Upgrade to Pro
— share decks privately, control downloads, hide ads and more …
Speaker Deck
Features
Speaker Deck
PRO
Sign in
Sign up for free
Search
Search
整数論と様々な数学
Search
Naoya Umezaki
October 06, 2018
0
750
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
Tweet
Share
More Decks by Naoya Umezaki
See All by Naoya Umezaki
ミケル点とべズーの定理
unaoya
0
710
すうがく徒のつどい@オンライン「ラマヌジャンのデルタ」
unaoya
0
580
合同式と幾何学
unaoya
0
2.2k
すうがく徒のつどい@オンライン「ヴェイユ予想とl進層のフーリエ変換」
unaoya
0
760
Egisonパターンマッチによる彩色
unaoya
1
550
関数等式と双対性
unaoya
1
700
直交多項式と表現論
unaoya
0
790
導来代数幾何入門
unaoya
0
900
作図と対称性
unaoya
0
200
Featured
See All Featured
How to train your dragon (web standard)
notwaldorf
85
5.6k
The Cult of Friendly URLs
andyhume
76
6k
Clear Off the Table
cherdarchuk
91
320k
The Invisible Customer
myddelton
119
13k
How GitHub (no longer) Works
holman
310
140k
Designing Experiences People Love
moore
138
23k
Fight the Zombie Pattern Library - RWD Summit 2016
marcelosomers
230
17k
KATA
mclloyd
27
13k
実際に使うSQLの書き方 徹底解説 / pgcon21j-tutorial
soudai
166
48k
Web development in the modern age
philhawksworth
205
10k
Designing on Purpose - Digital PM Summit 2013
jponch
114
6.8k
Easily Structure & Communicate Ideas using Wireframe
afnizarnur
190
16k
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