"Reciprocity law"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
imported>Pythagoras0
잔글 (찾아 바꾸기 – “4909919” 문자열을 “” 문자열로)
 
(사용자 2명의 중간 판 12개는 보이지 않습니다)
1번째 줄: 1번째 줄:
 
==introduction==
 
==introduction==
  
−
 
+
  
−
 
+
  
−
 
+
  
 
==example 1==
 
==example 1==
12번째 줄: 12번째 줄:
 
* <math>x^3=d</math>
 
* <math>x^3=d</math>
  
−
 
+
  
−
 
+
  
 
==example 2==
 
==example 2==
  
 
* Sums of sqaures of integers 126p
 
* Sums of sqaures of integers 126p
−
*  equation<br> number of solutions of <math>x^4-2x^2+2=0</math> in F_p = <math>1+(\frac{-1}{p})+a_p</math> where<br><math>q\prod_{n=1}^{\infty} (1-q^{2n})(1-q^{16n})=\sum_{n=1}^\infty a_nq^n</math><br>
+
*  equation number of solutions of <math>x^4-2x^2+2=0</math> in F_p = <math>1+(\frac{-1}{p})+a_p</math> where<math>q\prod_{n=1}^{\infty} (1-q^{2n})(1-q^{16n})=\sum_{n=1}^\infty a_nq^n</math>
  
−
#  Clear[g, p, M, a]<br> (*table of primes*)<br> Pr := Table[Prime[n], {n, 1, 20}]<br> (*equation*)<br> g[x_] := x^4 - 2 x^2 + 2<br> (*factorization of the discriminant & bad primes*)<br> FactorInteger[Discriminant[g[x], x]]<br> (* M[p] = number of solutions  for the equation g[x]=0 modulo p*)<br> M[n_] := 0<br> Do[For[i = 0, i < p, i++,<br>   M[p] = M[p] + If[Mod[PolynomialMod[g[i], p], p] == 0, 1, 0]], {p,<br>   Pr}]<br> (*modification of the number of solutions *)<br> a[p_] := 1 + JacobiSymbol[-1, p] + M[p]<br> (*modular form*)<br> f[q_] := Series[<br>   q*Product[(1 - q^(2 n))*(1 - q^(16 n)), {n, 1, 200}], {q, 0, 100}]<br> (*the coefficients of modular form f[q]*)<br> n[p_] := SeriesCoefficient[f[q], p]<br> (* output *)<br> title := {M_p, a_p, c_p};<br> TableForm[Table[{M[p], a[p], n[p]}, {p, Pr}] ,<br>  TableHeadings -> {Pr, title}]<br>
+
#  Clear[g, p, M, a] (*table of primes*) Pr := Table[Prime[n], {n, 1, 20}] (*equation*) g[x_] := x^4 - 2 x^2 + 2 (*factorization of the discriminant & bad primes*) FactorInteger[Discriminant[g[x], x]] (* M[p] = number of solutions  for the equation g[x]=0 modulo p*) M[n_] := 0 Do[For[i = 0, i < p, i++,   M[p] = M[p] + If[Mod[PolynomialMod[g[i], p], p] == 0, 1, 0]], {p,   Pr}] (*modification of the number of solutions *) a[p_] := 1 + JacobiSymbol[-1, p] + M[p] (*modular form*) f[q_] := Series[   q*Product[(1 - q^(2 n))*(1 - q^(16 n)), {n, 1, 200}], {q, 0, 100}] (*the coefficients of modular form f[q]*) n[p_] := SeriesCoefficient[f[q], p] (* output *) title := {M_p, a_p, c_p}; TableForm[Table[{M[p], a[p], n[p]}, {p, Pr}] , TableHeadings -> {Pr, title}]
  
−
 
+
  
 
==example 3==
 
==example 3==
29번째 줄: 29번째 줄:
 
* 1-2-3- of modular forms
 
* 1-2-3- of modular forms
  
−
 
+
  
−
 
+
  
−
 
+
−
 
 
−
==history==
 
−
 
 
−
* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
−
 
 
−
 
 
−
 
 
−
 
 
  
 
==related items==
 
==related items==
  
−
* [[mathematics of x^3-x+1=0]]<br>
+
* [[mathematics of x^3-x+1=0]]
−
* [[4817997|Taniyama-Shimura]]<br>
+
* [[4817997|Taniyama-Shimura]]
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
==encyclopedia==
 
−
 
 
−
* http://en.wikipedia.org/wiki/
 
−
* http://www.scholarpedia.org/
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
==books==
 
−
 
 
−
 
 
−
 
 
−
* [[2010년 books and articles]]<br>
 
−
* http://gigapedia.info/1/squares
 
−
* http://gigapedia.info/1/
 
−
* http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
==articles==
 
−
 
 
−
 
 
−
 
 
−
* http://www.ams.org/mathscinet
 
−
* [http://www.zentralblatt-math.org/zmath/en/ ]http://www.zentralblatt-math.org/zmath/en/
 
−
* [http://arxiv.org/ ]http://arxiv.org/
 
−
* http://pythagoras0.springnote.com/
 
−
* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html]
 
−
* http://dx.doi.org/
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
==question and answers(Math Overflow)==
 
−
 
 
−
* http://mathoverflow.net/search?q=
 
−
* http://mathoverflow.net/search?q=
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
==blogs==
 
  
−
*  구글 블로그 검색<br>
 
−
** http://blogsearch.google.com/blogsearch?q=
 
−
** http://blogsearch.google.com/blogsearch?q=
 
  
−
 
+
[[분류:개인노트]]
−
 
 
−
 
 
−
 
 
−
==experts on the field==
 
−
 
 
−
* http://arxiv.org/
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
==links==
 
−
 
 
−
* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
 
−
* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
 
−
* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
 
−
* http://functions.wolfram.com/
 
−
*[[분류:개인노트]]
 
−
[[분류:math and physics]]
 
 
[[분류:math and physics]]
 
[[분류:math and physics]]
 +
[[분류:math]]
 +
[[분류:automorphic forms]]
 +
[[분류:migrate]]

2020년 12월 28일 (월) 04:02 기준 최신판

introduction

example 1

  • Diamond & Shurman 155p
  • \(x^3=d\)



example 2

  • Sums of sqaures of integers 126p
  • equation number of solutions of \(x^4-2x^2+2=0\) in F_p = \(1+(\frac{-1}{p})+a_p\) where\(q\prod_{n=1}^{\infty} (1-q^{2n})(1-q^{16n})=\sum_{n=1}^\infty a_nq^n\)
  1. Clear[g, p, M, a] (*table of primes*) Pr := Table[Prime[n], {n, 1, 20}] (*equation*) g[x_] := x^4 - 2 x^2 + 2 (*factorization of the discriminant & bad primes*) FactorInteger[Discriminant[g[x], x]] (* M[p] = number of solutions for the equation g[x]=0 modulo p*) M[n_] := 0 Do[For[i = 0, i < p, i++, M[p] = M[p] + If[Mod[PolynomialMod[g[i], p], p] == 0, 1, 0]], {p, Pr}] (*modification of the number of solutions *) a[p_] := 1 + JacobiSymbol[-1, p] + M[p] (*modular form*) f[q_] := Series[ q*Product[(1 - q^(2 n))*(1 - q^(16 n)), {n, 1, 200}], {q, 0, 100}] (*the coefficients of modular form f[q]*) n[p_] := SeriesCoefficient[f[q], p] (* output *) title := {M_p, a_p, c_p}; TableForm[Table[{M[p], a[p], n[p]}, {p, Pr}] , TableHeadings -> {Pr, title}]


example 3

  • 1-2-3- of modular forms




related items

원본 주소 "https://wiki.mathnt.net/index.php?title=Reciprocity_law&oldid=48398"