t2 + t + 1 = 0 Ͱ྆ลΛ t3 Ͱׂ͔ͬͯ Β x = t + 1 t ͱஔ͍ͯΈΔɻ 19/36

ΑΓ࣍਺ͷߴ͍৔߹ ΑΓ࣍਺ͷߴ͍৔߹͸Ͳ͏͔ʁ Ωʔϫʔυ • ԁ෼ମ • ฏํ৒༨ͷ૬ޓ๏ଇ • ྨମ࿦ 20/36

x3 − 2 = 0 ͷղͷݸ਺ ࡾ࣍Ҏ্ͷํఔࣜͰ͸ ap ͕ p mod N Ͱܾ·Δέʔε͸࣮͸كɻ ໰୊ x3 − 2 = 0 mod p ͷ 0 ͔Β p − 1 ·Ͱͷ੔਺ղͷݸ਺ ap Λٻ ΊΑɻ ࣮ݧ͢Δɻίʔυ͸ͪ͜Βɻ ap = 0 p = 7, 13, 19, 37, 61, 67, 73, . . . ap = 1 p = 2, 3, 5, 11, 17, 23, 29, . . . ap = 3 p = 31, 43, 109, 127, 157, . . . ap = 1 ͕ͨ͘͞Μ͋Δͷ͕͖ͬ͞ͱҧ͏ɻp = 3 ·ͨ͸ p ≡ 2 mod 3 ͱ ap = 1 ͕ඞཁे෼ɻp ≡ 1 mod 3 ͷͱ͖͸ ap = 0, 3 ͷ ͍ͣΕ͔ɻ 21/36

൑ผࣜ ࡾ࣍ํఔࣜͷ൑ผࣜ D = b2c2 − 4ac3 − 4b3d − 27a2d2 + 18abcd x3 + x2 − 2x − 1 = 0 D = 49 x3 − 3x + 1 = 0 D = 81 x3 + x2 − 4x + 1 = 0 D = 169 x3 + x2 − 6x − 7 = 0 D = 361 x3 − 2 = 0 D = −108 22/36

อܕܗࣜ

Τʔλؔ਺ σσΩϯυͷΤʔλؔ਺ η(q) = q1/24 ∞ ∏ n=1 (1 − qn) = q1/24(1 − q)(1 − q2)(1 − q3) · · · ͜ΕΛల։ͯ͠ΈΔɻ ஫ҙ ௨ৗ͸ q = e2πiτ ͱͯ͠ η(τ) ͱ͔͔ΕΔɻ 23/36

ೋ߲ఆཧ ೋ߲ఆཧ (1 + q)n = (1 + q)(1 + q) · · · (1 + q) = nC0q0 + nC1q1 + · · · + nCnqn qk ͷ܎਺͸ n ݸ͔Β k ݸબͿ૊Έ߹Θͤͷ਺ɻ n ݸͷ 1 ͔Β k ݸબΜͰ࿨͕ k ͱͳΔ૊Έ߹Θͤͷ਺ͱߟ͑Δ͜ ͱ͕Ͱ͖Δɻ 24/36

Τʔλؔ਺ͷల։ ∞ ∏ n=1 (1 − qn) = (1 − q)(1 − q2)(1 − q3)(1 − q4) · · · Λల։͢Δɻ (1 − q)(1 − q2) = 1 − q − q2 + q3 (1 − q)(1 − q2)(1 − q3) = 1 − q − q2 + q4 + q5 − q6 (1 − q)(1 − q2)(1 − q3)(1 − q4) = 1 − q − q2 + 2q5 − q8 − q9 + q10 qn ͷ܎਺ an ͸ n Λ 1, 2, 3, 4, . . . Λ 1 ճ·Ͱ࢖ͬͯ࿨Ͱද͢૊Έ߹ ΘͤͰܭࢉͰ͖Δɻ߲ͨͩ͠ͷ਺ͷۮحͰ ±1 ഒ͢Δɻ 25/36

Τʔλؔ਺ͷల։ 3 = 2 + 1 a3 = −1 + 1 4 = 3 + 1 a4 = −1 + 1 5 = 4 + 1 = 3 + 2 a5 = −1 + 1 + 1 6 = 5 + 1 = 4 + 2 = 3 + 2 + 1 a6 = −1 + 1 + 1 − 1 7 = 6 + 1 = 5 + 2 = 4 + 3 = 4 + 2 + 1 a7 = −1 + 1 + 1 + 1 − 1 ∞ ∏ n=1 (1 − qn) = (1 − q)(1 − q2)(1 − q3)(1 − q4) · · · = 1 − q − q2 + q5 + q7 − q12 − q15 + q22 + q26 − q35 − q40 + q51 + q57 − q70 − q77 + q92 + . . . 26/36

ޒ֯਺ఆཧ ޒ֯਺ Pk = k(3k − 1) 2 P1 = 1, P2 = 5, P3 = 12, P4 = 22, . . . , P0 = 0, P−1 = 2, P−2 = 7, P−3 = 15, . . . Ͱ͋Δɻ 1 − q − q2 + q5 + q7 − q12 − q15 + q22 + q26 − q35 − q40 + q51 + q57 − q70 − q77 + q92 + . . . ͷ܎਺͕ 0 Ͱͳ͍ͷ͸ n ͕ޒ֯਺ͷͱ͖ɻ 27/36

Τʔλੵ ໰୊ η(q6)η(q18) = q ∞ ∏ n=1 (1 − q6n)(1 − q18n) = q(1 − q6)(1 − q12)(1 − q18)(1 − q18)(1 − q24)(1 − q30) · · · Λల։ͯ͠ qn ͷ܎਺ an Λௐ΂Δɻ ap ͸ p − 1 Λ 6, 12, 18, 24, 30, 36, . . . ͱ 18, 36, 54, . . . Λ 1 ճ·Ͱ ࢖ͬͯͷ࿨Ͱද͢૊Έ߹ΘͤͰܭࢉͰ͖Δɻ߲ͨͩ͠ͷ਺ͷۮح Ͱ ±1 ഒ͢Δɻ a7 = −1 a13 = −1 28/36

Τʔλੵͷల։ 18 = 18 = 12 + 6 a19 = −1 − 1 + 1 = −1 24 = 18 + 6 = 18 + 6 a25 = −1 + 1 + 1 30 = 24 + 6 = 18 + 12 = 18 + 12 a31 = −1 + 1 + 1 + 1 = 2 36 = 36 = 30 + 6 = 24 + 12 = 18 + 18 = 18 + 12 + 6 = 18 + 12 + 6 a37 = −1 − 1 + 1 + 1 + 1 − 1 − 1 = −1 42 = 42 = 36 + 6 = 36 + 6 = 30 + 12 = 24 + 18 = 24 + 18 = 24 + 12 + 6 = 18 + 18 + 6 a43 = −1 + 1 + 1 + 1 + 1 + 1 − 1 − 1 = 2 29/36

ೋͭͷ਺ྻͷରԠ η(q6)η(q18) Λల։͢Δͱ q − q7 − q13 − q19 + q25 + 2q31 − q37 + 2q43 − q61 − q67 − q73 − q79 + q91 − q97 − q103 + 2q109 + q121 + 2q127 + q133 − q139 − q151 + 2q157 − q163 − q175 − q181 − q193 − q199 − q211 − 2q217 + 2q223 + · · · ͱͳΔɻqn ͷ܎਺Λ an ͱ͢Δɻ x3 − 2 = 0 mod p ͷ 0 ͔Β p − 1 ·Ͱͷ੔਺ղͷݸ਺ bp ͱ͢Δɻ ap = 0 p = 7, 13, 19, 37, 61, 67, 73, . . . ap = 1 p = 2, 3, 5, 11, 17, 23, 29, . . . ap = 3 p = 31, 43, 109, 127, 157, . . . bp − 1 = ap 30/36

ΦΠϥʔੵ η(q6)η(q18) ʹରԠ͢ΔσΟϦΫϨڃ਺ 1 − 1 7s − 1 13s − 1 19s + 1 25s + 2 1 31s − 1 37s + 2 1 43s − 1 61s − 1 67s − · · · ͷΦΠϥʔੵදࣔ 1 × 1 × 1 1 − 5−2s × 1 1 + 7−s + 7−2s × 1 1 − 11−2s × · · · લͷ΋ͷͱҧͬͯ 2 ࣍ࣜʹͳ͍ͬͯΔɻ ϥϚψδϟϯͷ τ ؔ਺ ϥϚψδϟϯ͸ η(q)24 ͷ܎਺ʹ͍ͭͯௐ΂ɺ ͦΕʹରԠ͢ΔσΟϦΫϨڃ਺ͷΦΠϥʔੵΛ ಋ͍ͨɻ 31/36

Τʔλੵͱղͷݸ਺ͷରԠྫ Τʔλੵͷܗ͸ඇৗʹ௝͍͠ɻ • η(q)η(q23) ͱ x3 − x − 1 = 0 • η(q2)η(q22) ͱ x3 − x2 − x − 1 = 0 • η(q8)η(q16) ͱ x4 − 2x2 + 2 = 0 • η(q4)3η(q44)3 η(q2)η(q8)η(q22)η(q88) ͱ x3 − 4x + 4 = 0 ඞͣ͠΋௚઀Ұக͢Δͱ͍͏ରԠͰ͸ͳ͘ଟগͷमਖ਼͕ඞཁͱͳ Δ͜ͱʹ஫ҙɻίʔυ͸ͪ͜Βɻ 32/36

ପԁۂઢͱอܕܗࣜ 2 ม਺Ҏ্ͷଟ߲ࣜʹ͍ͭͯ΋ಉ༷ʹ ap Λߟ͑Δ͜ͱ͕Ͱ͖Δɻ ྫ ૉ਺ p ʹର͠ɺy2 + y = x3 − x2 ͷ mod p Ͱͷղͷݸ਺ ap ͱ ͨ͠ͱ͖ɺp − ap ͸ η(q)2η(q11)2 = q ∞ ∏ n=1 (1 − qn)2(1 − q11n)2 ͷ qp ͷ܎਺Ͱ͋Δɻ ίʔυ͸ͪ͜Βɻ 33/36

อܕܗࣜ ࢤଜ୩ࢁ༧૝ ΑΓҰൠʹପԁۂઢʹର্ͯ͠ͷҙຯͰରԠ͢Δอܕܗ͕ࣜଘ ࡏ͢Δɻ ͜ͷ༧૝ͷղܾ͕ϑΣϧϚʔͷ࠷ऴఆཧΛղܾ͢Δ伴ͱͳͬͨɻ ΑΓ޿͍࿮૊Έͱͯ͠ϥϯάϥϯζରԠͱ͍͏΋ͷ͕͋Δɻ͜Ε ͸ݱࡏ΋ݚڀ͕ਐΜͰ͍Δ໰୊ɻ อܕܗࣜ ඇৗʹߴ͍ରশੑΛ࣋ͭؔ਺Ͱؔ܎͕ࣜ๛෋ɻΤʔλੵ͸อܕ ܗࣜͷ͏ͪͰಛผͳ͘͝Ұ෦ɻ ࡾ֯ؔ਺͸ sin(θ + 2π) = sin θ, cos(θ + 2π) = cos θ ͱ͍͏ରশੑ ͕͋Γɺsin2 θ + cos2 θ = 1, sin(θ + π 2 ) = cos θ, (sin θ)′ = cos θ ͳ Ͳͷؔ܎͕ࣜ͋Δɻ͜Εͷߴ࣍ݩ൛ͷΑ͏ͳ΋ͷɻ 34/36

ۭؒͱؔ਺ ۭؒ ؔ਺ ԁ x2 + y2 = 1 ࡾ֯ؔ਺ cos θ, sin θ ੔਺શମ ৐๏తؔ਺ an Ϟδϡϥʔۂઢ อܕܗࣜ 35/36

ࢀߟจݙ ࡾࢬ༸ҰઌੜʹΑΔ༗ݶମ্ͷํఔࣜΛ௨ͯ͠ݟΔݱ୅੔਺࿦Ͱ ͸ɺಉ͡಺༰͕ߴߍੜ޲͚ʹΑΓৄ͘͠ղઆ͞Ε͍ͯ·͢ɻଞ ʹ͸ • ҏ౻఩࢙ɺฏํ਺ͷ࿨Ͱද͞ΕΔૉ਺ʹ͍ͭͯ • A. Weil, θʔλവ਺ͷҭ੒ʹ͍ͭͯ • T. Hiramatu, S. Saito, An Introduction to Non-abelian Class Field Theory • Sympy, https://www.sympy.org/en/index.html • LMFDB - The L-functions and Modular Forms Database, https://www.lmfdb.org/ 36/36