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整数論と様々な数学
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Naoya Umezaki
October 06, 2018
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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。
Naoya Umezaki
October 06, 2018
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Transcript
ͱ༷ʑͳֶ Akshay Venkateshͷۀհ ക࡚@unaoya ͢͏͕͘ͿΜ͔ MATHPOWER2018 10/6
डཧ༝ ͷ༷ʑͳΛ ▶ ྗֶܥ ▶ τϙϩδʔ ▶ දݱ ΛԠ༻ͯ͠ղܾɻ
ೋ࣍ܗࣜ ϥάϥϯδϡͷ࢛ฏํఆཧ x2 + y2 + z2 + w2 ͰશͯͷΛද͢ɻ
10 = 12 + 32 15 = 32 + 22 + 12 + 12
ೋ࣍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
ೋ࣍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
ೋ࣍ܗࣜ ͋Δೋ࣍ܗࣜQ ͕ɺଞͷೋ࣍ܗࣜP ͔Βม มͰදݱͰ͖Δ͔ʁmมͷP ͕nมͷ Q Λදݱ͢Δ͔ʁ ہॴେҬݪཧʢϋοηݪཧʣ p
ਐQp ͷൣғͱ࣮RͷൣғͰߟ͑Δɻ શͯͷp ٴͼRͰදݱͰ͖Ε༗ཧͷൣғ ͰදݱͰ͖Δ͔ʁ
ೋ࣍ܗࣜ ΤϨϯόʔά-ϰΣϯΧςγϡ Q ͕nมͷ࣌ɺશͯͷہॴతʹදݱՄೳͳ n − 7มҎԼͷೋ࣍ܗࣜQ′ Λදݱ͢Δɻ ূ໌ʹΤϧΰʔυཧɺྗֶܥΛ͏
ϦχοΫ༧ ੪࣍ଟ߲ࣜQ ʹର͠ɺQ(x) = d ͳΔx ͷू ߹ɻd ͰׂͬͯɺQ(x) =
1Ͱͷd → ∞Ͱͷ ͷ༷ࢠɻ Q(x) = x2 1 + x2 2 + · · · + x2 n Λߟ͑Δͱɺٿ໘্ ͷ༗ཧͷɻ ܈ͷ࡞༻͕͋Δ߹Λߟ͑ΔɻௐղੳͱΤ ϧΰʔυཧΛ͏ɻ
ΠσΞϧྨ܈ͷ ΠσΞϧྨ܈ͱʁͰͷૉҼղͷҰ ҙੑ 6 = 2 × 3 10 =
2 × 5 √ −5Λ͚Ճ͑Δͱ่ΕΔ 6 = 2 × 3 = (1 + √ −5)(1 − √ −5)
ΠσΞϧྨ܈ͷ ͜Εͷ่Ε۩߹ΛଌΔͷ͕ΠσΞϧྨ܈ɻ༗ ݶΞʔϕϧ܈ʹͳΔɻ ▶ Qͷ߹ɺΠσΞϧྨ܈1 ▶ Q( √ −5)ͷ߹ɺΠσΞϧྨ܈{±1}
ΠσΞϧྨ܈ͷ ৭ʑͳମQ(a)Λಈ͔ͨ͠ͱ͖ɺΠσΞ ϧྨ܈ʹͲͷΑ͏ͳ܈͕ݱΕΔ͔ʁ ίʔΤϯɺϨϯετϥͷΠσΞϧྨ܈ͷ ʹ͍ͭͯͷ؍ͱ༧ɻ
ΠσΞϧྨ܈ͷ ΤϨϯόʔά-ϰΣϯΧςγϡ-Σε λʔϥϯυ ίʔΤϯɺϨϯετϥ༧ͷؔମྨࣅΛূ ໌ͨ͠ɻ ؔମFp (x, a)༗ݶମ্ͷۂઢͷ༗ཧ ؔશͯूΊͨͷɻ͜Εಉ༷ʹΠσΞϧ ྨ܈ΛఆٛͰ͖Δɻ
ϑϧϏοπۭؒͷϗϞϩδʔ҆ఆੑΛͬͯ
ہॴରশۭؒ ϥϯάϥϯζରԠʹؔɻ ςΠϥʔɺϫΠϧζͷΨϩΞදݱͷߏΛࢤ ଜଟ༷ମ͕͑ͳ͍έʔεʹݚڀɻ ہॴରশۭؒͷίϗϞϩδʔΛදݱɺτϙ ϩδʔʹΑΓௐΔɻ