整数論と様々な数学

MATHPOWER2018での講演。フィールズ賞受賞者Akshay Venkateshの業績紹介。

Transcript

1. ੔਺࿦ͱ༷ʑͳ਺ֶ Akshay Venkateshͷۀ੷঺հ ക࡚௚໵@unaoya ͢͏͕͘ͿΜ͔ MATHPOWER2018 10/6
2. ड৆ཧ༝ ੔਺࿦ͷ༷ʑͳ໰୊Λ ▶ ྗֶܥ ▶ τϙϩδʔ ▶ දݱ࿦ ΛԠ༻ͯ͠ղܾɻ
3. ೋ࣍ܗࣜ ϥάϥϯδϡͷ࢛ฏํ࿨ఆཧ x2 + y2 + z2 + w2 Ͱશͯͷ੔਺Λද͢ɻ 10 = 12 + 32 15 = 32 + 22 + 12 + 12
4. ೋ࣍ܗࣜ ೋ࣍ܗࣜͷม׵ P(x1 , x2 , x3 ) = x2 1 + x2 2 + x2 3 Q(y1 , y2 ) = 2y2 1 + 2y1 y2 + 2y2 2 Λߟ͑Δɻ x1 = y1 + y2 , x2 = y1 , x3 = y2 ͱ͢Δɻ
5. ೋ࣍ܗࣜ P(x1 , x2 , x3 ) = x2 1 + x2 2 + x2 3 Q(y1 , y2 ) = 2y2 1 + 2y1 y2 + 2y2 2 P(x1 , x2 , x3 ) = (y1 + y2 )2 + y2 1 + y2 2 = 2y2 1 + 2y1 y2 + 2y2 2
6. ೋ࣍ܗࣜ ͋Δೋ࣍ܗࣜQ ͕ɺଞͷೋ࣍ܗࣜP ͔Βม਺ ม׵ͰදݱͰ͖Δ͔ʁmม਺ͷP ͕nม਺ͷ Q Λදݱ͢Δ͔ʁ ہॴେҬݪཧʢϋοηݪཧʣ p ਐ਺Qp ͷൣғͱ࣮਺RͷൣғͰߟ͑Δɻ શͯͷp ٴͼRͰදݱͰ͖Ε͹༗ཧ਺ͷൣғ Ͱ΋දݱͰ͖Δ͔ʁ
7. ೋ࣍ܗࣜ ΤϨϯόʔά-ϰΣϯΧςγϡ Q ͕nม਺ͷ࣌ɺશͯͷہॴతʹදݱՄೳͳ n − 7ม਺ҎԼͷೋ࣍ܗࣜQ′ Λදݱ͢Δɻ ূ໌ʹΤϧΰʔυཧ࿦ɺྗֶܥΛ࢖͏
8. ϦχοΫ༧૝ ੪࣍ଟ߲ࣜQ ʹର͠ɺQ(x) = d ͳΔx ͷू ߹ɻd ͰׂͬͯɺQ(x) = 1Ͱͷd → ∞Ͱͷ ෼෍ͷ༷ࢠɻ Q(x) = x2 1 + x2 2 + · · · + x2 n Λߟ͑Δͱɺٿ໘্ ͷ༗ཧ఺ͷ෼෍ɻ ܈ͷ࡞༻͕͋Δ৔߹Λߟ͑Δɻௐ࿨ղੳͱΤ ϧΰʔυཧ࿦Λ࢖͏ɻ
9. ΠσΞϧྨ܈ͷ෼෍ ΠσΞϧྨ܈ͱ͸ʁ੔਺ͰͷૉҼ਺෼ղͷҰ ҙੑ 6 = 2 × 3 10 = 2 × 5 √ −5Λ෇͚Ճ͑Δͱ่ΕΔ 6 = 2 × 3 = (1 + √ −5)(1 − √ −5)
10. ΠσΞϧྨ܈ͷ෼෍ ͜Εͷ่Ε۩߹ΛଌΔͷ͕ΠσΞϧྨ܈ɻ༗ ݶΞʔϕϧ܈ʹͳΔɻ ▶ Qͷ৔߹ɺΠσΞϧྨ܈͸1 ▶ Q( √ −5)ͷ৔߹ɺΠσΞϧྨ܈͸{±1}
11. ΠσΞϧྨ܈ͷ෼෍ ৭ʑͳ୅਺ମQ(a)Λಈ͔ͨ͠ͱ͖ɺΠσΞ ϧྨ܈ʹ͸ͲͷΑ͏ͳ܈͕ݱΕΔ͔ʁ ίʔΤϯɺϨϯετϥͷΠσΞϧྨ܈ͷ෼෍ ʹ͍ͭͯͷ؍࡯ͱ༧૝ɻ
12. ΠσΞϧྨ܈ͷ෼෍ ΤϨϯόʔά-ϰΣϯΧςγϡ-΢Σε λʔϥϯυ ίʔΤϯɺϨϯετϥ༧૝ͷؔ਺ମྨࣅΛূ ໌ͨ͠ɻ ؔ਺ମFp (x, a)͸༗ݶମ্ͷ୅਺ۂઢͷ༗ཧ ؔ਺શͯूΊͨ΋ͷɻ͜Ε΋ಉ༷ʹΠσΞϧ ྨ܈ΛఆٛͰ͖Δɻ ϑϧϏοπۭؒͷϗϞϩδʔ҆ఆੑΛ࢖ͬͯ
13. ہॴରশۭؒ ϥϯάϥϯζରԠʹؔ܎ɻ ςΠϥʔɺϫΠϧζͷΨϩΞදݱͷߏ੒Λࢤ ଜଟ༷ମ͕࢖͑ͳ͍έʔεʹ΋ݚڀɻ ہॴରশۭؒͷίϗϞϩδʔΛදݱ࿦ɺτϙ ϩδʔʹΑΓௐ΂Δɻ