Reciprocity law
imported>Pythagoras0님의 2013년 2월 8일 (금) 14:09 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
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- http://gigapedia.info/1/squares
- http://gigapedia.info/1/
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articles
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- [1]http://www.zentralblatt-math.org/zmath/en/
- [2]http://arxiv.org/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html
- http://dx.doi.org/
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