"Reciprocity law"의 두 판 사이의 차이

수학노트
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==introduction==
 
==introduction==
  
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==example 1==
 
==example 1==
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* <math>x^3=d</math>
 
* <math>x^3=d</math>
  
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==example 2==
 
==example 2==
  
 
* Sums of sqaures of integers 126p
 
* Sums of sqaures of integers 126p
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*  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>
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*  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>
  
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#  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}]
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#  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}]
  
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==example 3==
 
==example 3==
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* 1-2-3- of modular forms
 
* 1-2-3- of modular forms
  
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==history==
 
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
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==related items==
 
==related items==
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* [[4817997|Taniyama-Shimura]]
 
* [[4817997|Taniyama-Shimura]]
  
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==encyclopedia==
 
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* http://en.wikipedia.org/wiki/
 
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* http://www.scholarpedia.org/
 
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==books==
 
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* [[2010년 books and articles]]
 
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* http://gigapedia.info/1/squares
 
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* http://gigapedia.info/1/
 
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* http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
 
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==articles==
 
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* http://www.ams.org/mathscinet
 
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* [http://www.zentralblatt-math.org/zmath/en/ ]http://www.zentralblatt-math.org/zmath/en/
 
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* [http://arxiv.org/ ]http://arxiv.org/
 
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* http://pythagoras0.springnote.com/
 
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* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html]
 
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* http://dx.doi.org/
 
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==question and answers(Math Overflow)==
 
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* http://mathoverflow.net/search?q=
 
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* http://mathoverflow.net/search?q=
 
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==blogs==
 
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*  구글 블로그 검색
 
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** http://blogsearch.google.com/blogsearch?q=
 
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** http://blogsearch.google.com/blogsearch?q=
 
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==experts on the field==
 
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* http://arxiv.org/
 
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==links==
 
  
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* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
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[[분류:개인노트]]
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* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
 
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
 
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* http://functions.wolfram.com/
 
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*[[분류:개인노트]]
 
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[[분류:math and physics]]
 
 
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[[분류:automorphic forms]]
 
[[분류:automorphic forms]]
 
[[분류:migrate]]
 
[[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

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