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Pythagoras0 (토론 | 기여) |
Pythagoras0 (토론 | 기여) |
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| (같은 사용자의 중간 판 2개는 보이지 않습니다) | |||
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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 | ||
| − | * equation number of solutions | + | * 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] (*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 | + | # 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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==related items== | ==related items== | ||
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* [[4817997|Taniyama-Shimura]] | * [[4817997|Taniyama-Shimura]] | ||
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[[분류:math and physics]] | [[분류:math and physics]] | ||
[[분류:math]] | [[분류:math]] | ||
[[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\(q\prod_{n=1}^{\infty} (1-q^{2n})(1-q^{16n})=\sum_{n=1}^\infty a_nq^n\)
- 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