Reciprocity law
Pythagoras0 (토론 | 기여)님의 2020년 11월 16일 (월) 01:51 판
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
history
encyclopedia
books
- 2010년 books and articles
- 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
- [1]http://www.zentralblatt-math.org/zmath/en/
- [2]http://arxiv.org/
- http://pythagoras0.springnote.com/
question and answers(Math Overflow)
blogs
experts on the field