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
1
730
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
520
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
470
合同式と幾何学
unaoya
0
2.1k
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
730
Egisonパターンマッチによる彩色
unaoya
1
510
関数等式と双対性
unaoya
1
650
直交多項式と表現論
unaoya
1
710
導来代数幾何入門
unaoya
0
840
作図と対称性
unaoya
1
180
Featured
See All Featured
Helping Users Find Their Own Way: Creating Modern Search Experiences
danielanewman
19
1.9k
What’s in a name? Adding method to the madness
productmarketing
PRO
14
2.6k
Embracing the Ebb and Flow
colly
78
4.1k
Navigating Team Friction
lara
177
13k
4 Signs Your Business is Dying
shpigford
174
21k
The Pragmatic Product Professional
lauravandoore
24
5.7k
The MySQL Ecosystem @ GitHub 2015
samlambert
242
12k
Let's Do A Bunch of Simple Stuff to Make Websites Faster
chriscoyier
501
140k
Practical Orchestrator
shlominoach
180
9.7k
Intergalactic Javascript Robots from Outer Space
tanoku
266
26k
Mobile First: as difficult as doing things right
swwweet
215
8.5k
Automating Front-end Workflow
addyosmani
1353
200k
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